Aralıklar

Aralıklar

Gercel sayıların, aralık olarak adlandırılan bazı kumeleri kalkuluste sık sıkkullanılır ve geometrik olarak dogru parcalarına karsılık gelir. Ornegin,a < b ise, a’dan b’ye acık aralık a ile b arasındaki tum sayıları kapsar ve(a, b) ile gosterilir. Kume gosterimini kullanarak

(a, b) = {x ∈ R∣∣ a < x < b}

yazabiliriz.

MAT 1009 Kalkulus I 1 / 79

Aralıklar

(a, b) = {x|a < x < b}

Aralıgın uc noktaları olan a ve b’nin kapsanmadıgına dikkat ediniz. Budurum yuvarlak parantez ( ) kullanılarak ve Sekildeki ici bos dairelerlegosterilmistir.

Notation Set description Picture Notation Set description Picture

�x � a � x � b��a, b�

�x � a � x � b��a, b�

�x � a � x � b��a, b�

�x � a � x � b��a, b�

Intervals, Inequalities, and Absolute Values � � � � � � � �

Certain sets of real numbers, called intervals, occur frequently in calculus and corre-spond geometrically to line segments. For example, if , the open interval from

to consists of all numbers between and and is denoted by the symbol .Using set-builder notation, we can write

Notice that the endpoints of the interval—namely, and —are excluded. This isindicated by the round brackets and by the open dots in Figure 1. The closed inter-val from to is the set

Here the endpoints of the interval are included. This is indicated by the square brack-ets and by the solid dots in Figure 2. It is also possible to include only one endpointin an interval, as shown in Table 1.

We also need to consider infinite intervals such as

This does not mean that (“infinity”) is a number. The notation stands for theset of all numbers that are greater than , so the symbol simply indicates that theinterval extends indefinitely far in the positive direction.

Table of Intervals

Inequalities

When working with inequalities, note the following rules.

Rules for Inequalities

1. If , then .

2. If and , then .

3. If and , then .

4. If and , then .

5. If , then .1�a � 1�b0 � a � b

ac � bcc � 0a � b

ac � bcc � 0a � b

a � c � b � dc � da � b

a � c � b � ca � b

1

�a�a, ���

�a, �� � �x � x � a�

� �

�a, b� � �x � a � x � b�

ba� �

ba

�a, b� � �x � a � x � b�

�a, b�babaa � b

A

A2 � APPENDIX A INTERVALS, INEQUALITIES, AND ABSOLUTE VALUES

� Table 1 lists the nine possible typesof intervals. When these intervals arediscussed, it is always assumed that

.a � b

a b

FIGURE 1Open interval (a, b)

a b

FIGURE 2Closed interval [a, b]

(set of all real numbers)����, ��

�x � x � b����, b�

�x � x � b����, b�

�x � x � a��a, ��

�x � x � a��a, ��a b

a b

a b

a b

a

a

b

b

MAT 1009 Kalkulus I 2 / 79

Aralıklar

Tanım 1

a’dan b’ye kapalı aralık

[a, b] = {x ∈ R∣∣ a ≤ x ≤ b}

kumesidir. Burada uc noktaların her ikisi de kapsanmıstır. Bu durum koseliparantez [ ] kullanılarak ve Sekildeki ici dolu dairelerle gosterilmistir.

Notation Set description Picture Notation Set description Picture

�x � a � x � b��a, b�

�x � a � x � b��a, b�

�x � a � x � b��a, b�

�x � a � x � b��a, b�

Intervals, Inequalities, and Absolute Values � � � � � � � �

Certain sets of real numbers, called intervals, occur frequently in calculus and corre-spond geometrically to line segments. For example, if , the open interval from

to consists of all numbers between and and is denoted by the symbol .Using set-builder notation, we can write

Notice that the endpoints of the interval—namely, and —are excluded. This isindicated by the round brackets and by the open dots in Figure 1. The closed inter-val from to is the set

Here the endpoints of the interval are included. This is indicated by the square brack-ets and by the solid dots in Figure 2. It is also possible to include only one endpointin an interval, as shown in Table 1.

We also need to consider infinite intervals such as

This does not mean that (“infinity”) is a number. The notation stands for theset of all numbers that are greater than , so the symbol simply indicates that theinterval extends indefinitely far in the positive direction.

Table of Intervals

Inequalities

When working with inequalities, note the following rules.

Rules for Inequalities

1. If , then .

2. If and , then .

3. If and , then .

4. If and , then .

5. If , then .1�a � 1�b0 � a � b

ac � bcc � 0a � b

ac � bcc � 0a � b

a � c � b � dc � da � b

a � c � b � ca � b

1

�a�a, ���

�a, �� � �x � x � a�

� �

�a, b� � �x � a � x � b�

ba� �

ba

�a, b� � �x � a � x � b�

�a, b�babaa � b

A

A2 � APPENDIX A INTERVALS, INEQUALITIES, AND ABSOLUTE VALUES

� Table 1 lists the nine possible typesof intervals. When these intervals arediscussed, it is always assumed that

.a � b

a b

FIGURE 1Open interval (a, b)

a b

FIGURE 2Closed interval [a, b]

(set of all real numbers)����, ��

�x � x � b����, b�

�x � x � b����, b�

�x � x � a��a, ��

�x � x � a��a, ��a b

a b

a b

a b

a

a

b

b

MAT 1009 Kalkulus I 3 / 79

Aralıklar

Bir aralık, Tablo 1’de gosterildigi gibi, uc noktalardan yalnızca birini dekapsayabilir.

(a,∞) = {x ∈ R∣∣ x > a}

gibi sonsuz aralıkları da ele almamız gerekir. Bu, ∞’un (“sonsuz”) bir sayıoldugu anlamına gelmez. (a,∞) ifadesi a’dan buyuk tum sayılarınkumesidir. Burada ∞ simgesi yalnızca aralıgın pozitif yonde sınırsız olarakgenisledigini gosterir.

MAT 1009 Kalkulus I 4 / 79

Aralıklar

Aralık Gosterimi Kume Gosterimi Sekil

(a, b) {x : a < x < b}Notation Set description Picture Notation Set description Picture

�x � a � x � b��a, b�

�x � a � x � b��a, b�

�x � a � x � b��a, b�

�x � a � x � b��a, b�

Intervals, Inequalities, and Absolute Values � � � � � � � �

Certain sets of real numbers, called intervals, occur frequently in calculus and corre-spond geometrically to line segments. For example, if , the open interval from

to consists of all numbers between and and is denoted by the symbol .Using set-builder notation, we can write

Notice that the endpoints of the interval—namely, and —are excluded. This isindicated by the round brackets and by the open dots in Figure 1. The closed inter-val from to is the set

Here the endpoints of the interval are included. This is indicated by the square brack-ets and by the solid dots in Figure 2. It is also possible to include only one endpointin an interval, as shown in Table 1.

We also need to consider infinite intervals such as

This does not mean that (“infinity”) is a number. The notation stands for theset of all numbers that are greater than , so the symbol simply indicates that theinterval extends indefinitely far in the positive direction.

Table of Intervals

Inequalities

When working with inequalities, note the following rules.

Rules for Inequalities

1. If , then .

2. If and , then .

3. If and , then .

4. If and , then .

5. If , then .1�a � 1�b0 � a � b

ac � bcc � 0a � b

ac � bcc � 0a � b

a � c � b � dc � da � b

a � c � b � ca � b

1

�a�a, ���

�a, �� � �x � x � a�

� �

�a, b� � �x � a � x � b�

ba� �

ba

�a, b� � �x � a � x � b�

�a, b�babaa � b

A

A2 � APPENDIX A INTERVALS, INEQUALITIES, AND ABSOLUTE VALUES

� Table 1 lists the nine possible typesof intervals. When these intervals arediscussed, it is always assumed that

.a � b

a b

FIGURE 1Open interval (a, b)

a b

FIGURE 2Closed interval [a, b]

(set of all real numbers)����, ��

�x � x � b����, b�

�x � x � b����, b�

�x � x � a��a, ��

�x � x � a��a, ��a b

a b

a b

a b

a

a

b

b

[a, b] {x : a ≤ x ≤ b}

Notation Set description Picture Notation Set description Picture

�x � a � x � b��a, b�

�x � a � x � b��a, b�

�x � a � x � b��a, b�

�x � a � x � b��a, b�

Intervals, Inequalities, and Absolute Values � � � � � � � �

Certain sets of real numbers, called intervals, occur frequently in calculus and corre-spond geometrically to line segments. For example, if , the open interval from

to consists of all numbers between and and is denoted by the symbol .Using set-builder notation, we can write

Notice that the endpoints of the interval—namely, and —are excluded. This isindicated by the round brackets and by the open dots in Figure 1. The closed inter-val from to is the set

Here the endpoints of the interval are included. This is indicated by the square brack-ets and by the solid dots in Figure 2. It is also possible to include only one endpointin an interval, as shown in Table 1.

We also need to consider infinite intervals such as

This does not mean that (“infinity”) is a number. The notation stands for theset of all numbers that are greater than , so the symbol simply indicates that theinterval extends indefinitely far in the positive direction.

Table of Intervals

Inequalities

When working with inequalities, note the following rules.

Rules for Inequalities

1. If , then .

2. If and , then .

3. If and , then .

4. If and , then .

5. If , then .1�a � 1�b0 � a � b

ac � bcc � 0a � b

ac � bcc � 0a � b

a � c � b � dc � da � b

a � c � b � ca � b

1

�a�a, ���

�a, �� � �x � x � a�

� �

�a, b� � �x � a � x � b�

ba� �

ba

�a, b� � �x � a � x � b�

�a, b�babaa � b

A

A2 � APPENDIX A INTERVALS, INEQUALITIES, AND ABSOLUTE VALUES

� Table 1 lists the nine possible typesof intervals. When these intervals arediscussed, it is always assumed that

.a � b

a b

FIGURE 1Open interval (a, b)

a b

FIGURE 2Closed interval [a, b]

(set of all real numbers)����, ��

�x � x � b����, b�

�x � x � b����, b�

�x � x � a��a, ��

�x � x � a��a, ��a b

a b

a b

a b

a

a

b

b

[a, b) {x : a ≤ x < b}

Notation Set description Picture Notation Set description Picture

�x � a � x � b��a, b�

�x � a � x � b��a, b�

�x � a � x � b��a, b�

�x � a � x � b��a, b�

Intervals, Inequalities, and Absolute Values � � � � � � � �

Certain sets of real numbers, called intervals, occur frequently in calculus and corre-spond geometrically to line segments. For example, if , the open interval from

to consists of all numbers between and and is denoted by the symbol .Using set-builder notation, we can write

Notice that the endpoints of the interval—namely, and —are excluded. This isindicated by the round brackets and by the open dots in Figure 1. The closed inter-val from to is the set

Here the endpoints of the interval are included. This is indicated by the square brack-ets and by the solid dots in Figure 2. It is also possible to include only one endpointin an interval, as shown in Table 1.

We also need to consider infinite intervals such as

This does not mean that (“infinity”) is a number. The notation stands for theset of all numbers that are greater than , so the symbol simply indicates that theinterval extends indefinitely far in the positive direction.

Table of Intervals

Inequalities

When working with inequalities, note the following rules.

Rules for Inequalities

1. If , then .

2. If and , then .

3. If and , then .

4. If and , then .

5. If , then .1�a � 1�b0 � a � b

ac � bcc � 0a � b

ac � bcc � 0a � b

a � c � b � dc � da � b

a � c � b � ca � b

1

�a�a, ���

�a, �� � �x � x � a�

� �

�a, b� � �x � a � x � b�

ba� �

ba

�a, b� � �x � a � x � b�

�a, b�babaa � b

A

A2 � APPENDIX A INTERVALS, INEQUALITIES, AND ABSOLUTE VALUES

� Table 1 lists the nine possible typesof intervals. When these intervals arediscussed, it is always assumed that

.a � b

a b

FIGURE 1Open interval (a, b)

a b

FIGURE 2Closed interval [a, b]

(set of all real numbers)����, ��

�x � x � b����, b�

�x � x � b����, b�

�x � x � a��a, ��

�x � x � a��a, ��a b

a b

a b

a b

a

a

b

b

(a, b] {x : a < x ≤ b}

Notation Set description Picture Notation Set description Picture

�x � a � x � b��a, b�

�x � a � x � b��a, b�

�x � a � x � b��a, b�

�x � a � x � b��a, b�

Intervals, Inequalities, and Absolute Values � � � � � � � �

Certain sets of real numbers, called intervals, occur frequently in calculus and corre-spond geometrically to line segments. For example, if , the open interval from

to consists of all numbers between and and is denoted by the symbol .Using set-builder notation, we can write

Notice that the endpoints of the interval—namely, and —are excluded. This isindicated by the round brackets and by the open dots in Figure 1. The closed inter-val from to is the set

Here the endpoints of the interval are included. This is indicated by the square brack-ets and by the solid dots in Figure 2. It is also possible to include only one endpointin an interval, as shown in Table 1.

We also need to consider infinite intervals such as

This does not mean that (“infinity”) is a number. The notation stands for theset of all numbers that are greater than , so the symbol simply indicates that theinterval extends indefinitely far in the positive direction.

Table of Intervals

Inequalities

When working with inequalities, note the following rules.

Rules for Inequalities

1. If , then .

2. If and , then .

3. If and , then .

4. If and , then .

5. If , then .1�a � 1�b0 � a � b

ac � bcc � 0a � b

ac � bcc � 0a � b

a � c � b � dc � da � b

a � c � b � ca � b

1

�a�a, ���

�a, �� � �x � x � a�

� �

�a, b� � �x � a � x � b�

ba� �

ba

�a, b� � �x � a � x � b�

�a, b�babaa � b

A

A2 � APPENDIX A INTERVALS, INEQUALITIES, AND ABSOLUTE VALUES

� Table 1 lists the nine possible typesof intervals. When these intervals arediscussed, it is always assumed that

.a � b

a b

FIGURE 1Open interval (a, b)

a b

FIGURE 2Closed interval [a, b]

(set of all real numbers)����, ��

�x � x � b����, b�

�x � x � b����, b�

�x � x � a��a, ��

�x � x � a��a, ��a b

a b

a b

a b

a

a

b

b

(a,∞) {x : x > a}

Notation Set description Picture Notation Set description Picture

�x � a � x � b��a, b�

�x � a � x � b��a, b�

�x � a � x � b��a, b�

�x � a � x � b��a, b�

Intervals, Inequalities, and Absolute Values � � � � � � � �

Certain sets of real numbers, called intervals, occur frequently in calculus and corre-spond geometrically to line segments. For example, if , the open interval from

to consists of all numbers between and and is denoted by the symbol .Using set-builder notation, we can write

Notice that the endpoints of the interval—namely, and —are excluded. This isindicated by the round brackets and by the open dots in Figure 1. The closed inter-val from to is the set

Here the endpoints of the interval are included. This is indicated by the square brack-ets and by the solid dots in Figure 2. It is also possible to include only one endpointin an interval, as shown in Table 1.

We also need to consider infinite intervals such as

This does not mean that (“infinity”) is a number. The notation stands for theset of all numbers that are greater than , so the symbol simply indicates that theinterval extends indefinitely far in the positive direction.

Table of Intervals

Inequalities

When working with inequalities, note the following rules.

Rules for Inequalities

1. If , then .

2. If and , then .

3. If and , then .

4. If and , then .

5. If , then .1�a � 1�b0 � a � b

ac � bcc � 0a � b

ac � bcc � 0a � b

a � c � b � dc � da � b

a � c � b � ca � b

1

�a�a, ���

�a, �� � �x � x � a�

� �

�a, b� � �x � a � x � b�

ba� �

ba

�a, b� � �x � a � x � b�

�a, b�babaa � b

A

A2 � APPENDIX A INTERVALS, INEQUALITIES, AND ABSOLUTE VALUES

� Table 1 lists the nine possible typesof intervals. When these intervals arediscussed, it is always assumed that

.a � b

a b

FIGURE 1Open interval (a, b)

a b

FIGURE 2Closed interval [a, b]

(set of all real numbers)����, ��

�x � x � b����, b�

�x � x � b����, b�

�x � x � a��a, ��

�x � x � a��a, ��a b

a b

a b

a b

a

a

b

b

[a,∞) {x : x ≥ a}

Notation Set description Picture Notation Set description Picture

�x � a � x � b��a, b�

�x � a � x � b��a, b�

�x � a � x � b��a, b�

�x � a � x � b��a, b�

Intervals, Inequalities, and Absolute Values � � � � � � � �

Certain sets of real numbers, called intervals, occur frequently in calculus and corre-spond geometrically to line segments. For example, if , the open interval from

to consists of all numbers between and and is denoted by the symbol .Using set-builder notation, we can write

Notice that the endpoints of the interval—namely, and —are excluded. This isindicated by the round brackets and by the open dots in Figure 1. The closed inter-val from to is the set

Here the endpoints of the interval are included. This is indicated by the square brack-ets and by the solid dots in Figure 2. It is also possible to include only one endpointin an interval, as shown in Table 1.

We also need to consider infinite intervals such as

This does not mean that (“infinity”) is a number. The notation stands for theset of all numbers that are greater than , so the symbol simply indicates that theinterval extends indefinitely far in the positive direction.

Table of Intervals

Inequalities

When working with inequalities, note the following rules.

Rules for Inequalities

1. If , then .

2. If and , then .

3. If and , then .

4. If and , then .

5. If , then .1�a � 1�b0 � a � b

ac � bcc � 0a � b

ac � bcc � 0a � b

a � c � b � dc � da � b

a � c � b � ca � b

1

�a�a, ���

�a, �� � �x � x � a�

� �

�a, b� � �x � a � x � b�

ba� �

ba

�a, b� � �x � a � x � b�

�a, b�babaa � b

A

A2 � APPENDIX A INTERVALS, INEQUALITIES, AND ABSOLUTE VALUES

� Table 1 lists the nine possible typesof intervals. When these intervals arediscussed, it is always assumed that

.a � b

a b

FIGURE 1Open interval (a, b)

a b

FIGURE 2Closed interval [a, b]

(set of all real numbers)����, ��

�x � x � b����, b�

�x � x � b����, b�

�x � x � a��a, ��

�x � x � a��a, ��a b

a b

a b

a b

a

a

b

b

(−∞, b) {x : x < b}

Notation Set description Picture Notation Set description Picture

�x � a � x � b��a, b�

�x � a � x � b��a, b�

�x � a � x � b��a, b�

�x � a � x � b��a, b�

Intervals, Inequalities, and Absolute Values � � � � � � � �

Certain sets of real numbers, called intervals, occur frequently in calculus and corre-spond geometrically to line segments. For example, if , the open interval from

to consists of all numbers between and and is denoted by the symbol .Using set-builder notation, we can write

Notice that the endpoints of the interval—namely, and —are excluded. This isindicated by the round brackets and by the open dots in Figure 1. The closed inter-val from to is the set

Here the endpoints of the interval are included. This is indicated by the square brack-ets and by the solid dots in Figure 2. It is also possible to include only one endpointin an interval, as shown in Table 1.

We also need to consider infinite intervals such as

This does not mean that (“infinity”) is a number. The notation stands for theset of all numbers that are greater than , so the symbol simply indicates that theinterval extends indefinitely far in the positive direction.

Table of Intervals

Inequalities

When working with inequalities, note the following rules.

Rules for Inequalities

1. If , then .

2. If and , then .

3. If and , then .

4. If and , then .

5. If , then .1�a � 1�b0 � a � b

ac � bcc � 0a � b

ac � bcc � 0a � b

a � c � b � dc � da � b

a � c � b � ca � b

1

�a�a, ���

�a, �� � �x � x � a�

� �

�a, b� � �x � a � x � b�

ba� �

ba

�a, b� � �x � a � x � b�

�a, b�babaa � b

A

A2 � APPENDIX A INTERVALS, INEQUALITIES, AND ABSOLUTE VALUES

� Table 1 lists the nine possible typesof intervals. When these intervals arediscussed, it is always assumed that

.a � b

a b

FIGURE 1Open interval (a, b)

a b

FIGURE 2Closed interval [a, b]

(set of all real numbers)����, ��

�x � x � b����, b�

�x � x � b����, b�

�x � x � a��a, ��

�x � x � a��a, ��a b

a b

a b

a b

a

a

b

b(−∞, b] {x : x ≤ b}

Notation Set description Picture Notation Set description Picture

�x � a � x � b��a, b�

�x � a � x � b��a, b�

�x � a � x � b��a, b�

�x � a � x � b��a, b�

Intervals, Inequalities, and Absolute Values � � � � � � � �

Certain sets of real numbers, called intervals, occur frequently in calculus and corre-spond geometrically to line segments. For example, if , the open interval from

to consists of all numbers between and and is denoted by the symbol .Using set-builder notation, we can write

Notice that the endpoints of the interval—namely, and —are excluded. This isindicated by the round brackets and by the open dots in Figure 1. The closed inter-val from to is the set

Here the endpoints of the interval are included. This is indicated by the square brack-ets and by the solid dots in Figure 2. It is also possible to include only one endpointin an interval, as shown in Table 1.

We also need to consider infinite intervals such as

This does not mean that (“infinity”) is a number. The notation stands for theset of all numbers that are greater than , so the symbol simply indicates that theinterval extends indefinitely far in the positive direction.

Table of Intervals

Inequalities

When working with inequalities, note the following rules.

Rules for Inequalities

1. If , then .

2. If and , then .

3. If and , then .

4. If and , then .

5. If , then .1�a � 1�b0 � a � b

ac � bcc � 0a � b

ac � bcc � 0a � b

a � c � b � dc � da � b

a � c � b � ca � b

1

�a�a, ���

�a, �� � �x � x � a�

� �

�a, b� � �x � a � x � b�

ba� �

ba

�a, b� � �x � a � x � b�

�a, b�babaa � b

A

A2 � APPENDIX A INTERVALS, INEQUALITIES, AND ABSOLUTE VALUES

� Table 1 lists the nine possible typesof intervals. When these intervals arediscussed, it is always assumed that

.a � b

a b

FIGURE 1Open interval (a, b)

a b

FIGURE 2Closed interval [a, b]

(set of all real numbers)����, ��

�x � x � b����, b�

�x � x � b����, b�

�x � x � a��a, ��

�x � x � a��a, ��a b

a b

a b

a b

a

a

b

b

(−∞,∞) R

Notation Set description Picture Notation Set description Picture

�x � a � x � b��a, b�

�x � a � x � b��a, b�

�x � a � x � b��a, b�

�x � a � x � b��a, b�

Intervals, Inequalities, and Absolute Values � � � � � � � �

Certain sets of real numbers, called intervals, occur frequently in calculus and corre-spond geometrically to line segments. For example, if , the open interval from

to consists of all numbers between and and is denoted by the symbol .Using set-builder notation, we can write

Notice that the endpoints of the interval—namely, and —are excluded. This isindicated by the round brackets and by the open dots in Figure 1. The closed inter-val from to is the set

Here the endpoints of the interval are included. This is indicated by the square brack-ets and by the solid dots in Figure 2. It is also possible to include only one endpointin an interval, as shown in Table 1.

We also need to consider infinite intervals such as

This does not mean that (“infinity”) is a number. The notation stands for theset of all numbers that are greater than , so the symbol simply indicates that theinterval extends indefinitely far in the positive direction.

Table of Intervals

Inequalities

When working with inequalities, note the following rules.

Rules for Inequalities

1. If , then .

2. If and , then .

3. If and , then .

4. If and , then .

5. If , then .1�a � 1�b0 � a � b

ac � bcc � 0a � b

ac � bcc � 0a � b

a � c � b � dc � da � b

a � c � b � ca � b

1

�a�a, ���

�a, �� � �x � x � a�

� �

�a, b� � �x � a � x � b�

ba� �

ba

�a, b� � �x � a � x � b�

�a, b�babaa � b

A

A2 � APPENDIX A INTERVALS, INEQUALITIES, AND ABSOLUTE VALUES

� Table 1 lists the nine possible typesof intervals. When these intervals arediscussed, it is always assumed that

.a � b

a b

FIGURE 1Open interval (a, b)

a b

FIGURE 2Closed interval [a, b]

(set of all real numbers)����, ��

�x � x � b����, b�

�x � x � b����, b�

�x � x � a��a, ��

�x � x � a��a, ��a b

a b

a b

a b

a

a

b

b

Tablo 1: Aralıklar Tablosu

MAT 1009 Kalkulus I 5 / 79

Esitsizlikler

Esitsizlikler

Esitsizlik Kuralları

1 a < b ise a+ c < b+ c’dir.

2 a < b ve c < d ise a+ c < b+ d’dir.

3 a < b ve c > 0 ise ac < bc’dir.

4 a < b ve c < 0 ise ac > bc’dir.

5 0 < a < b ise1

a>

1

bsaglanır.

MAT 1009 Kalkulus I 6 / 79

Esitsizlikler

Ornek 2

x2 − 5x+ 6 ≤ 0 esitsizligini cozunuz.

Cozum.

Once sol yanı carpanlarına ayırırız.

(x− 2)(x− 3) ≤ 0

(x− 2)(x− 3) = 0 denkleminin 2 ve 3 oldugunu biliyoruz. 2 ve 3 gercelsayı dogrusunu uc parcaya boler:

(−∞, 2) (2, 3) (3,∞)

MAT 1009 Kalkulus I 7 / 79

Esitsizlikler

Cozum (devamı).

Bu aralıkların her biri icin carpanların isaretlerini belirleriz.

x

(x − 2)

(x − 3)

(x − 2)(x − 3)

2 3

− 0 + +

− − 0 +

+ − +

Sonra, (x− 2)(x− 3) ifadesinin 2 < x < 3 icin negatif oldugunu tablodanyorumlarız. Dolayısıyla (x− 2)(x− 3) ≤ 0 esitsizliginin cozumu

{x ∈ R∣∣ 2 ≤ x ≤ 3} = [2, 3]

aralıgıdır.

Carpımın negatif ya da sıfır oldugu x degerlerini aradıgımız icin, uc noktalar olan 2 ve 3un kapsandıgına dikkat ediniz.

MAT 1009 Kalkulus I 8 / 79

Esitsizlikler

Ornek 3

x3 + 3x2 > 4x esitsizligini cozunuz.

Cozum.

Ilk olarak tum sıfırdan farklı terimleri bir yana toplar ve elde edilen ifadeyicarpanlanna ayırınz:

x3 + 3x2 − 4x > 0 x(x− 1)(x+ 4) > 0

Onceki ornekte oldugu gibi, karsıgelen x(x− 1)(x+ 4) = 0 denkleminicozer ve x = −4, x = 0 ve x = 1 cozumlerini kullanarak gercel sayıdogrusunu (∞,−4), (−4, 0), (0, 1) ve (1,∞) aralıklarına ayırırız.

MAT 1009 Kalkulus I 9 / 79

Esitsizlikler

Cozum (devamı).

Her bir aralıkta carpanların isaretlerini belirleriz:

x

x

(x − 1)

(x + 4)

x(x − 1)(x + 4)

−4 0 1

− − 0 + +

− − − 0 +

− 0 + + +

− + − +

Sonra cozum kumesini, tablodan okuyarak asagıdaki sekilde elde ederiz:

{x ∈ R∣∣ − 4 < x < 0 veya x > 1} = (−4, 0) ∪ (1,∞)

MAT 1009 Kalkulus I 10 / 79

Bir Fonksiyonun Dort Farklı Gosterimi

Fonksiyonlar ve ModellerBir Fonksiyonun Dort Farklı Gosterimi

Bir nicelik bir digerine baglı oldugunda ortaya fonksiyonlar cıkar. Asagıdakidort durumu dusunelim:

1 Bir dairenin alanı A, yarıcapı r’ye baglıdır. Bu baglılık A = πr2

esitligi ile gosterilir. Her pozitif r degerine karsılık bir A degeri vardırve bu A’nın r’nin fonksiyonu olması ile ifade edilir.

MAT 1009 Kalkulus I 11 / 79

Bir Fonksiyonun Dort Farklı Gosterimi

2 P ile gosterilen dunya nufusu, t zamanına baglıdır.Tablo, dunyanufusu P (t)’yi t yıllarında yaklasık olarak vermektedir.

Ornegin,

P (1950) ≈ 2.560.000.000

Zaman t’nin her degerine karsılık ge-len bir P degeri oldugundan P ’ninzaman t’nin fonksiyonu oldugunusoyleriz.

Yıl Nufus (milyon)

1900 16501910 17501920 18601930 20701940 23001950 25601960 30401970 37101980 44501990 52802000 6070

MAT 1009 Kalkulus I 12 / 79

Bir Fonksiyonun Dort Farklı Gosterimi

3 Bir mektubun posta ucreti C, agırlıgı w’ye baglıdır. w ile C arasındakolayca ifade edilebilecek basit bir formul olmamasına karsın postaidareleri w bilindiginde C’yi belirleyen kurallar kullanırlar.

MAT 1009 Kalkulus I 13 / 79

Bir Fonksiyonun Dort Farklı Gosterimi

4 Bir depremde yer kabugunun dusey ivmesi a, sismograflar tarafındangecen t suresinin fonksiyonu olarak bellege kaydedilmektedir. Sekilden,1994’te Los Angeles kentindeki sismik hareketin grafigi verilmektedir.Verilen t degerine karsılık gelen a degerini grafikten okuyabiliriz.

Four Ways to Represent a Function � � � � � � � � � � �

Functions arise whenever one quantity depends on another. Consider the followingfour situations.

A. The area of a circle depends on the radius of the circle. The rule that con-nects and is given by the equation . With each positive number there is associated one value of , and we say that is a function of .

B. The human population of the world depends on the time . The table gives esti-mates of the world population at time for certain years. For instance,

But for each value of the time there is a corresponding value of and we saythat is a function of .

C. The cost of mailing a first-class letter depends on the weight of the letter.Although there is no simple formula that connects and , the post office has arule for determining when is known.

D. The vertical acceleration of the ground as measured by a seismograph duringan earthquake is a function of the elapsed time Figure 1 shows a graph gener-ated by seismic activity during the Northridge earthquake that shook Los Angelesin 1994. For a given value of the graph provides a corresponding value of .

Each of these examples describes a rule whereby, given a number ( , , , or ),another number ( , , , or ) is assigned. In each case we say that the second num-ber is a function of the first number.

aCPAtwtr

FIGURE 1Vertical ground acceleration during

the Northridge earthquake

{cm/s@}

(seconds)

Calif. Dept. of Mines and Geology

5

50

10 15 20 25

a

t

100

30

_50

at,

t.a

wCCw

wC

tPP,t

P�1950� � 2,560,000,000

t,P�t�tP

rAArA � �r 2Ar

rA

1.1

11

The fundamental objects that we deal with in calculusare functions. This chapter prepares the way for calcu-lus by discussing the basic ideas concerning functions,their graphs, and ways of transforming and combiningthem. We stress that a function can be represented indifferent ways: by an equation, in a table, by a graph,or in words. We look at the main types of functions

that occur in calculus and describe the process ofusing these functions as mathematical models of real-world phenomena. We also discuss the use of graph-ing calculators and graphing software for computersand see that parametric equations provide the bestmethod for graphing certain types of curves.

PopulationYear (millions)

1900 16501910 17501920 18601930 20701940 23001950 25601960 30401970 37101980 44501990 52802000 6070

Sekil 1: Northridge depreminde dusey yer ivmeleri

MAT 1009 Kalkulus I 14 / 79

Bir Fonksiyonun Dort Farklı Gosterimi

Bu orneklerin tumu, verilen bir sayıya (r, t, w veya t) karsılık diger birsayıyı veren (A, P , C veya a) bir kural betimler. Her bir durumda ikincisayı birincisinin fonksiyonudur.

MAT 1009 Kalkulus I 15 / 79

Fonksiyon

Fonksiyon

Tanım 4

Bir f fonksiyonu, bir A kumesinin x ogesini, bir B kumesinin tek bir f(x)ogesine tasıyan bir kuraldır.

Genellikle A ve B kumelerinin gercel sayıların kumeleri oldugu fonksiyonlarıdusunecegiz.

A kumesine fonksiyonun tanım kumesi denir.

f(x) sayısına f fonksiyonunun x deki degeri denir.

x sayısı A kumesi icinde degisirken f(x) in tum olası degerlerinin kumesi fnin goruntu kumesidir.

MAT 1009 Kalkulus I 16 / 79

Fonksiyon

f nin tanım kuınesinin herhangi bir ogesini ifade eden sembole, bagımsızdegisken denir.

Goruntu kumesinin herhangi bir ogesini temsil eden sembole bagımlıdegisken denir.

MAT 1009 Kalkulus I 17 / 79

Fonksiyon

Bir fonksiyonu anlamanın en yaygın yolu onun grafigidir. Tanım kumesi Aolan bir f fonksiyonunun grafigi

{(x, f(x)) ∈ R2∣∣ x ∈ A}

ile betimlenen sıralı ikililer kumesidir.

Baska bir deyisle, f nin grafigi, xtanım kumesinde ve y = f(x) ol-mak kosulu ile duzlemdeki (x, y) nok-talarının kumesidir.

A function is a rule that assigns to each element in a set exactly oneelement, called , in a set .

We usually consider functions for which the sets and are sets of real numbers.The set is called the domain of the function. The number is the value of at and is read “ of .” The range of is the set of all possible values of as varies throughout the domain. A symbol that represents an arbitrary number in thedomain of a function is called an independent variable. A symbol that representsa number in the range of is called a dependent variable. In Example A, forinstance, r is the independent variable and A is the dependent variable.

It’s helpful to think of a function as a machine (see Figure 2). If is in the domainof the function then when enters the machine, it’s accepted as an input and themachine produces an output according to the rule of the function. Thus, we canthink of the domain as the set of all possible inputs and the range as the set of all pos-sible outputs.

The preprogrammed functions in a calculator are good examples of a function as amachine. For example, the square root key on your calculator is such a function. Youpress the key labeled (or ) and enter the input x. If , then is not in thedomain of this function; that is, is not an acceptable input, and the calculator willindicate an error. If , then an approximation to will appear in the display.Thus, the key on your calculator is not quite the same as the exact mathematicalfunction defined by .

Another way to picture a function is by an arrow diagram as in Figure 3. Eacharrow connects an element of to an element of . The arrow indicates that isassociated with is associated with , and so on.

The most common method for visualizing a function is its graph. If is a functionwith domain , then its graph is the set of ordered pairs

(Notice that these are input-output pairs.) In other words, the graph of consists of allpoints in the coordinate plane such that and is in the domain of .

The graph of a function gives us a useful picture of the behavior or “life history”of a function. Since the -coordinate of any point on the graph is , wecan read the value of from the graph as being the height of the graph above thepoint (see Figure 4). The graph of also allows us to picture the domain of on the -axis and its range on the -axis as in Figure 5.

FIGURE 4

{x, ƒ}

ƒ

f(1)f(2)

x

y

0 1 2 x

FIGURE 5

0 x

y � ƒ(x)

domain

range

y

yxffx

f �x�y � f �x��x, y�y

ffxy � f �x��x, y�

f

��x, f �x�� � x � A�

Af

af �a�x,f �x�BA

f �x� � sxfsx

sxx � 0x

xx � 0sxs

f �x�xf,

x

ff

xf �x�fxfxff �x�A

BA

Bf �x�Axf

12 � CHAPTER 1 FUNCTIONS AND MODELS

FIGURE 2Machine diagram for a function ƒ

x(input)

ƒ(output)

f

fA B

ƒ

f(a)a

x

FIGURE 3Arrow diagram for ƒ

MAT 1009 Kalkulus I 18 / 79

Fonksiyon

Grafik, f nin tanım ve goruntu kumelerini, sırası ile x− ve y−ekseniuzerinde, Sekildeki gibi gostermemize de yardımcı olur.

A function is a rule that assigns to each element in a set exactly oneelement, called , in a set .

We usually consider functions for which the sets and are sets of real numbers.The set is called the domain of the function. The number is the value of at and is read “ of .” The range of is the set of all possible values of as varies throughout the domain. A symbol that represents an arbitrary number in thedomain of a function is called an independent variable. A symbol that representsa number in the range of is called a dependent variable. In Example A, forinstance, r is the independent variable and A is the dependent variable.

It’s helpful to think of a function as a machine (see Figure 2). If is in the domainof the function then when enters the machine, it’s accepted as an input and themachine produces an output according to the rule of the function. Thus, we canthink of the domain as the set of all possible inputs and the range as the set of all pos-sible outputs.

The preprogrammed functions in a calculator are good examples of a function as amachine. For example, the square root key on your calculator is such a function. Youpress the key labeled (or ) and enter the input x. If , then is not in thedomain of this function; that is, is not an acceptable input, and the calculator willindicate an error. If , then an approximation to will appear in the display.Thus, the key on your calculator is not quite the same as the exact mathematicalfunction defined by .

Another way to picture a function is by an arrow diagram as in Figure 3. Eacharrow connects an element of to an element of . The arrow indicates that isassociated with is associated with , and so on.

The most common method for visualizing a function is its graph. If is a functionwith domain , then its graph is the set of ordered pairs

(Notice that these are input-output pairs.) In other words, the graph of consists of allpoints in the coordinate plane such that and is in the domain of .

The graph of a function gives us a useful picture of the behavior or “life history”of a function. Since the -coordinate of any point on the graph is , wecan read the value of from the graph as being the height of the graph above thepoint (see Figure 4). The graph of also allows us to picture the domain of on the -axis and its range on the -axis as in Figure 5.

FIGURE 4

{x, ƒ}

ƒ

f(1)f(2)

x

y

0 1 2 x

FIGURE 5

0 x

y � ƒ(x)

domain

range

y

yxffx

f �x�y � f �x��x, y�y

ffxy � f �x��x, y�

f

��x, f �x�� � x � A�

Af

af �a�x,f �x�BA

f �x� � sxfsx

sxx � 0x

xx � 0sxs

f �x�xf,

x

ff

xf �x�fxfxff �x�A

BA

Bf �x�Axf

12 � CHAPTER 1 FUNCTIONS AND MODELS

FIGURE 2Machine diagram for a function ƒ

x(input)

ƒ(output)

f

fA B

ƒ

f(a)a

x

FIGURE 3Arrow diagram for ƒ

MAT 1009 Kalkulus I 19 / 79

Fonksiyon

Ornek 5

EXAMPLE 1 The graph of a function is shown in Figure 6.(a) Find the values of and .(b) What are the domain and range of ?

SOLUTION(a) We see from Figure 6 that the point lies on the graph of , so the value of

at 1 is . (In other words, the point on the graph that lies above x � 1 isthree units above the x-axis.)

When x � 5, the graph lies about 0.7 unit below the x-axis, so we estimate that.

(b) We see that is defined when , so the domain of is the closedinterval . Notice that takes on all values from �2 to 4, so the range of is

EXAMPLE 2 Sketch the graph and find the domain and range of each function.(a) (b)

SOLUTION(a) The equation of the graph is , and we recognize this as being theequation of a line with slope 2 and y-intercept �1. (Recall the slope-intercept formof the equation of a line: . See Appendix B.) This enables us to sketchthe graph of in Figure 7. The expression is defined for all real numbers, sothe domain of is the set of all real numbers, which we denote by �. The graphshows that the range is also �.

(b) Since and , we could plot the points and , together with a few other points on the graph, and join them to producethe graph (Figure 8). The equation of the graph is , which represents aparabola (see Appendix B). The domain of t is �. The range of t consists of allvalues of , that is, all numbers of the form . But for all numbers x andany positive number y is a square. So the range of t is . This canalso be seen from Figure 8.

�y � y � 0� � �0, ��x 2 � 0x 2

t�x�

y � x 2��1, 1�

�2, 4�t��1� � ��1�2 � 1t�2� � 22 � 4

f2x � 1f

y � mx � b

y � 2x � 1

t�x� � x 2f�x� � 2x � 1

�y � �2 � y � 4� � ��2, 4�

ff�0, 7�f0 � x � 7f �x�

f �5� � �0.7

f �1� � 3ff�1, 3�

FIGURE 6

x

y

0

1

1

ff �5�f �1�

f

SECTION 1.1 FOUR WAYS TO REPRESENT A FUNCTION � 13

FIGURE 7

x

y=2x-1

0

-1

12

y

� The notation for intervals is given inAppendix A.

(_1, 1)

(2, 4)

0

y

1

x1

y=≈

FIGURE 8

Sekilde bir f fonksiyonunun grafigi verilmistir.

a) f(1) ve f(5) degerleri ile,

b) f nin tanım ve goruntu kumelerini bulunuz.

MAT 1009 Kalkulus I 20 / 79

Fonksiyon

Cozum.

a) Sekilden (1, 3) noktasının grafikte oldugunu, dolayısıyla f(1) = 3oldugunu buluruz. (Baska bir deyisle, x = 1 de grafikteki noktax−ekseninden 3 birim yukarıdadır.)x = 5 icin grafik x ekseninin yaklasık 0.7 birim altındadır. Bu nedenlef(5) ≈ −0.7 buluruz.

b) f(x) degerleri yalnızca 0 ≤ x ≤ 7 olan x degerleri icintanımlandıgından, f nin tanım kumesi [0, 7] aralıgıdır. Grafikten, fnin −2 ile 4 arasındaki tum degerleri aldıgını goruyoruz. Dolayısıyla fnin goruntu kumesi

{y| − 2 ≤ y ≤ 4} = [−2, 4]

olur.

MAT 1009 Kalkulus I 21 / 79

Fonksiyon Fonksiyonların Gosterimi

Fonksiyonların Gosterimi

Fonksiyonların Gosterimi

• Sozel olarak (sozcukler ile ifade)

• Sayısal olarak (degerler tablosu ile)

• Gorsel olarak (grafik ile)

• Cebirsel olarak (acık bir formul ile)

MAT 1009 Kalkulus I 22 / 79

Fonksiyon Fonksiyonların Gosterimi

Ornek 6

Ustu acık dikdortgenler prizması seklindeki bir kutunun hacmi 10m3’tur.Tabanın uzun kenarı, kısa kenarın iki katıdır. Tabanda kullanılacakmalzemenin metrekaresi 10TL, yan yuzlerde kullanılacak malzemeninmetrekaresi 6TL ise, maliyet fonksiyonunu kısa kenarın fonksiyonu olarakbulunuz.

Cozum.

Sekilde kısa kenar w, uzun kenar 2w veyukseklik h olarak gosterilmistir.

A more accurate graph of the function in Example 3 could be obtained by using athermometer to measure the temperature of the water at 10-second intervals. In gen-eral, scientists collect experimental data and use them to sketch the graphs of func-tions, as the next example illustrates.

EXAMPLE 4 The data shown in the margin come from an experiment on the lactoni-zation of hydroxyvaleric acid at 25 C. They give the concentration of this acid(in moles per liter) after minutes. Use these data to draw an approximation to thegraph of the concentration function. Then use this graph to estimate the concentra-tion after 5 minutes.

SOLUTION We plot the five points corresponding to the data from the table in Fig-ure 14. The curve-fitting methods of Section 1.2 could be used to choose a modeland graph it. But the data points in Figure 14 look quite well behaved, so we simplydraw a smooth curve through them by hand as in Figure 15.

Then we use the graph to estimate that the concentration after 5 minutes is

moleliter

In the following example we start with a verbal description of a function in a phys-ical situation and obtain an explicit algebraic formula. The ability to do this is a use-ful skill in solving calculus problems that ask for the maximum or minimum values ofquantities.

EXAMPLE 5 A rectangular storage container with an open top has a volume of 10 m .The length of its base is twice its width. Material for the base costs $10 per squaremeter; material for the sides costs $6 per square meter. Express the cost of materialsas a function of the width of the base.

SOLUTION We draw a diagram as in Figure 16 and introduce notation by letting andbe the width and length of the base, respectively, and be the height.The area of the base is , so the cost, in dollars, of the material for

the base is . Two of the sides have area and the other two have area , so the cost of the material for the sides is . The total cost is

therefore

To express as a function of alone, we need to eliminate and we do so by usingthe fact that the volume is 10 m . Thus

w�2w�h � 10

3hwC

C � 10�2w2 � � 6�2�wh� � 2�2wh�� � 20w2 � 36wh

6�2�wh� � 2�2wh��2whwh10�2w2 �

�2w�w � 2w2h2w

w

3

C�5� � 0.035

FIGURE 14

C(t )

0.08

0.06

0.04

0.02

0 1 2 3 4 5 6 7 8 t t

0.02

0.04

0.06

C(t )

0.08

1 2 30 4 5 6 7 8

FIGURE 15

t C�t�

16 � CHAPTER 1 FUNCTIONS AND MODELS

t

0 0.08002 0.05704 0.04086 0.02958 0.0210

C�t�

w

2w

h

FIGURE 16

MAT 1009 Kalkulus I 23 / 79

Fonksiyon Fonksiyonların Gosterimi

Cozum (devamı).

Taban alanı (2w)w = 2w2 oldugundan, taban maliyeti 10(2w2) TLolacaktır.Iki yan yuzun alanı wh, diger ikisinin alanı ise 2wh dır. Buradan yanyuzlerin toplam alanı [2(wh) + 2(2wh)] olur. Dolayısıyla yan yuzlerinmaliyeti 6[2(wh) + 2(2wh)] dir.Toplam maliyet ise

C = 10(2w2) + 6[2(wh) + 2(2wh)] = 20w2 + 36wh

olur.

MAT 1009 Kalkulus I 24 / 79

Fonksiyon Fonksiyonların Gosterimi

Cozum (devamı).

C yi w nin bir fonksiyonu olarak ifade edebilmek icin h yi yok etmemizgerekir. Hacim 10m3 oldugu icin,

w(2w)h = 10

ve dolayısıyla

h =10

2w2=

5

w2

dir. Bunu C ifadesinde yerine koyarak

C = 20w2 + 36w

(5

w2

)= 20w2 +

180

w

elde ederiz.

C(w) = 20w2 +180

w, w > 0

denklemi, C yi w nin fonksiyonu olarak ifade eder.

MAT 1009 Kalkulus I 25 / 79

Fonksiyon Dusey Dogru Olcutu

Dusey Dogru Olcutu

Bir fonksiyonun grafigi xy−duzleminde bir egridir. Bu durumda akla birsoru geliyor: xy−duzlemindeki hangi egriler bir fonksiyonun grafigidir? Busorunun yanıtı asagıdaki olcuttur.

MAT 1009 Kalkulus I 26 / 79

Fonksiyon Dusey Dogru Olcutu

xy−duzlemindeki bir egrinin x’in bir fonksiyonunun grafigi olması icingerekli ve yeterli kosul her dusey dogrunun bu egriyi en fazla bir noktadakesmesidir.

Eger x = a gibi dusey bir dogru egriyi yalnızca bir kez (a, b) noktasındakesiyorsa f(a) = b ile fonksiyon tanımlanabilir. Ancak, x = a dogrusu,egriyi (a, b) ve (a, c) noktalarında kesiyorsa bu egri bir fonksiyonun grafigiolamaz cunku bir fonksiyon x = a degerinde b ve c gibi iki farklı degeralamaz.

which gives

Substituting this into the expression for , we have

Therefore, the equation

expresses as a function of .

EXAMPLE 6 Find the domain of each function.

(a) (b)

SOLUTION(a) Because the square root of a negative number is not defined (as a real number),the domain of consists of all values of x such that . This is equivalent to

, so the domain is the interval .

(b) Since

and division by is not allowed, we see that is not defined when or. Thus, the domain of is

which could also be written in interval notation as

The graph of a function is a curve in the -plane. But the question arises: Whichcurves in the -plane are graphs of functions? This is answered by the following test.

The Vertical Line Test A curve in the -plane is the graph of a function of ifand only if no vertical line intersects the curve more than once.

The reason for the truth of the Vertical Line Test can be seen in Figure 17. If eachvertical line intersects a curve only once, at , then exactly one functionalvalue is defined by . But if a line intersects the curve twice, at and , then the curve can’t represent a function because a function can’t assigntwo different values to .

FIGURE 17xa

y

(a, c)

(a, b)

x=a

0xa

yx=a

(a, b)

0

a�a, c�

�a, b�x � af �a� � b�a, b�x � a

xxy

xyxy

���, 0� � �0, 1� � �1, ��

�x � x � 0, x � 1�

tx � 1x � 0t�x�0

t�x� �1

x 2 � x�

1

x�x � 1�

��2, ��x � �2x � 2 � 0f

t�x� �1

x 2 � xf �x� � sx � 2

wC

w 0C�w� � 20w2 �180

w

C � 20w2 � 36w 5

w2� � 20w2 �180

w

C

h �10

2w2 �5

w2

SECTION 1.1 FOUR WAYS TO REPRESENT A FUNCTION � 17

� In setting up applied functions as inExample 5, it may be useful to reviewthe principles of problem solving as dis-cussed on page 88, particularly Step 1:Understand the Problem.

� If a function is given by a formulaand the domain is not stated explicitly,the convention is that the domain is theset of all numbers for which the formulamakes sense and defines a real number.

which gives

Substituting this into the expression for , we have

Therefore, the equation

expresses as a function of .

EXAMPLE 6 Find the domain of each function.

(a) (b)

SOLUTION(a) Because the square root of a negative number is not defined (as a real number),the domain of consists of all values of x such that . This is equivalent to

, so the domain is the interval .

(b) Since

and division by is not allowed, we see that is not defined when or. Thus, the domain of is

which could also be written in interval notation as

The graph of a function is a curve in the -plane. But the question arises: Whichcurves in the -plane are graphs of functions? This is answered by the following test.

The Vertical Line Test A curve in the -plane is the graph of a function of ifand only if no vertical line intersects the curve more than once.

The reason for the truth of the Vertical Line Test can be seen in Figure 17. If eachvertical line intersects a curve only once, at , then exactly one functionalvalue is defined by . But if a line intersects the curve twice, at and , then the curve can’t represent a function because a function can’t assigntwo different values to .

FIGURE 17xa

y

(a, c)

(a, b)

x=a

0xa

yx=a

(a, b)

0

a�a, c�

�a, b�x � af �a� � b�a, b�x � a

xxy

xyxy

���, 0� � �0, 1� � �1, ��

�x � x � 0, x � 1�

tx � 1x � 0t�x�0

t�x� �1

x 2 � x�

1

x�x � 1�

��2, ��x � �2x � 2 � 0f

t�x� �1

x 2 � xf �x� � sx � 2

wC

w 0C�w� � 20w2 �180

w

C � 20w2 � 36w 5

w2� � 20w2 �180

w

C

h �10

2w2 �5

w2

SECTION 1.1 FOUR WAYS TO REPRESENT A FUNCTION � 17

� In setting up applied functions as inExample 5, it may be useful to reviewthe principles of problem solving as dis-cussed on page 88, particularly Step 1:Understand the Problem.

� If a function is given by a formulaand the domain is not stated explicitly,the convention is that the domain is theset of all numbers for which the formulamakes sense and defines a real number.

MAT 1009 Kalkulus I 27 / 79

Fonksiyon Parcalı Tanımlı Fonksiyonlar

Parcalı Tanımlı Fonksiyonlar

Asagıdaki orneklerde, tanım kumelerinin farklı parcalarında farklı bicimdetanımlanmıs fonksiyonlar verilmektedir.

Ornek 7

Bir f fonksiyonu,

f(x) =

{1− x, x ≤ 1

x2, x > 1

ile tanımlanmıstır. f(0), f(1) ve f(2) degerlerini bulunuz ve grafigi ciziniz.

MAT 1009 Kalkulus I 28 / 79

Fonksiyon Parcalı Tanımlı Fonksiyonlar

Cozum.

Eger x ≤ 1 ise f(x) in degeri 1− x dir. Eger x > 1 ise, f(x) in degeri x2

dir.

For example, the parabola shown in Figure 18(a) is not the graph of afunction of because, as you can see, there are vertical lines that intersect the parabolatwice. The parabola, however, does contain the graphs of two functions of . Noticethat implies , so . So the upper and lowerhalves of the parabola are the graphs of the functions [from Example6(a)] and . [See Figures 18(b) and (c).] We observe that if we reversethe roles of and , then the equation does define as a functionof (with as the independent variable and as the dependent variable) and theparabola now appears as the graph of the function .

Piecewise Defined Functions

The functions in the following four examples are defined by different formulas in dif-ferent parts of their domains.

EXAMPLE 7 A function is defined by

Evaluate , , and and sketch the graph.

SOLUTION Remember that a function is a rule. For this particular function the rule isthe following: First look at the value of the input . If it happens that , then thevalue of is . On the other hand, if , then the value of is .

How do we draw the graph of ? We observe that if , then ,so the part of the graph of that lies to the left of the vertical line must coin-cide with the line , which has slope and -intercept 1. If , then

, so the part of the graph of that lies to the right of the line mustcoincide with the graph of , which is a parabola. This enables us to sketch thegraph in Figure l9. The solid dot indicates that the point is included on thegraph; the open dot indicates that the point is excluded from the graph.�1, 1�

�1, 0�y � x 2

x � 1ff �x� � x 2x 1y�1y � 1 � x

x � 1ff �x� � 1 � xx � 1f

Since 2 1, we have f �2� � 22 � 4.

Since 1 � 1, we have f �1� � 1 � 1 � 0.

Since 0 � 1, we have f �0� � 1 � 0 � 1.

x 2f �x�x 11 � xf �x�� 1xx

f �2�f �1�f �0�

f �x� � �1 � x

x 2

if x � 1

if x 1

f

FIGURE 18

(_2, 0)

(a) x=¥-2

0 x

y

(c) y=_œ„„„„x+2

_2

0 x

y

(b) y=œ„„„„x+2

_2 0 x

y

hxyy

xx � h�y� � y 2 � 2yxt�x� � �sx � 2

f �x� � sx � 2y � �sx � 2y 2 � x � 2x � y 2 � 2

xx

x � y 2 � 2

18 � CHAPTER 1 FUNCTIONS AND MODELS

FIGURE 19

x

y

1

1

MAT 1009 Kalkulus I 29 / 79

Fonksiyon Parcalı Tanımlı Fonksiyonlar

Ornek 8

w agırlıgındaki mektubun posta ucreti C(w) oldugunu kabul edelim. Bubir parcalı tanımlı fonksiyondur, cunku fonksiyonun degerlerini verentablodan asagıdakileri elde ederiz.

C(w) =

0.39, 0 < w ≤ 1 ise,

0.63, 1 < w ≤ 2 ise,

0.87, 2 < w ≤ 3 ise,...

We also see that the graph of coincides with the -axis for . Putting thisinformation together, we have the following three-piece formula for :

EXAMPLE 10 In Example C at the beginning of this section we considered the costof mailing a first-class letter with weight . In effect, this is a piecewise

defined function because, from the table of values, we have

The graph is shown in Figure 22. You can see why functions similar to this one arecalled step functions—they jump from one value to the next. Such functions will bestudied in Chapter 2.

Symmetry

If a function satisfies for every number in its domain, then iscalled an even function. For instance, the function is even because

The geometric significance of an even function is that its graph is symmetric withrespect to the -axis (see Figure 23). This means that if we have plotted the graph of

for , we obtain the entire graph simply by reflecting about the -axis. If satisfies for every number in its domain, then is called an

odd function. For example, the function is odd because

The graph of an odd function is symmetric about the origin (see Figure 24). If wealready have the graph of for , we can obtain the entire graph by rotatingthrough about the origin.

EXAMPLE 11 Determine whether each of the following functions is even, odd, or neither even nor odd.(a) (b) (c)

SOLUTION

(a)

Therefore, is an odd function.

(b)

So is even.t

t��x� � 1 � ��x�4 � 1 � x 4 � t�x�

f

� �f �x�

� �x 5 � x � ��x 5 � x�

f ��x� � ��x�5 � ��x� � ��1�5x 5 � ��x�

h�x� � 2x � x 2t�x� � 1 � x 4f �x� � x 5 � x

180x � 0f

f ��x� � ��x�3 � �x 3 � �f �x�

f �x� � x 3fxf ��x� � �f �x�f

yx � 0fy

f ��x� � ��x�2 � x 2 � f �x�

f �x� � x 2fxf ��x� � f �x�f

0.34

0.56

0.78

1.00

if 0 � w � 1

if 1 � w � 2

if 2 � w � 3

if 3 � w � 4

C�w� �

wC�w�

f �x� � �x

2 � x

0

if 0 � x � 1

if 1 � x � 2

if x 2

fx 2xf

20 � CHAPTER 1 FUNCTIONS AND MODELS

FIGURE 22

C

1

1

0 2 3 4 5 w

x0

y

x_x

f(_x) ƒ

FIGURE 23An even function

x0

y

x

_x ƒ

FIGURE 24An odd function

MAT 1009 Kalkulus I 30 / 79

Fonksiyon Mutlak Deger

Mutlak Deger

Bir a sayısının mutlak degeri, |a| ile gosterilir ve gercel sayı dogrusundaa’nın sıfıra olan uzaklıgıdır.

Uzaklıklar her zaman pozitif ya da 0 oldugundan

her a sayısı icin |a| ≥ 0

saglanır.

Ornegin,

|3| = 3 | − 3| = 3 |0| = 0

|√2− 1| =

√2− 1 |3− π| = π − 3

MAT 1009 Kalkulus I 31 / 79

Fonksiyon Mutlak Deger

Genel olarak,

a ≥ 0 ise |a| = a

a < 0 ise |a| = −a

olur.√

simgesinin “pozitif karekok” oldugunu anımsayınız. Bu nedenle√r = s’nin anlamı s2 = r ve s ≥ 0’dır. Dolayısıyla,

√a2 = a denklemi her

zaman dogru degildir. Yalnızca a ≥ 0 icin dogrudur. a < 0 ise −a > 0olup

√a2 = −a saglanır. Buradan, her a degeri icin dogru olan

√a2 = |a|

denklemi elde edilir.

MAT 1009 Kalkulus I 32 / 79

Fonksiyon Mutlak Degerin ozellikleri

Mutlak Degerin ozellikleri

Mutlak Degerin ozellikleri

a ve b gercel sayı ve n tamsayı olsun. Boylece, asagıdakiler gecerlidir.

1 |ab| = |a|.|b|

2

∣∣∣ab

∣∣∣ = |a||b| (b 6= 0)

3 |an| = |a|n

a > 0 ise asagıdakiler gecerlidir.

4 |x| = a ancak ve ancak x = ∓a5 |x| < a ancak ve ancak −a < x < a

6 |x| > a ancak ve ancak x > a veya x < −a

MAT 1009 Kalkulus I 33 / 79

Fonksiyon Mutlak Degerin ozellikleri

Ornek 9

|3x+ 2| ≥ 4 esitsizligini cozunuz.

Cozum.

Ozellik 4 ve 6 dan, |3x+ 2| ≥ 4 esitsizligi

3x+ 2 ≥ 4 ya da 3x+ 2 ≤ −4

ifadesine denktir. Birinci durumda 3x ≥ 2 olur, bu ise x ≥ 23 verir. Ikinci

durumda 3x ≤ −6 olur, bu ise x ≤ −2 verir. Boylece cozum kumesi{x ∈ R

∣∣ x ≤ −2 ya da x ≥ 2

3

}= (−∞,−2] ∪

[2

3,∞)

olur.

MAT 1009 Kalkulus I 34 / 79

Fonksiyon Simetri

Simetri

Tanım 10

Tanım kumesindeki her x icin f(−x) = f(x) kosulunu saglayan ffonksiyonuna cift fonksiyon denir.

Ornegin, f(x) = x2 fonksiyonu icin

f(−x) = (−x)2 = x2 = f(x)

saglandıgından f cifttir.

MAT 1009 Kalkulus I 35 / 79

Fonksiyon Simetri

Bu fonksiyonların onemi, grafiklerinin y−eksenine gore simetrik olmasıdır.Yalnızca x ≥ 0 icin grafik cizildiginde, tum grafik y−eksenine gore simetrialınarak bulunur.

We also see that the graph of coincides with the -axis for . Putting thisinformation together, we have the following three-piece formula for :

EXAMPLE 10 In Example C at the beginning of this section we considered the costof mailing a first-class letter with weight . In effect, this is a piecewise

defined function because, from the table of values, we have

The graph is shown in Figure 22. You can see why functions similar to this one arecalled step functions—they jump from one value to the next. Such functions will bestudied in Chapter 2.

Symmetry

If a function satisfies for every number in its domain, then iscalled an even function. For instance, the function is even because

The geometric significance of an even function is that its graph is symmetric withrespect to the -axis (see Figure 23). This means that if we have plotted the graph of

for , we obtain the entire graph simply by reflecting about the -axis. If satisfies for every number in its domain, then is called an

odd function. For example, the function is odd because

The graph of an odd function is symmetric about the origin (see Figure 24). If wealready have the graph of for , we can obtain the entire graph by rotatingthrough about the origin.

EXAMPLE 11 Determine whether each of the following functions is even, odd, or neither even nor odd.(a) (b) (c)

SOLUTION

(a)

Therefore, is an odd function.

(b)

So is even.t

t��x� � 1 � ��x�4 � 1 � x 4 � t�x�

f

� �f �x�

� �x 5 � x � ��x 5 � x�

f ��x� � ��x�5 � ��x� � ��1�5x 5 � ��x�

h�x� � 2x � x 2t�x� � 1 � x 4f �x� � x 5 � x

180x � 0f

f ��x� � ��x�3 � �x 3 � �f �x�

f �x� � x 3fxf ��x� � �f �x�f

yx � 0fy

f ��x� � ��x�2 � x 2 � f �x�

f �x� � x 2fxf ��x� � f �x�f

0.34

0.56

0.78

1.00

if 0 � w � 1

if 1 � w � 2

if 2 � w � 3

if 3 � w � 4

C�w� �

wC�w�

f �x� � �x

2 � x

0

if 0 � x � 1

if 1 � x � 2

if x 2

fx 2xf

20 � CHAPTER 1 FUNCTIONS AND MODELS

FIGURE 22

C

1

1

0 2 3 4 5 w

x0

y

x_x

f(_x) ƒ

FIGURE 23An even function

x0

y

x

_x ƒ

FIGURE 24An odd function

MAT 1009 Kalkulus I 36 / 79

Fonksiyon Simetri

Tanım 11

Tanım kumesindeki her x icin f(−x) = −f(x) kosulunu saglayanfonksiyonlara tek fonksiyon denir.

Ornegin, f(x) = x3 fonksiyonu tektir cunku

f(−x) = (−x)3 = −x3 = −f(x)

saglanır.

MAT 1009 Kalkulus I 37 / 79

Fonksiyon Simetri

Tek fonksiyonların grafikleri baslangıc noktasına gore simetriktir. Egerx ≥ 0 degerleri icin grafik biliniyorsa tum grafik eldeki grafigin baslangıcnoktası etrafında 180◦ dondurulmesi ile elde edilir.

We also see that the graph of coincides with the -axis for . Putting thisinformation together, we have the following three-piece formula for :

EXAMPLE 10 In Example C at the beginning of this section we considered the costof mailing a first-class letter with weight . In effect, this is a piecewise

defined function because, from the table of values, we have

The graph is shown in Figure 22. You can see why functions similar to this one arecalled step functions—they jump from one value to the next. Such functions will bestudied in Chapter 2.

Symmetry

If a function satisfies for every number in its domain, then iscalled an even function. For instance, the function is even because

The geometric significance of an even function is that its graph is symmetric withrespect to the -axis (see Figure 23). This means that if we have plotted the graph of

for , we obtain the entire graph simply by reflecting about the -axis. If satisfies for every number in its domain, then is called an

odd function. For example, the function is odd because

The graph of an odd function is symmetric about the origin (see Figure 24). If wealready have the graph of for , we can obtain the entire graph by rotatingthrough about the origin.

EXAMPLE 11 Determine whether each of the following functions is even, odd, or neither even nor odd.(a) (b) (c)

SOLUTION

(a)

Therefore, is an odd function.

(b)

So is even.t

t��x� � 1 � ��x�4 � 1 � x 4 � t�x�

f

� �f �x�

� �x 5 � x � ��x 5 � x�

f ��x� � ��x�5 � ��x� � ��1�5x 5 � ��x�

h�x� � 2x � x 2t�x� � 1 � x 4f �x� � x 5 � x

180x � 0f

f ��x� � ��x�3 � �x 3 � �f �x�

f �x� � x 3fxf ��x� � �f �x�f

yx � 0fy

f ��x� � ��x�2 � x 2 � f �x�

f �x� � x 2fxf ��x� � f �x�f

0.34

0.56

0.78

1.00

if 0 � w � 1

if 1 � w � 2

if 2 � w � 3

if 3 � w � 4

C�w� �

wC�w�

f �x� � �x

2 � x

0

if 0 � x � 1

if 1 � x � 2

if x 2

fx 2xf

20 � CHAPTER 1 FUNCTIONS AND MODELS

FIGURE 22

C

1

1

0 2 3 4 5 w

x0

y

x_x

f(_x) ƒ

FIGURE 23An even function

x0

y

x

_x ƒ

FIGURE 24An odd functionMAT 1009 Kalkulus I 38 / 79

Fonksiyon Simetri

Ornek 12

Asagıdaki fonksiyonların teklik ve ciftlik durumlarını arastırınız.(a) f(x) = x5 + x (b) g(x) = 1− x4 (c) h(x) = 2x− x2

Cozum.

(a) f(−x) = (−x)5 + (−x) = (−1)5x5 + (−x)= −x5 − x = −(x5 + x)

= −f(x)oldugundan f tektir.

(b) g(−x) = 1− (−x)4 = 1− x4 = g(x) oldugundan g cift fonksiyondur.

(c) h(−x) = 2(−x)− (−x)2 = −2x− x2. Dolayısı ile h(−x) 6= h(x) veh(−x) 6= −h(x) oldugundan h ne tek, ne de cift bir fonksiyondur.

MAT 1009 Kalkulus I 39 / 79

Fonksiyon Simetri

Cozum (devamı).

(c)

Since and , we conclude that is neither even norodd.

The graphs of the functions in Example 11 are shown in Figure 25. Notice that thegraph of h is symmetric neither about the y-axis nor about the origin.

Increasing and Decreasing Functions

The graph shown in Figure 26 rises from to , falls from to , and rises againfrom to . The function is said to be increasing on the interval , decreasingon , and increasing again on . Notice that if and are any two numbersbetween and with , then . We use this as the defining prop-erty of an increasing function.

A function is called increasing on an interval if

It is called decreasing on if

In the definition of an increasing function it is important to realize that the inequal-ity must be satisfied for every pair of numbers and in with

.You can see from Figure 27 that the function is decreasing on the inter-

val and increasing on the interval .�0, �����, 0�f �x� � x 2

x1 � x2

Ix2x1f �x1 � � f �x2 �

whenever x1 � x2 in If �x1 � f �x2 �

I

whenever x1 � x2 in If �x1 � � f �x2 �

If

A

B

C

D

y=ƒ

f(x¡)

f(x™)

a

y

0 xx¡ x™ b c dFIGURE 26

f �x1 � � f �x2 �x1 � x2bax2x1�c, d ��b, c�

�a, b�fDCCBBA

1

1 x

y

h1

1

y

x

g1

_1

1

y

x

f

_1

(a) (b) (c)FIGURE 25

hh��x� � �h�x�h��x� � h�x�

h��x� � 2��x� � ��x�2 � �2x � x 2

SECTION 1.1 FOUR WAYS TO REPRESENT A FUNCTION � 21

0

y

x

y=≈

FIGURE 27

MAT 1009 Kalkulus I 40 / 79

Matematiksel Modeller

Matematiksel Modeller

Bir matematiksel model; nufus artısı, urune talep, dusen bir nesneninhızı, kimyasal tepkimedeki bir kimyasalın konsantrasyonu, bir bebeginyasam suresi beklentisi gibi olayların fonksiyonlar veya esitlikler ilematematiksel ifadesidir.

Modelin amacı olayları anlamak ve belki de gelecekteki gelisimleri hakkındabir ongorude bulunmaktır.

MAT 1009 Kalkulus I 41 / 79

Matematiksel Modeller

a chemical reaction, the life expectancy of a person at birth, or the cost of emissionreductions. The purpose of the model is to understand the phenomenon and perhapsto make predictions about future behavior.

Figure 1 illustrates the process of mathematical modeling. Given a real-world prob-lem, our first task is to formulate a mathematical model by identifying and naming theindependent and dependent variables and making assumptions that simplify the phe-nomenon enough to make it mathematically tractable. We use our knowledge of thephysical situation and our mathematical skills to obtain equations that relate the vari-ables. In situations where there is no physical law to guide us, we may need to collectdata (either from a library or the Internet or by conducting our own experiments) andexamine the data in the form of a table in order to discern patterns. From this numeri-cal representation of a function we may wish to obtain a graphical representation byplotting the data. The graph might even suggest a suitable algebraic formula in somecases.

The second stage is to apply the mathematics that we know (such as the calculusthat will be developed throughout this book) to the mathematical model that we haveformulated in order to derive mathematical conclusions. Then, in the third stage, wetake those mathematical conclusions and interpret them as information about the orig-inal real-world phenomenon by way of offering explanations or making predictions.The final step is to test our predictions by checking against new real data. If the pre-dictions don’t compare well with reality, we need to refine our model or to formulatea new model and start the cycle again.

A mathematical model is never a completely accurate representation of a physicalsituation—it is an idealization. A good model simplifies reality enough to permitmathematical calculations but is accurate enough to provide valuable conclusions. Itis important to realize the limitations of the model. In the end, Mother Nature has thefinal say.

There are many different types of functions that can be used to model relationshipsobserved in the real world. In what follows, we discuss the behavior and graphs of these functions and give examples of situations appropriately modeled by suchfunctions.

Linear Models

When we say that y is a linear function of x, we mean that the graph of the functionis a line, so we can use the slope-intercept form of the equation of a line to write a for-mula for the function as

where m is the slope of the line and b is the y-intercept.

y � f �x� � mx � b

� The coordinate geometry of lines isreviewed in Appendix B.

FIGURE 1The modeling process

Real-worldproblem

Mathematicalmodel

Real-worldpredictions

Mathematicalconclusions

Formulate

Interpret

SolveTest

SECTION 1.2 MATHEMATICAL MODELS � 25

MAT 1009 Kalkulus I 42 / 79

Matematiksel Modeller Dogrusal Modeller

Dogrusal Modeller

y, x’in dogrusal fonksiyonudur diyerek kastettigimiz, fonksiyonungrafiginin bir dogru olmasıdır.

Dolayısıyla, dogrunun egimi m ve y−keseni b ise dogrunun egim-kesenbicimini kullanarak fonksiyonun formulunu

y = f(x) = mx+ b

olarak yazabiliriz.

MAT 1009 Kalkulus I 43 / 79

Matematiksel Modeller Dogrusal Modeller

Dogrusal fonksiyonların ozgun ozelligi degisim hızlarının sabit olmasıdır.

Sekilde f(x) = 3x− 2 dogrusal fonksiyonunun grafigi ve bazı noktalardaaldıgı degerlerin tablosu verilmistir. x degiskeni 0.1 kadar arttıgında,f(x)’in 0.3 kadar arttıgına dikkat ediniz. Dolayısı ile f(x), x’ten uc katdaha hızlı artar. Bu nedenle y = 3x− 2 grafiginin egimi olan 3, aynızamanda y’nin x’e gore degisim hızı olarak da yorumlanabilir.

A characteristic feature of linear functions is that they grow at a constant rate. Forinstance, Figure 2 shows a graph of the linear function and a table ofsample values. Notice that whenever x increases by 0.1, the value of increases by0.3. So increases three times as fast as x. Thus, the slope of the graph ,namely 3, can be interpreted as the rate of change of y with respect to x.

EXAMPLE 1(a) As dry air moves upward, it expands and cools. If the ground temperature is

and the temperature at a height of 1 km is , express the temperature T(in °C) as a function of the height h (in kilometers), assuming that a linear model isappropriate.(b) Draw the graph of the function in part (a). What does the slope represent?(c) What is the temperature at a height of 2.5 km?

SOLUTION(a) Because we are assuming that T is a linear function of h, we can write

We are given that when , so

In other words, the y-intercept is .We are also given that when , so

The slope of the line is therefore and the required linear func-tion is

(b) The graph is sketched in Figure 3. The slope is , and this repre-sents the rate of change of temperature with respect to height.

(c) At a height of , the temperature is

If there is no physical law or principle to help us formulate a model, we constructan empirical model, which is based entirely on collected data. We seek a curve that“fits” the data in the sense that it captures the basic trend of the data points.

T � �10�2.5� � 20 � �5 C

h � 2.5 km

m � �10 Ckm

T � �10h � 20

m � 10 � 20 � �10

10 � m � 1 � 20

h � 1T � 10b � 20

20 � m � 0 � b � b

h � 0T � 20

T � mh � b

10 C20 C

FIGURE 2

x

y

0

y=3x-2

_2

y � 3x � 2f �x�f �x�

f �x� � 3x � 2

26 � CHAPTER 1 FUNCTIONS AND MODELS

x

1.0 1.01.1 1.31.2 1.61.3 1.91.4 2.21.5 2.5

f �x� � 3x � 2

FIGURE 3

T

h0

10

20

1 3

T=_10h+20

A characteristic feature of linear functions is that they grow at a constant rate. Forinstance, Figure 2 shows a graph of the linear function and a table ofsample values. Notice that whenever x increases by 0.1, the value of increases by0.3. So increases three times as fast as x. Thus, the slope of the graph ,namely 3, can be interpreted as the rate of change of y with respect to x.

EXAMPLE 1(a) As dry air moves upward, it expands and cools. If the ground temperature is

and the temperature at a height of 1 km is , express the temperature T(in °C) as a function of the height h (in kilometers), assuming that a linear model isappropriate.(b) Draw the graph of the function in part (a). What does the slope represent?(c) What is the temperature at a height of 2.5 km?

SOLUTION(a) Because we are assuming that T is a linear function of h, we can write

We are given that when , so

In other words, the y-intercept is .We are also given that when , so

The slope of the line is therefore and the required linear func-tion is

(b) The graph is sketched in Figure 3. The slope is , and this repre-sents the rate of change of temperature with respect to height.

(c) At a height of , the temperature is

If there is no physical law or principle to help us formulate a model, we constructan empirical model, which is based entirely on collected data. We seek a curve that“fits” the data in the sense that it captures the basic trend of the data points.

T � �10�2.5� � 20 � �5 C

h � 2.5 km

m � �10 Ckm

T � �10h � 20

m � 10 � 20 � �10

10 � m � 1 � 20

h � 1T � 10b � 20

20 � m � 0 � b � b

h � 0T � 20

T � mh � b

10 C20 C

FIGURE 2

x

y

0

y=3x-2

_2

y � 3x � 2f �x�f �x�

f �x� � 3x � 2

26 � CHAPTER 1 FUNCTIONS AND MODELS

x

1.0 1.01.1 1.31.2 1.61.3 1.91.4 2.21.5 2.5

f �x� � 3x � 2

FIGURE 3

T

h0

10

20

1 3

T=_10h+20

MAT 1009 Kalkulus I 44 / 79

Matematiksel Modeller Dogrusal Modeller

Ornek 13

(a) Kuru hava yeryuzunden yukarıya dogru hareket ettiginde, genlesir vesogur. Yeryuzu sıcaklıgı 20◦C, 1 km yukseklikte sıcaklık 10◦C ise,dogrusal bir modelin gecerli oldugu varsayımı ile sıcaklık T yi (◦Colarak), yukseklik h nin (km olarak) bir fonksiyonu olarak bulunuz.

(b) (a) sıkkındaki fonksiyonu ciziniz ve egimin ne ifade ettigini acıklayınız.

(c) 2.5 km yukseklikteki sıcaklıgı bulunuz.

MAT 1009 Kalkulus I 45 / 79

Matematiksel Modeller Dogrusal Modeller

Cozum.

(a) T nin, h nin dogrusal bir fonksiyonu oldugu varsayıldıgından

T = mh+ b

yazabiliriz.h = 0 icin T = 20 verildiginden

20 = m · 0 + b = b

esitliginden, y−keseni b = 20 olarak bulunur. h = 1 icin T = 10verildiginden

10 = m · 1 + 20

esitligi, dogrının egimini m = 10− 20 = −10 olarak verir ve istenilendogrusal fonksiyon

T = −10h+ 20

olarak bulunur.

MAT 1009 Kalkulus I 46 / 79

Matematiksel Modeller Dogrusal Modeller

Cozum (devamı).

(b) Grafik Sekilde verilmistir. m = −10◦C/km olan egim sıcaklıgınyukseklige gore degisim hızını verir.

A characteristic feature of linear functions is that they grow at a constant rate. Forinstance, Figure 2 shows a graph of the linear function and a table ofsample values. Notice that whenever x increases by 0.1, the value of increases by0.3. So increases three times as fast as x. Thus, the slope of the graph ,namely 3, can be interpreted as the rate of change of y with respect to x.

EXAMPLE 1(a) As dry air moves upward, it expands and cools. If the ground temperature is

and the temperature at a height of 1 km is , express the temperature T(in °C) as a function of the height h (in kilometers), assuming that a linear model isappropriate.(b) Draw the graph of the function in part (a). What does the slope represent?(c) What is the temperature at a height of 2.5 km?

SOLUTION(a) Because we are assuming that T is a linear function of h, we can write

We are given that when , so

In other words, the y-intercept is .We are also given that when , so

The slope of the line is therefore and the required linear func-tion is

(b) The graph is sketched in Figure 3. The slope is , and this repre-sents the rate of change of temperature with respect to height.

(c) At a height of , the temperature is

If there is no physical law or principle to help us formulate a model, we constructan empirical model, which is based entirely on collected data. We seek a curve that“fits” the data in the sense that it captures the basic trend of the data points.

T � �10�2.5� � 20 � �5 C

h � 2.5 km

m � �10 Ckm

T � �10h � 20

m � 10 � 20 � �10

10 � m � 1 � 20

h � 1T � 10b � 20

20 � m � 0 � b � b

h � 0T � 20

T � mh � b

10 C20 C

FIGURE 2

x

y

0

y=3x-2

_2

y � 3x � 2f �x�f �x�

f �x� � 3x � 2

26 � CHAPTER 1 FUNCTIONS AND MODELS

x

1.0 1.01.1 1.31.2 1.61.3 1.91.4 2.21.5 2.5

f �x� � 3x � 2

FIGURE 3

T

h0

10

20

1 3

T=_10h+20

(c) h = 2.5 km oldugunda sıcaklık

T = −10(2.5) + 20 = −5◦C

olur.

MAT 1009 Kalkulus I 47 / 79

Matematiksel Modeller Polinomlar

Polinomlar

Tanım 14

n bir tamsayı ve a0, a1, · · · , an sabit gercel sayılar olmak uzere

P (x) = anxn + an−1x

n−1 + . . .+ a2x2 + a1x+ a0

seklindeki fonksiyonlara polinom denir. a0, a1, · · · , an sayılarınapolinomun katsayıları denir.

Her polinomun tanım kumesi R = (−∞,∞)’dir.

Ilk katsayı an 6= 0 ise n sayısına polinomun derecesi denir.

MAT 1009 Kalkulus I 48 / 79

Matematiksel Modeller Polinomlar

Ornegin,

P (x) = 2x6 − x4 + 2

5x3 +

√2

derecesi 6 olan bir polinomdur.

Derecesi 1 olan polinom P (x) = mx+ b biciminde olacagından, dogrusalbir fonksiyondur.

Derecesi 2 olan bir polinom P (x) = ax2 + bx+ c bicimindedir vekuadratik fonksiyon (veya ikinci dereceden polinom) adını tasır.

MAT 1009 Kalkulus I 49 / 79

Matematiksel Modeller Polinomlar

Ikinci dereceden polinomların grafigi parabololur ve grafikler bir sonrakibolumde gorecegimiz gibi y = ax2 parabolunun kaydırılması ile elde edilir.a > 0 ise parabolun agzı yukarıya, a < 0 ise asagıya dogru bakar.

We therefore predict that the level will exceed 400 ppm by the year 2020. This prediction is somewhat risky because it involves a time quite remote from ourobservations.

Polynomials

A function is called a polynomial if

where is a nonnegative integer and the numbers are constants,which are called the coefficients of the polynomial. The domain of any polynomial is

If the leading coefficient , then the degree of the polynomial is . For example, the function

is a polynomial of degree 6.A polynomial of degree 1 is of the form and so it is a linear func-

tion. A polynomial of degree 2 is of the form and is called aquadratic function. The graph of P is always a parabola obtained by shifting theparabola , as we will see in the next section. The parabola opens upward if

and downward if . (See Figure 7.)

A polynomial of degree 3 is of the form

and is called a cubic function. Figure 8 shows the graph of a cubic function in part(a) and graphs of polynomials of degrees 4 and 5 in parts (b) and (c). We will see laterwhy the graphs have these shapes.

FIGURE 8 (a) y=˛-x+1 (b) y=x$-3≈+x (c) y=3x%-25˛+60x

x

1

y

10 x

2

y

1

x

20

y

1

P�x� � ax 3 � bx 2 � cx � d

FIGURE 7The graphs of quadratic functions are parabolas.

y

2

x1

(b) y=_2≈+3x+1

0

y

2

x1

(a) y=≈+x+1

a � 0a 0y � ax 2

P�x� � ax 2 � bx � cP�x� � mx � b

P�x� � 2x 6 � x 4 �25 x 3 � s2

nan � 0� � ���, ��.

a0, a1, a2, . . . , ann

P�x� � an xn � an�1xn�1 � � � � � a2x 2 � a1x � a0

P

CO2

SECTION 1.2 MATHEMATICAL MODELS � 29

We therefore predict that the level will exceed 400 ppm by the year 2020. This prediction is somewhat risky because it involves a time quite remote from ourobservations.

Polynomials

A function is called a polynomial if

where is a nonnegative integer and the numbers are constants,which are called the coefficients of the polynomial. The domain of any polynomial is

If the leading coefficient , then the degree of the polynomial is . For example, the function

is a polynomial of degree 6.A polynomial of degree 1 is of the form and so it is a linear func-

tion. A polynomial of degree 2 is of the form and is called aquadratic function. The graph of P is always a parabola obtained by shifting theparabola , as we will see in the next section. The parabola opens upward if

and downward if . (See Figure 7.)

A polynomial of degree 3 is of the form

and is called a cubic function. Figure 8 shows the graph of a cubic function in part(a) and graphs of polynomials of degrees 4 and 5 in parts (b) and (c). We will see laterwhy the graphs have these shapes.

FIGURE 8 (a) y=˛-x+1 (b) y=x$-3≈+x (c) y=3x%-25˛+60x

x

1

y

10 x

2

y

1

x

20

y

1

P�x� � ax 3 � bx 2 � cx � d

FIGURE 7The graphs of quadratic functions are parabolas.

y

2

x1

(b) y=_2≈+3x+1

0

y

2

x1

(a) y=≈+x+1

a � 0a 0y � ax 2

P�x� � ax 2 � bx � cP�x� � mx � b

P�x� � 2x 6 � x 4 �25 x 3 � s2

nan � 0� � ���, ��.

a0, a1, a2, . . . , ann

P�x� � an xn � an�1xn�1 � � � � � a2x 2 � a1 x � a0

P

CO2

SECTION 1.2 MATHEMATICAL MODELS � 29

Sekil 2: y = x2 + x+ 1 y = −2x2 + 3x+ 1

MAT 1009 Kalkulus I 50 / 79

Matematiksel Modeller Polinomlar

Derecesi 3 olan bir polinom

P (x) = ax3 + bx2 + cx+ d

bicimindedir ve kubik fonksiyon adını tasır.

MAT 1009 Kalkulus I 51 / 79

Matematiksel Modeller Kuvvet Fonksiyonları

Kuvvet Fonksiyonları

Tanım 15

a sabit bir sayı olmak uzere

f(x) = xa

bicimindeki fonksiyonlara kuvvet fonksiyonu denir.

MAT 1009 Kalkulus I 52 / 79

Matematiksel Modeller Kuvvet Fonksiyonları

Bazı ozel durumları dusunelim:

i) n pozitif bir tamsayı olmak uzere a = n olsun. n = 3, 4 ve 5oldugunda f(x) = xn fonksiyonlarının grafikleri Sekilde cizilmistir(bunlar yalnızca bir terimi olan polinomlardır).

The general shape of the graph of depends on whether is even orodd. If is even, then is an even function and its graph is similar to theparabola . If is odd, then is an odd function and its graph is simi-lar to that of . Notice from Figure 12, however, that as increases, the graphof becomes flatter near 0 and steeper when . (If is small, then issmaller, is even smaller, is smaller still, and so on.)

( i i ) , where n is a positive integerThe function is a root function. For it is the square rootfunction , whose domain is and whose graph is the upper half ofthe parabola . [See Figure 13(a).] For other even values of n, the graph of

is similar to that of . For we have the cube root functionwhose domain is (recall that every real number has a cube root) and

whose graph is shown in Figure 13(b). The graph of for n odd issimilar to that of .

(b) ƒ=#œ„x

x

y

0

(1, 1)

(a) ƒ=œ„x

x

y

0

(1, 1)

FIGURE 13Graphs of root functions

y � s3 x

�n 3�y � sn x

�f �x� � s3 x

n � 3y � sxy � sn x

x � y 2�0, ��f �x� � sx

n � 2f �x� � x 1n � sn x

a � 1n

0

y

x

y=x$

(1, 1)(_1, 1)

y=x^

y=≈

FIGURE 12Families of power functions

x

y

0

y=x#

y=x%

(_1, _1)

(1, 1)

x 4x 3x 2x� x � � 1y � xn

ny � x 3f �x� � xnny � x 2

f �x� � xnnnf �x� � xn

Graphs of ƒ=xn for n=1, 2, 3, 4, 5

x

1

y

10

y=x%

x

1

y

10

y=x#

x

1

y

10

y=≈

x

1

y

10

y=x

x

1

y

10

y=x$

FIGURE 11

SECTION 1.2 MATHEMATICAL MODELS � 31

The general shape of the graph of depends on whether is even orodd. If is even, then is an even function and its graph is similar to theparabola . If is odd, then is an odd function and its graph is simi-lar to that of . Notice from Figure 12, however, that as increases, the graphof becomes flatter near 0 and steeper when . (If is small, then issmaller, is even smaller, is smaller still, and so on.)

( i i ) , where n is a positive integerThe function is a root function. For it is the square rootfunction , whose domain is and whose graph is the upper half ofthe parabola . [See Figure 13(a).] For other even values of n, the graph of

is similar to that of . For we have the cube root functionwhose domain is (recall that every real number has a cube root) and

whose graph is shown in Figure 13(b). The graph of for n odd issimilar to that of .

(b) ƒ=#œ„x

x

y

0

(1, 1)

(a) ƒ=œ„x

x

y

0

(1, 1)

FIGURE 13Graphs of root functions

y � s3 x

�n 3�y � sn x

�f �x� � s3 x

n � 3y � sxy � sn x

x � y 2�0, ��f �x� � sx

n � 2f �x� � x 1n � sn x

a � 1n

0

y

x

y=x$

(1, 1)(_1, 1)

y=x^

y=≈

FIGURE 12Families of power functions

x

y

0

y=x#

y=x%

(_1, _1)

(1, 1)

x 4x 3x 2x� x � � 1y � xn

ny � x 3f �x� � xnny � x 2

f �x� � xnnnf �x� � xn

Graphs of ƒ=xn for n=1, 2, 3, 4, 5

x

1

y

10

y=x%

x

1

y

10

y=x#

x

1

y

10

y=≈

x

1

y

10

y=x

x

1

y

10

y=x$

FIGURE 11

SECTION 1.2 MATHEMATICAL MODELS � 31

The general shape of the graph of depends on whether is even orodd. If is even, then is an even function and its graph is similar to theparabola . If is odd, then is an odd function and its graph is simi-lar to that of . Notice from Figure 12, however, that as increases, the graphof becomes flatter near 0 and steeper when . (If is small, then issmaller, is even smaller, is smaller still, and so on.)

( i i ) , where n is a positive integerThe function is a root function. For it is the square rootfunction , whose domain is and whose graph is the upper half ofthe parabola . [See Figure 13(a).] For other even values of n, the graph of

is similar to that of . For we have the cube root functionwhose domain is (recall that every real number has a cube root) and

whose graph is shown in Figure 13(b). The graph of for n odd issimilar to that of .

(b) ƒ=#œ„x

x

y

0

(1, 1)

(a) ƒ=œ„x

x

y

0

(1, 1)

FIGURE 13Graphs of root functions

y � s3 x

�n 3�y � sn x

�f �x� � s3 x

n � 3y � sxy � sn x

x � y 2�0, ��f �x� � sx

n � 2f �x� � x 1n � sn x

a � 1n

0

y

x

y=x$

(1, 1)(_1, 1)

y=x^

y=≈

FIGURE 12Families of power functions

x

y

0

y=x#

y=x%

(_1, _1)

(1, 1)

x 4x 3x 2x� x � � 1y � xn

ny � x 3f �x� � xnny � x 2

f �x� � xnnnf �x� � xn

Graphs of ƒ=xn for n=1, 2, 3, 4, 5

x

1

y

10

y=x%

x

1

y

10

y=x#

x

1

y

10

y=≈

x

1

y

10

y=x

x

1

y

10

y=x$

FIGURE 11

SECTION 1.2 MATHEMATICAL MODELS � 31

MAT 1009 Kalkulus I 53 / 79

Matematiksel Modeller Kuvvet Fonksiyonları

Sekilden gorulecegi gibi n artarken f(x) = xn, 0 yakınındaduzlesmekte, |x| ≥ 1 icin diklesmektedir.

The general shape of the graph of depends on whether is even orodd. If is even, then is an even function and its graph is similar to theparabola . If is odd, then is an odd function and its graph is simi-lar to that of . Notice from Figure 12, however, that as increases, the graphof becomes flatter near 0 and steeper when . (If is small, then issmaller, is even smaller, is smaller still, and so on.)

( i i ) , where n is a positive integerThe function is a root function. For it is the square rootfunction , whose domain is and whose graph is the upper half ofthe parabola . [See Figure 13(a).] For other even values of n, the graph of

is similar to that of . For we have the cube root functionwhose domain is (recall that every real number has a cube root) and

whose graph is shown in Figure 13(b). The graph of for n odd issimilar to that of .

(b) ƒ=#œ„x

x

y

0

(1, 1)

(a) ƒ=œ„x

x

y

0

(1, 1)

FIGURE 13Graphs of root functions

y � s3 x

�n 3�y � sn x

�f �x� � s3 x

n � 3y � sxy � sn x

x � y 2�0, ��f �x� � sx

n � 2f �x� � x 1n � sn x

a � 1n

0

y

x

y=x$

(1, 1)(_1, 1)

y=x^

y=≈

FIGURE 12Families of power functions

x

y

0

y=x#

y=x%

(_1, _1)

(1, 1)

x 4x 3x 2x� x � � 1y � xn

ny � x 3f �x� � xnny � x 2

f �x� � xnnnf �x� � xn

Graphs of ƒ=xn for n=1, 2, 3, 4, 5

x

1

y

10

y=x%

x

1

y

10

y=x#

x

1

y

10

y=≈

x

1

y

10

y=x

x

1

y

10

y=x$

FIGURE 11

SECTION 1.2 MATHEMATICAL MODELS � 31

The general shape of the graph of depends on whether is even orodd. If is even, then is an even function and its graph is similar to theparabola . If is odd, then is an odd function and its graph is simi-lar to that of . Notice from Figure 12, however, that as increases, the graphof becomes flatter near 0 and steeper when . (If is small, then issmaller, is even smaller, is smaller still, and so on.)

( i i ) , where n is a positive integerThe function is a root function. For it is the square rootfunction , whose domain is and whose graph is the upper half ofthe parabola . [See Figure 13(a).] For other even values of n, the graph of

is similar to that of . For we have the cube root functionwhose domain is (recall that every real number has a cube root) and

whose graph is shown in Figure 13(b). The graph of for n odd issimilar to that of .

(b) ƒ=#œ„x

x

y

0

(1, 1)

(a) ƒ=œ„x

x

y

0

(1, 1)

FIGURE 13Graphs of root functions

y � s3 x

�n 3�y � sn x

�f �x� � s3 x

n � 3y � sxy � sn x

x � y 2�0, ��f �x� � sx

n � 2f �x� � x 1n � sn x

a � 1n

0

y

x

y=x$

(1, 1)(_1, 1)

y=x^

y=≈

FIGURE 12Families of power functions

x

y

0

y=x#

y=x%

(_1, _1)

(1, 1)

x 4x 3x 2x� x � � 1y � xn

ny � x 3f �x� � xnny � x 2

f �x� � xnnnf �x� � xn

Graphs of ƒ=xn for n=1, 2, 3, 4, 5

x

1

y

10

y=x%

x

1

y

10

y=x#

x

1

y

10

y=≈

x

1

y

10

y=x

x

1

y

10

y=x$

FIGURE 11

SECTION 1.2 MATHEMATICAL MODELS � 31

(x kucukse, x2 daha kucuk, x3 daha da kucuk, x4 ondan da kucuk,v.b. olacaktır).

MAT 1009 Kalkulus I 54 / 79

Matematiksel Modeller Kuvvet Fonksiyonları

ii) n pozitif bir tamsayı olmak uzere a = 1n olsun. f(x) = x

1n = n

√x

fonksiyonuna kok fonksiyonu denir. n = 2 ise f(x) =√x, tanım

kumesi [0,∞), grafigi ise x = y2 parabolunun ust kolu olan kare-kokfonksiyonudur.

The general shape of the graph of depends on whether is even orodd. If is even, then is an even function and its graph is similar to theparabola . If is odd, then is an odd function and its graph is simi-lar to that of . Notice from Figure 12, however, that as increases, the graphof becomes flatter near 0 and steeper when . (If is small, then issmaller, is even smaller, is smaller still, and so on.)

( i i ) , where n is a positive integerThe function is a root function. For it is the square rootfunction , whose domain is and whose graph is the upper half ofthe parabola . [See Figure 13(a).] For other even values of n, the graph of

is similar to that of . For we have the cube root functionwhose domain is (recall that every real number has a cube root) and

whose graph is shown in Figure 13(b). The graph of for n odd issimilar to that of .

(b) ƒ=#œ„x

x

y

0

(1, 1)

(a) ƒ=œ„x

x

y

0

(1, 1)

FIGURE 13Graphs of root functions

y � s3 x

�n 3�y � sn x

�f �x� � s3 x

n � 3y � sxy � sn x

x � y 2�0, ��f �x� � sx

n � 2f �x� � x 1n � sn x

a � 1n

0

y

x

y=x$

(1, 1)(_1, 1)

y=x^

y=≈

FIGURE 12Families of power functions

x

y

0

y=x#

y=x%

(_1, _1)

(1, 1)

x 4x 3x 2x� x � � 1y � xn

ny � x 3f �x� � xnny � x 2

f �x� � xnnnf �x� � xn

Graphs of ƒ=xn for n=1, 2, 3, 4, 5

x

1

y

10

y=x%

x

1

y

10

y=x#

x

1

y

10

y=≈

x

1

y

10

y=x

x

1

y

10

y=x$

FIGURE 11

SECTION 1.2 MATHEMATICAL MODELS � 31

n tamsayısının cift olması durumunday = n

√x fonksiyonunun grafigi y =√

x fonksiyonunun grafigine benzer.

MAT 1009 Kalkulus I 55 / 79

Matematiksel Modeller Kuvvet Fonksiyonları

n = 3 durumunda f(x) = 3√x, tanım kumesi R olan (her gercel

sayının kup-koku vardır) kup-kok fonksiyonudur ve grafigi asagıdaverilmistir.

The general shape of the graph of depends on whether is even orodd. If is even, then is an even function and its graph is similar to theparabola . If is odd, then is an odd function and its graph is simi-lar to that of . Notice from Figure 12, however, that as increases, the graphof becomes flatter near 0 and steeper when . (If is small, then issmaller, is even smaller, is smaller still, and so on.)

( i i ) , where n is a positive integerThe function is a root function. For it is the square rootfunction , whose domain is and whose graph is the upper half ofthe parabola . [See Figure 13(a).] For other even values of n, the graph of

is similar to that of . For we have the cube root functionwhose domain is (recall that every real number has a cube root) and

whose graph is shown in Figure 13(b). The graph of for n odd issimilar to that of .

(b) ƒ=#œ„x

x

y

0

(1, 1)

(a) ƒ=œ„x

x

y

0

(1, 1)

FIGURE 13Graphs of root functions

y � s3 x

�n 3�y � sn x

�f �x� � s3 x

n � 3y � sxy � sn x

x � y 2�0, ��f �x� � sx

n � 2f �x� � x 1n � sn x

a � 1n

0

y

x

y=x$

(1, 1)(_1, 1)

y=x^

y=≈

FIGURE 12Families of power functions

x

y

0

y=x#

y=x%

(_1, _1)

(1, 1)

x 4x 3x 2x� x � � 1y � xn

ny � x 3f �x� � xnny � x 2

f �x� � xnnnf �x� � xn

Graphs of ƒ=xn for n=1, 2, 3, 4, 5

x

1

y

10

y=x%

x

1

y

10

y=x#

x

1

y

10

y=≈

x

1

y

10

y=x

x

1

y

10

y=x$

FIGURE 11

SECTION 1.2 MATHEMATICAL MODELS � 31

n tek ise, (n > 3) y = n√x’nin

grafigi y = 3√x fonksiyonunkine

benzer.

MAT 1009 Kalkulus I 56 / 79

Matematiksel Modeller Kuvvet Fonksiyonları

a = −1 olsun. Sekilde bir bolusu fonksiyonu f(x) = x−1 =1

x’in

grafigi verilmistir. Denklemi y =1

xya da xy = 1 olan grafik,

koordinat eksenlerini asimptot kabul eden bir hiperboldur.

( i i i )The graph of the reciprocal function is shown in Figure 14. Itsgraph has the equation , or , and is a hyperbola with the coordinateaxes as its asymptotes.

This function arises in physics and chemistry in connection with Boyle’s Law,which says that, when the temperature is constant, the volume of a gas is inverselyproportional to the pressure:

where C is a constant. Thus, the graph of V as a function of P (see Figure 15) hasthe same general shape as the right half of Figure 14.

Another instance in which a power function is used to model a physical phenom-enon is discussed in Exercise 20.

Rational Functions

A rational function is a ratio of two polynomials:

where and are polynomials. The domain consists of all values of such that. A simple example of a rational function is the function , whose

domain is ; this is the reciprocal function graphed in Figure 14. The function

is a rational function with domain . Its graph is shown in Figure 16.

Algebraic Functions

A function is called an algebraic function if it can be constructed using algebraicoperations (such as addition, subtraction, multiplication, division, and taking roots)starting with polynomials. Any rational function is automatically an algebraic func-tion. Here are two more examples:

t�x� �x 4 � 16x 2

x � sx� �x � 2�s

3 x � 1f �x� � sx 2 � 1

f

�x � x � �2�

f �x� �2x 4 � x 2 � 1

x 2 � 4

�x � x � 0�f �x� � 1xQ�x� � 0

xQP

f �x� �P�x�Q�x�

f

P

V

0

FIGURE 15Volume as a function of pressure

at constant temperature

V �C

P

xy � 1y � 1xf �x� � x�1 � 1x

a � �1

32 � CHAPTER 1 FUNCTIONS AND MODELS

FIGURE 14The reciprocal function

x

1

y

10

y=∆

FIGURE 16

ƒ=2x$-≈+1

≈-4

x

20

y

20

MAT 1009 Kalkulus I 57 / 79

Matematiksel Modeller Rasyonel Fonksiyonlar

Rasyonel Fonksiyonlar

Tanım 16

P ve Q gibi iki polinomun oranı olarak ifade edilebilen

f(x) =P (x)

Q(x)

fonksiyonuna rasyonel (kesirli) fonksiyon denir

Tanım kumesi, Q(x) 6= 0 olan tum x sayılarıdır.

Ornegin, tanım kumesi {x : x 6= 0} olan f(x) = 1x fonksiyonu da rasyonel

bir fonksiyondur.

MAT 1009 Kalkulus I 58 / 79

Matematiksel Modeller Rasyonel Fonksiyonlar

f(x) =2x4 − x2 + 1

x2 − 4

fonksiyonu da tanım kumesi {x|x 6= ±2} olan bir rasyonel fonksiyondur.

( i i i )The graph of the reciprocal function is shown in Figure 14. Itsgraph has the equation , or , and is a hyperbola with the coordinateaxes as its asymptotes.

This function arises in physics and chemistry in connection with Boyle’s Law,which says that, when the temperature is constant, the volume of a gas is inverselyproportional to the pressure:

where C is a constant. Thus, the graph of V as a function of P (see Figure 15) hasthe same general shape as the right half of Figure 14.

Another instance in which a power function is used to model a physical phenom-enon is discussed in Exercise 20.

Rational Functions

A rational function is a ratio of two polynomials:

where and are polynomials. The domain consists of all values of such that. A simple example of a rational function is the function , whose

domain is ; this is the reciprocal function graphed in Figure 14. The function

is a rational function with domain . Its graph is shown in Figure 16.

Algebraic Functions

A function is called an algebraic function if it can be constructed using algebraicoperations (such as addition, subtraction, multiplication, division, and taking roots)starting with polynomials. Any rational function is automatically an algebraic func-tion. Here are two more examples:

t�x� �x 4 � 16x 2

x � sx� �x � 2�s

3 x � 1f �x� � sx 2 � 1

f

�x � x � �2�

f �x� �2x 4 � x 2 � 1

x 2 � 4

�x � x � 0�f �x� � 1xQ�x� � 0

xQP

f �x� �P�x�Q�x�

f

P

V

0

FIGURE 15Volume as a function of pressure

at constant temperature

V �C

P

xy � 1y � 1xf �x� � x�1 � 1x

a � �1

32 � CHAPTER 1 FUNCTIONS AND MODELS

FIGURE 14The reciprocal function

x

1

y

10

y=∆

FIGURE 16

ƒ=2x$-≈+1

≈-4

x

20

y

20

MAT 1009 Kalkulus I 59 / 79

Matematiksel Modeller Cebirsel Fonksiyonlar

Cebirsel Fonksiyonlar

Tanım 17

Polinomlardan (toplama, cıkarma, carpma, bolme ve kok alma gibi)cebirsel islemler ile elde edilebilen f fonksiyonuna cebirsel fonksiyon denir.

Rasyonel fonksiyonlar cebirsel fonksiyonlardır.

Asagıdaki fonksiyonlar da birer cebirsel fonksiyon ornegidir.

f(x) =√x2 + 1 g(x) =

x4 − 16x2

x+√x

+ (x− 2) 3√x+ 1

MAT 1009 Kalkulus I 60 / 79

Eski Fonksiyonlardan Yenilerini Elde Etmek

Eski Fonksiyonlardan Yenilerini Elde Etmek

Bu bolumde, onceki bolumlerde ogrendigimiz temel fonksiyonlardanbaslayacagız ve grafiklerini kaydırarak, gererek veya yansıtarak yenifonksiyonlar elde edecegiz. Ayrıca, bir fonksiyon ciftinin standart aritmetikislemler ve bileskeyle nasıl birlestirildigini gosterecegiz.

MAT 1009 Kalkulus I 61 / 79

Eski Fonksiyonlardan Yenilerini Elde Etmek Fonksiyonların Donusumleri

Fonksiyonların Donusumleri

Bir fonksiyonun grafigine donusumler uygulayarak yeni fonksiyonlar eldeedebiliriz.

Bu fikirler bize bir cok fonksiyonun grafigini hızlıca cizebilme yeteneginikazandıracaktır.

Aynı zamanda, verilen grafiklerin denklemlerini bulabilecegiz.

MAT 1009 Kalkulus I 62 / 79

Eski Fonksiyonlardan Yenilerini Elde Etmek Fonksiyonların Donusumleri

Once otelemeleri dusunelim.

Eger c pozitif bir sayı ise y = f(x) + c fonksiyonunun grafigi y = f(x)fonksiyonunun grafiginin yukarı dogru c birim kaydırılması ile elde edilir.(Bunun nedeni tum y−koordinatlarının c kadar arttırılmasıdır.)

Benzer bicimde, c > 0 icin g(x) = f(x− c) ile tanımlanan gfonksiyonunun x sayısındaki degeri, f ’nin (x− c) sayısındaki degeridir(baska bir deyisle x’in c birim solundaki deger). Bu nedenle, y = f(x− c)fonksiyonunun grafigi, y = f(x) grafiginin c birim saga kaydırılmıs halidir.

MAT 1009 Kalkulus I 63 / 79

Eski Fonksiyonlardan Yenilerini Elde Etmek Fonksiyonların Donusumleri

Yatay ve Dusey Kaydırmalar

Kabul edelim ki, c > 0 olsun.

• y = f(x) + c’nin grafigini elde etmek icin y = f(x) grafigini yukarıdogru c birim kaydırınız.

• y = f(x)− c’nin grafigini elde etmek icin y = f(x) grafigini asagıyadogru c birim kaydırınız.

• y = f(x− c)’nin grafigini elde etmek icin y = f(x) grafigini sagadogru c birim kaydırınız.

• y = f(x+ c)’nin grafigini elde etmek icin y = f(x) grafigini soladogru c birim kaydırınız.

MAT 1009 Kalkulus I 64 / 79

Eski Fonksiyonlardan Yenilerini Elde Etmek Fonksiyonların Donusumleri

Kabul edelim ki, c > 0 olsun.

of is the graph of reflected about the -axis because the pointis replaced by the point . (See Figure 2 and the following chart, where

the results of other stretching, compressing, and reflecting transformations are alsogiven.)

Vertical and Horizontal Stretching and Reflecting Suppose . To obtain the graph of

Figure 3 illustrates these stretching transformations when applied to the cosinefunction with . For instance, to get the graph of we multiply the y-coordinate of each point on the graph of by 2. This means that the graphof gets stretched vertically by a factor of 2.

FIGURE 3

x

1

2

y

0

y=Ł x

y=Ł 2x

y=Ł 2 x12

x

1

2

y

0

y=2 Ł x

y=Ł x

y=   Ł x12

y � cos xy � cos x

y � 2 cos xc � 2

y � f ��x�, reflect the graph of y � f �x� about the y-axis

y � �f �x�, reflect the graph of y � f �x� about the x-axis

y � f �xc�, stretch the graph of y � f �x� horizontally by a factor of c

y � f �cx�, compress the graph of y � f �x� horizontally by a factor of c

y � �1c� f �x�, compress the graph of y � f �x� vertically by a factor of c

y � cf �x�, stretch the graph of y � f �x� vertically by a factor of c

c 1

�x, �y��x, y�xy � f �x�y � �f �x�

FIGURE 2Stretching and reflecting the graph of ƒ

y= ƒ1c

x

y

0

y=f(_x)

y=ƒ

y=_ƒ

y=cƒ(c>1)

FIGURE 1Translating the graph of ƒ

x

y

0

y=f(x-c)y=f(x+c) y =ƒ

y=ƒ-c

y=ƒ+c

c

c

c c

SECTION 1.3 NEW FUNCTIONS FROM OLD FUNCTIONS � 39

In Module1.3 you can see theeffect of combining the transfor-

mations of this section.

MAT 1009 Kalkulus I 65 / 79

Eski Fonksiyonlardan Yenilerini Elde Etmek Fonksiyonların Donusumleri

Simdi germe ve yansıma donusumlerini ele alalım.

c > 1 ise, y = cf(x) fonksiyonunun grafigi, y = f(x) fonksiyonunungrafiginin dusey dogrultuda c kadar gerilmesi ile elde edilir (cunku hery−koordinatı aynı c sayısı ile carpılmıstır).

y = −f(x) fonksiyonunun grafigi, y = f(x) grafiginin x−eksenine goreyansımasıdır, cunku (x, y) noktası (x,−y) noktası ile yer degistirmektedir.

MAT 1009 Kalkulus I 66 / 79

Eski Fonksiyonlardan Yenilerini Elde Etmek Fonksiyonların Donusumleri

Germe ve Yansımalar

Kabul edelim ki, c > 0 olsun.

• y = cf(x)’in grafigini elde etmek icin y = f(x)’in grafigini duseyolarak c kadar geriniz.

• y = 1cf(x)’in grafigini elde etmek icin y = f(x)’in grafigini dusey

olarak c kadar buzunuz.

• y = f(cx)’in grafigini elde etmek icin y = f(x)’in grafigini yatayolarak c kadar buzunuz.

• y = f(xc )’nin grafigini elde etmek icin y = f(x)’in grafigini yatayolarak c kadar geriniz.

• y = −f(x)’in grafigini elde etmek icin y = f(x)’in grafigininx-ekseninde yansıması alınız.

• y = f(−x) grafigini elde etmek icin y = f(x)’in grafigininy-ekseninde yansıması alınız.

MAT 1009 Kalkulus I 67 / 79

Eski Fonksiyonlardan Yenilerini Elde Etmek Fonksiyonların Donusumleri

Kabul edelim ki, c > 0 olsun.

of is the graph of reflected about the -axis because the pointis replaced by the point . (See Figure 2 and the following chart, where

the results of other stretching, compressing, and reflecting transformations are alsogiven.)

Vertical and Horizontal Stretching and Reflecting Suppose . To obtain the graph of

Figure 3 illustrates these stretching transformations when applied to the cosinefunction with . For instance, to get the graph of we multiply the y-coordinate of each point on the graph of by 2. This means that the graphof gets stretched vertically by a factor of 2.

FIGURE 3

x

1

2

y

0

y=Ł x

y=Ł 2x

y=Ł 2 x12

x

1

2

y

0

y=2 Ł x

y=Ł x

y=   Ł x12

y � cos xy � cos x

y � 2 cos xc � 2

y � f ��x�, reflect the graph of y � f �x� about the y-axis

y � �f �x�, reflect the graph of y � f �x� about the x-axis

y � f �xc�, stretch the graph of y � f �x� horizontally by a factor of c

y � f �cx�, compress the graph of y � f �x� horizontally by a factor of c

y � �1c� f �x�, compress the graph of y � f �x� vertically by a factor of c

y � cf �x�, stretch the graph of y � f �x� vertically by a factor of c

c 1

�x, �y��x, y�xy � f �x�y � �f �x�

FIGURE 2Stretching and reflecting the graph of ƒ

y= ƒ1c

x

y

0

y=f(_x)

y=ƒ

y=_ƒ

y=cƒ(c>1)

FIGURE 1Translating the graph of ƒ

x

y

0

y=f(x-c)y=f(x+c) y =ƒ

y=ƒ-c

y=ƒ+c

c

c

c c

SECTION 1.3 NEW FUNCTIONS FROM OLD FUNCTIONS � 39

In Module1.3 you can see theeffect of combining the transfor-

mations of this section.

MAT 1009 Kalkulus I 68 / 79

Eski Fonksiyonlardan Yenilerini Elde Etmek Fonksiyonların Donusumleri

Ornek 18

Verilen y =√x’in grafigine donusumler uygulanarak y =

√x− 2,

y =√x− 2, y = −

√x, y = 2

√x ve y =

√−x fonksiyonların grafiklerini

ciziniz.

Cozum.

Karekok fonksiyonu y =√x in grafigi Sekilde verilmistir.

EXAMPLE 1 Given the graph of , use transformations to graph ,, , , and .

SOLUTION The graph of the square root function , obtained from Figure 13 in Section 1.2, is shown in Figure 4(a). In the other parts of the figure we sketch

by shifting 2 units downward, by shifting 2 units to theright, by reflecting about the -axis, by stretching vertically by afactor of 2, and by reflecting about the -axis.

EXAMPLE 2 Sketch the graph of the function

SOLUTION Completing the square, we write the equation of the graph as

This means we obtain the desired graph by starting with the parabola andshifting 3 units to the left and then 1 unit upward (see Figure 5).

EXAMPLE 3 Sketch the graphs of the following functions.(a) (b)

SOLUTION(a) We obtain the graph of from that of by compressing hori-zontally by a factor of 2 (see Figures 6 and 7). Thus, whereas the period of is , the period of is .

FIGURE 6

x0

y

1

π2

π

y=sin x

FIGURE 7

x0

y

1

π2

π4

π

y=sin 2x

2�2 � �y � sin 2x2�y � sin x

y � sin xy � sin 2x

y � 1 � sin xy � sin 2x

FIGURE 5 (a) y=≈ (b) y=(x+3)@+1

x0_1_3

1

y

(_3, 1)

x0

y

y � x 2

y � x 2 � 6x � 10 � �x � 3�2 � 1

x 2 � 6x � 10.f �x� �

(a) y=œ„x (b) y=œ„-2x (c) y=œ„„„„x-2 (d) y=_œ„x (e) y=2œ„x (f ) y=œ„„_x

0 x

y

0 x

y

0 x

y

20 x

y

_2

0 x

y

1

10 x

y

yy � s�xy � 2sxxy � �sx

y � sx � 2y � sx � 2

y � sx

y � s�xy � 2sxy � �sxy � sx � 2y � sx � 2y � sx

40 � CHAPTER 1 FUNCTIONS AND MODELS

FIGURE 4

MAT 1009 Kalkulus I 69 / 79

Eski Fonksiyonlardan Yenilerini Elde Etmek Fonksiyonların Donusumleri

Cozum (devamı).

Grafigi 2 birim a sagı kaydırarak y =√x− 2 nin grafigini,

EXAMPLE 1 Given the graph of , use transformations to graph ,, , , and .

SOLUTION The graph of the square root function , obtained from Figure 13 in Section 1.2, is shown in Figure 4(a). In the other parts of the figure we sketch

by shifting 2 units downward, by shifting 2 units to theright, by reflecting about the -axis, by stretching vertically by afactor of 2, and by reflecting about the -axis.

EXAMPLE 2 Sketch the graph of the function

SOLUTION Completing the square, we write the equation of the graph as

This means we obtain the desired graph by starting with the parabola andshifting 3 units to the left and then 1 unit upward (see Figure 5).

EXAMPLE 3 Sketch the graphs of the following functions.(a) (b)

SOLUTION(a) We obtain the graph of from that of by compressing hori-zontally by a factor of 2 (see Figures 6 and 7). Thus, whereas the period of is , the period of is .

FIGURE 6

x0

y

1

π2

π

y=sin x

FIGURE 7

x0

y

1

π2

π4

π

y=sin 2x

2�2 � �y � sin 2x2�y � sin x

y � sin xy � sin 2x

y � 1 � sin xy � sin 2x

FIGURE 5 (a) y=≈ (b) y=(x+3)@+1

x0_1_3

1

y

(_3, 1)

x0

y

y � x 2

y � x 2 � 6x � 10 � �x � 3�2 � 1

x 2 � 6x � 10.f �x� �

(a) y=œ„x (b) y=œ„-2x (c) y=œ„„„„x-2 (d) y=_œ„x (e) y=2œ„x (f ) y=œ„„_x

0 x

y

0 x

y

0 x

y

20 x

y

_2

0 x

y

1

10 x

y

yy � s�xy � 2sxxy � �sx

y � sx � 2y � sx � 2

y � sx

y � s�xy � 2sxy � �sxy � sx � 2y � sx � 2y � sx

40 � CHAPTER 1 FUNCTIONS AND MODELS

FIGURE 4

EXAMPLE 1 Given the graph of , use transformations to graph ,, , , and .

SOLUTION The graph of the square root function , obtained from Figure 13 in Section 1.2, is shown in Figure 4(a). In the other parts of the figure we sketch

by shifting 2 units downward, by shifting 2 units to theright, by reflecting about the -axis, by stretching vertically by afactor of 2, and by reflecting about the -axis.

EXAMPLE 2 Sketch the graph of the function

SOLUTION Completing the square, we write the equation of the graph as

This means we obtain the desired graph by starting with the parabola andshifting 3 units to the left and then 1 unit upward (see Figure 5).

EXAMPLE 3 Sketch the graphs of the following functions.(a) (b)

SOLUTION(a) We obtain the graph of from that of by compressing hori-zontally by a factor of 2 (see Figures 6 and 7). Thus, whereas the period of is , the period of is .

FIGURE 6

x0

y

1

π2

π

y=sin x

FIGURE 7

x0

y

1

π2

π4

π

y=sin 2x

2�2 � �y � sin 2x2�y � sin x

y � sin xy � sin 2x

y � 1 � sin xy � sin 2x

FIGURE 5 (a) y=≈ (b) y=(x+3)@+1

x0_1_3

1

y

(_3, 1)

x0

y

y � x 2

y � x 2 � 6x � 10 � �x � 3�2 � 1

x 2 � 6x � 10.f �x� �

(a) y=œ„x (b) y=œ„-2x (c) y=œ„„„„x-2 (d) y=_œ„x (e) y=2œ„x (f ) y=œ„„_x

0 x

y

0 x

y

0 x

y

20 x

y

_2

0 x

y

1

10 x

y

yy � s�xy � 2sxxy � �sx

y � sx � 2y � sx � 2

y � sx

y � s�xy � 2sxy � �sxy � sx � 2y � sx � 2y � sx

40 � CHAPTER 1 FUNCTIONS AND MODELS

FIGURE 4MAT 1009 Kalkulus I 70 / 79

Eski Fonksiyonlardan Yenilerini Elde Etmek Fonksiyonların Donusumleri

Cozum (devamı).

2 birim saga kaydırarak y =√x− 2 nin grafigini,

EXAMPLE 1 Given the graph of , use transformations to graph ,, , , and .

SOLUTION The graph of the square root function , obtained from Figure 13 in Section 1.2, is shown in Figure 4(a). In the other parts of the figure we sketch

by shifting 2 units downward, by shifting 2 units to theright, by reflecting about the -axis, by stretching vertically by afactor of 2, and by reflecting about the -axis.

EXAMPLE 2 Sketch the graph of the function

SOLUTION Completing the square, we write the equation of the graph as

This means we obtain the desired graph by starting with the parabola andshifting 3 units to the left and then 1 unit upward (see Figure 5).

EXAMPLE 3 Sketch the graphs of the following functions.(a) (b)

SOLUTION(a) We obtain the graph of from that of by compressing hori-zontally by a factor of 2 (see Figures 6 and 7). Thus, whereas the period of is , the period of is .

FIGURE 6

x0

y

1

π2

π

y=sin x

FIGURE 7

x0

y

1

π2

π4

π

y=sin 2x

2�2 � �y � sin 2x2�y � sin x

y � sin xy � sin 2x

y � 1 � sin xy � sin 2x

FIGURE 5 (a) y=≈ (b) y=(x+3)@+1

x0_1_3

1

y

(_3, 1)

x0

y

y � x 2

y � x 2 � 6x � 10 � �x � 3�2 � 1

x 2 � 6x � 10.f �x� �

(a) y=œ„x (b) y=œ„-2x (c) y=œ„„„„x-2 (d) y=_œ„x (e) y=2œ„x (f ) y=œ„„_x

0 x

y

0 x

y

0 x

y

20 x

y

_2

0 x

y

1

10 x

y

yy � s�xy � 2sxxy � �sx

y � sx � 2y � sx � 2

y � sx

y � s�xy � 2sxy � �sxy � sx � 2y � sx � 2y � sx

40 � CHAPTER 1 FUNCTIONS AND MODELS

FIGURE 4

EXAMPLE 1 Given the graph of , use transformations to graph ,, , , and .

SOLUTION The graph of the square root function , obtained from Figure 13 in Section 1.2, is shown in Figure 4(a). In the other parts of the figure we sketch

by shifting 2 units downward, by shifting 2 units to theright, by reflecting about the -axis, by stretching vertically by afactor of 2, and by reflecting about the -axis.

EXAMPLE 2 Sketch the graph of the function

SOLUTION Completing the square, we write the equation of the graph as

This means we obtain the desired graph by starting with the parabola andshifting 3 units to the left and then 1 unit upward (see Figure 5).

EXAMPLE 3 Sketch the graphs of the following functions.(a) (b)

SOLUTION(a) We obtain the graph of from that of by compressing hori-zontally by a factor of 2 (see Figures 6 and 7). Thus, whereas the period of is , the period of is .

FIGURE 6

x0

y

1

π2

π

y=sin x

FIGURE 7

x0

y

1

π2

π4

π

y=sin 2x

2�2 � �y � sin 2x2�y � sin x

y � sin xy � sin 2x

y � 1 � sin xy � sin 2x

FIGURE 5 (a) y=≈ (b) y=(x+3)@+1

x0_1_3

1

y

(_3, 1)

x0

y

y � x 2

y � x 2 � 6x � 10 � �x � 3�2 � 1

x 2 � 6x � 10.f �x� �

(a) y=œ„x (b) y=œ„-2x (c) y=œ„„„„x-2 (d) y=_œ„x (e) y=2œ„x (f ) y=œ„„_x

0 x

y

0 x

y

0 x

y

20 x

y

_2

0 x

y

1

10 x

y

yy � s�xy � 2sxxy � �sx

y � sx � 2y � sx � 2

y � sx

y � s�xy � 2sxy � �sxy � sx � 2y � sx � 2y � sx

40 � CHAPTER 1 FUNCTIONS AND MODELS

FIGURE 4 MAT 1009 Kalkulus I 71 / 79

Eski Fonksiyonlardan Yenilerini Elde Etmek Fonksiyonların Donusumleri

Cozum (devamı).

x−ekseninde yansıma alarak y = −√x nin grafigini,

EXAMPLE 1 Given the graph of , use transformations to graph ,, , , and .

SOLUTION The graph of the square root function , obtained from Figure 13 in Section 1.2, is shown in Figure 4(a). In the other parts of the figure we sketch

by shifting 2 units downward, by shifting 2 units to theright, by reflecting about the -axis, by stretching vertically by afactor of 2, and by reflecting about the -axis.

EXAMPLE 2 Sketch the graph of the function

SOLUTION Completing the square, we write the equation of the graph as

This means we obtain the desired graph by starting with the parabola andshifting 3 units to the left and then 1 unit upward (see Figure 5).

EXAMPLE 3 Sketch the graphs of the following functions.(a) (b)

SOLUTION(a) We obtain the graph of from that of by compressing hori-zontally by a factor of 2 (see Figures 6 and 7). Thus, whereas the period of is , the period of is .

FIGURE 6

x0

y

1

π2

π

y=sin x

FIGURE 7

x0

y

1

π2

π4

π

y=sin 2x

2�2 � �y � sin 2x2�y � sin x

y � sin xy � sin 2x

y � 1 � sin xy � sin 2x

FIGURE 5 (a) y=≈ (b) y=(x+3)@+1

x0_1_3

1

y

(_3, 1)

x0

y

y � x 2

y � x 2 � 6x � 10 � �x � 3�2 � 1

x 2 � 6x � 10.f �x� �

(a) y=œ„x (b) y=œ„-2x (c) y=œ„„„„x-2 (d) y=_œ„x (e) y=2œ„x (f ) y=œ„„_x

0 x

y

0 x

y

0 x

y

20 x

y

_2

0 x

y

1

10 x

y

yy � s�xy � 2sxxy � �sx

y � sx � 2y � sx � 2

y � sx

y � s�xy � 2sxy � �sxy � sx � 2y � sx � 2y � sx

40 � CHAPTER 1 FUNCTIONS AND MODELS

FIGURE 4

EXAMPLE 1 Given the graph of , use transformations to graph ,, , , and .

SOLUTION The graph of the square root function , obtained from Figure 13 in Section 1.2, is shown in Figure 4(a). In the other parts of the figure we sketch

by shifting 2 units downward, by shifting 2 units to theright, by reflecting about the -axis, by stretching vertically by afactor of 2, and by reflecting about the -axis.

EXAMPLE 2 Sketch the graph of the function

SOLUTION Completing the square, we write the equation of the graph as

This means we obtain the desired graph by starting with the parabola andshifting 3 units to the left and then 1 unit upward (see Figure 5).

EXAMPLE 3 Sketch the graphs of the following functions.(a) (b)

SOLUTION(a) We obtain the graph of from that of by compressing hori-zontally by a factor of 2 (see Figures 6 and 7). Thus, whereas the period of is , the period of is .

FIGURE 6

x0

y

1

π2

π

y=sin x

FIGURE 7

x0

y

1

π2

π4

π

y=sin 2x

2�2 � �y � sin 2x2�y � sin x

y � sin xy � sin 2x

y � 1 � sin xy � sin 2x

FIGURE 5 (a) y=≈ (b) y=(x+3)@+1

x0_1_3

1

y

(_3, 1)

x0

y

y � x 2

y � x 2 � 6x � 10 � �x � 3�2 � 1

x 2 � 6x � 10.f �x� �

(a) y=œ„x (b) y=œ„-2x (c) y=œ„„„„x-2 (d) y=_œ„x (e) y=2œ„x (f ) y=œ„„_x

0 x

y

0 x

y

0 x

y

20 x

y

_2

0 x

y

1

10 x

y

yy � s�xy � 2sxxy � �sx

y � sx � 2y � sx � 2

y � sx

y � s�xy � 2sxy � �sxy � sx � 2y � sx � 2y � sx

40 � CHAPTER 1 FUNCTIONS AND MODELS

FIGURE 4 MAT 1009 Kalkulus I 72 / 79

Eski Fonksiyonlardan Yenilerini Elde Etmek Fonksiyonların Donusumleri

Cozum (devamı).

dusey yonde 2 birim gererek y = 2√x nin grafigini,

EXAMPLE 1 Given the graph of , use transformations to graph ,, , , and .

SOLUTION The graph of the square root function , obtained from Figure 13 in Section 1.2, is shown in Figure 4(a). In the other parts of the figure we sketch

by shifting 2 units downward, by shifting 2 units to theright, by reflecting about the -axis, by stretching vertically by afactor of 2, and by reflecting about the -axis.

EXAMPLE 2 Sketch the graph of the function

SOLUTION Completing the square, we write the equation of the graph as

This means we obtain the desired graph by starting with the parabola andshifting 3 units to the left and then 1 unit upward (see Figure 5).

EXAMPLE 3 Sketch the graphs of the following functions.(a) (b)

SOLUTION(a) We obtain the graph of from that of by compressing hori-zontally by a factor of 2 (see Figures 6 and 7). Thus, whereas the period of is , the period of is .

FIGURE 6

x0

y

1

π2

π

y=sin x

FIGURE 7

x0

y

1

π2

π4

π

y=sin 2x

2�2 � �y � sin 2x2�y � sin x

y � sin xy � sin 2x

y � 1 � sin xy � sin 2x

FIGURE 5 (a) y=≈ (b) y=(x+3)@+1

x0_1_3

1

y

(_3, 1)

x0

y

y � x 2

y � x 2 � 6x � 10 � �x � 3�2 � 1

x 2 � 6x � 10.f �x� �

(a) y=œ„x (b) y=œ„-2x (c) y=œ„„„„x-2 (d) y=_œ„x (e) y=2œ„x (f ) y=œ„„_x

0 x

y

0 x

y

0 x

y

20 x

y

_2

0 x

y

1

10 x

y

yy � s�xy � 2sxxy � �sx

y � sx � 2y � sx � 2

y � sx

y � s�xy � 2sxy � �sxy � sx � 2y � sx � 2y � sx

40 � CHAPTER 1 FUNCTIONS AND MODELS

FIGURE 4

EXAMPLE 1 Given the graph of , use transformations to graph ,, , , and .

SOLUTION The graph of the square root function , obtained from Figure 13 in Section 1.2, is shown in Figure 4(a). In the other parts of the figure we sketch

by shifting 2 units downward, by shifting 2 units to theright, by reflecting about the -axis, by stretching vertically by afactor of 2, and by reflecting about the -axis.

EXAMPLE 2 Sketch the graph of the function

SOLUTION Completing the square, we write the equation of the graph as

This means we obtain the desired graph by starting with the parabola andshifting 3 units to the left and then 1 unit upward (see Figure 5).

EXAMPLE 3 Sketch the graphs of the following functions.(a) (b)

SOLUTION(a) We obtain the graph of from that of by compressing hori-zontally by a factor of 2 (see Figures 6 and 7). Thus, whereas the period of is , the period of is .

FIGURE 6

x0

y

1

π2

π

y=sin x

FIGURE 7

x0

y

1

π2

π4

π

y=sin 2x

2�2 � �y � sin 2x2�y � sin x

y � sin xy � sin 2x

y � 1 � sin xy � sin 2x

FIGURE 5 (a) y=≈ (b) y=(x+3)@+1

x0_1_3

1

y

(_3, 1)

x0

y

y � x 2

y � x 2 � 6x � 10 � �x � 3�2 � 1

x 2 � 6x � 10.f �x� �

(a) y=œ„x (b) y=œ„-2x (c) y=œ„„„„x-2 (d) y=_œ„x (e) y=2œ„x (f ) y=œ„„_x

0 x

y

0 x

y

0 x

y

20 x

y

_2

0 x

y

1

10 x

y

yy � s�xy � 2sxxy � �sx

y � sx � 2y � sx � 2

y � sx

y � s�xy � 2sxy � �sxy � sx � 2y � sx � 2y � sx

40 � CHAPTER 1 FUNCTIONS AND MODELS

FIGURE 4 MAT 1009 Kalkulus I 73 / 79

Eski Fonksiyonlardan Yenilerini Elde Etmek Fonksiyonların Donusumleri

Cozum (devamı).

y−ekseninde yansıma alarak y =√−x nin grafigini cizebiliriz.

EXAMPLE 1 Given the graph of , use transformations to graph ,, , , and .

SOLUTION The graph of the square root function , obtained from Figure 13 in Section 1.2, is shown in Figure 4(a). In the other parts of the figure we sketch

by shifting 2 units downward, by shifting 2 units to theright, by reflecting about the -axis, by stretching vertically by afactor of 2, and by reflecting about the -axis.

EXAMPLE 2 Sketch the graph of the function

SOLUTION Completing the square, we write the equation of the graph as

This means we obtain the desired graph by starting with the parabola andshifting 3 units to the left and then 1 unit upward (see Figure 5).

EXAMPLE 3 Sketch the graphs of the following functions.(a) (b)

SOLUTION(a) We obtain the graph of from that of by compressing hori-zontally by a factor of 2 (see Figures 6 and 7). Thus, whereas the period of is , the period of is .

FIGURE 6

x0

y

1

π2

π

y=sin x

FIGURE 7

x0

y

1

π2

π4

π

y=sin 2x

2�2 � �y � sin 2x2�y � sin x

y � sin xy � sin 2x

y � 1 � sin xy � sin 2x

FIGURE 5 (a) y=≈ (b) y=(x+3)@+1

x0_1_3

1

y

(_3, 1)

x0

y

y � x 2

y � x 2 � 6x � 10 � �x � 3�2 � 1

x 2 � 6x � 10.f �x� �

(a) y=œ„x (b) y=œ„-2x (c) y=œ„„„„x-2 (d) y=_œ„x (e) y=2œ„x (f ) y=œ„„_x

0 x

y

0 x

y

0 x

y

20 x

y

_2

0 x

y

1

10 x

y

yy � s�xy � 2sxxy � �sx

y � sx � 2y � sx � 2

y � sx

y � s�xy � 2sxy � �sxy � sx � 2y � sx � 2y � sx

40 � CHAPTER 1 FUNCTIONS AND MODELS

FIGURE 4

EXAMPLE 1 Given the graph of , use transformations to graph ,, , , and .

SOLUTION The graph of the square root function , obtained from Figure 13 in Section 1.2, is shown in Figure 4(a). In the other parts of the figure we sketch

by shifting 2 units downward, by shifting 2 units to theright, by reflecting about the -axis, by stretching vertically by afactor of 2, and by reflecting about the -axis.

EXAMPLE 2 Sketch the graph of the function

SOLUTION Completing the square, we write the equation of the graph as

This means we obtain the desired graph by starting with the parabola andshifting 3 units to the left and then 1 unit upward (see Figure 5).

EXAMPLE 3 Sketch the graphs of the following functions.(a) (b)

SOLUTION(a) We obtain the graph of from that of by compressing hori-zontally by a factor of 2 (see Figures 6 and 7). Thus, whereas the period of is , the period of is .

FIGURE 6

x0

y

1

π2

π

y=sin x

FIGURE 7

x0

y

1

π2

π4

π

y=sin 2x

2�2 � �y � sin 2x2�y � sin x

y � sin xy � sin 2x

y � 1 � sin xy � sin 2x

FIGURE 5 (a) y=≈ (b) y=(x+3)@+1

x0_1_3

1

y

(_3, 1)

x0

y

y � x 2

y � x 2 � 6x � 10 � �x � 3�2 � 1

x 2 � 6x � 10.f �x� �

(a) y=œ„x (b) y=œ„-2x (c) y=œ„„„„x-2 (d) y=_œ„x (e) y=2œ„x (f ) y=œ„„_x

0 x

y

0 x

y

0 x

y

20 x

y

_2

0 x

y

1

10 x

y

yy � s�xy � 2sxxy � �sx

y � sx � 2y � sx � 2

y � sx

y � s�xy � 2sxy � �sxy � sx � 2y � sx � 2y � sx

40 � CHAPTER 1 FUNCTIONS AND MODELS

FIGURE 4 MAT 1009 Kalkulus I 74 / 79

Eski Fonksiyonlardan Yenilerini Elde Etmek Fonksiyonların Donusumleri

Ornek 19

f(x) = x2 + 6x+ 10 fonksiyonunun grafigini ciziniz.

Cozum.

Tam kareye tamamlayarak, grafigin denklemini

y = x2 + 6x+ 10 = (x+ 3)2 + 1

olarak yazarız. Istenilen grafigi, y = x2 parabolunu once 3 birim sola,sonra 1 birim yukarıya kaydırarak buluruz.

EXAMPLE 1 Given the graph of , use transformations to graph ,, , , and .

SOLUTION The graph of the square root function , obtained from Figure 13 in Section 1.2, is shown in Figure 4(a). In the other parts of the figure we sketch

by shifting 2 units downward, by shifting 2 units to theright, by reflecting about the -axis, by stretching vertically by afactor of 2, and by reflecting about the -axis.

EXAMPLE 2 Sketch the graph of the function

SOLUTION Completing the square, we write the equation of the graph as

This means we obtain the desired graph by starting with the parabola andshifting 3 units to the left and then 1 unit upward (see Figure 5).

EXAMPLE 3 Sketch the graphs of the following functions.(a) (b)

SOLUTION(a) We obtain the graph of from that of by compressing hori-zontally by a factor of 2 (see Figures 6 and 7). Thus, whereas the period of is , the period of is .

FIGURE 6

x0

y

1

π2

π

y=sin x

FIGURE 7

x0

y

1

π2

π4

π

y=sin 2x

2�2 � �y � sin 2x2�y � sin x

y � sin xy � sin 2x

y � 1 � sin xy � sin 2x

FIGURE 5 (a) y=≈ (b) y=(x+3)@+1

x0_1_3

1

y

(_3, 1)

x0

y

y � x 2

y � x 2 � 6x � 10 � �x � 3�2 � 1

x 2 � 6x � 10.f �x� �

(a) y=œ„x (b) y=œ„-2x (c) y=œ„„„„x-2 (d) y=_œ„x (e) y=2œ„x (f ) y=œ„„_x

0 x

y

0 x

y

0 x

y

20 x

y

_2

0 x

y

1

10 x

y

yy � s�xy � 2sxxy � �sx

y � sx � 2y � sx � 2

y � sx

y � s�xy � 2sxy � �sxy � sx � 2y � sx � 2y � sx

40 � CHAPTER 1 FUNCTIONS AND MODELS

FIGURE 4

EXAMPLE 1 Given the graph of , use transformations to graph ,, , , and .

SOLUTION The graph of the square root function , obtained from Figure 13 in Section 1.2, is shown in Figure 4(a). In the other parts of the figure we sketch

by shifting 2 units downward, by shifting 2 units to theright, by reflecting about the -axis, by stretching vertically by afactor of 2, and by reflecting about the -axis.

EXAMPLE 2 Sketch the graph of the function

SOLUTION Completing the square, we write the equation of the graph as

This means we obtain the desired graph by starting with the parabola andshifting 3 units to the left and then 1 unit upward (see Figure 5).

EXAMPLE 3 Sketch the graphs of the following functions.(a) (b)

SOLUTION(a) We obtain the graph of from that of by compressing hori-zontally by a factor of 2 (see Figures 6 and 7). Thus, whereas the period of is , the period of is .

FIGURE 6

x0

y

1

π2

π

y=sin x

FIGURE 7

x0

y

1

π2

π4

π

y=sin 2x

2�2 � �y � sin 2x2�y � sin x

y � sin xy � sin 2x

y � 1 � sin xy � sin 2x

FIGURE 5 (a) y=≈ (b) y=(x+3)@+1

x0_1_3

1

y

(_3, 1)

x0

y

y � x 2

y � x 2 � 6x � 10 � �x � 3�2 � 1

x 2 � 6x � 10.f �x� �

(a) y=œ„x (b) y=œ„-2x (c) y=œ„„„„x-2 (d) y=_œ„x (e) y=2œ„x (f ) y=œ„„_x

0 x

y

0 x

y

0 x

y

20 x

y

_2

0 x

y

1

10 x

y

yy � s�xy � 2sxxy � �sx

y � sx � 2y � sx � 2

y � sx

y � s�xy � 2sxy � �sxy � sx � 2y � sx � 2y � sx

40 � CHAPTER 1 FUNCTIONS AND MODELS

FIGURE 4

MAT 1009 Kalkulus I 75 / 79

Eski Fonksiyonlardan Yenilerini Elde Etmek Fonksiyonların Donusumleri

Ornek 20

y = |x2 − 1| fonksiyonunun grafigini ciziniz.

Cozum.

Once y = x2 − 1 parabolunu cizeriz. Bu, y = x2 parabolunun 1 birimasagı kaydırılmasıyla elde edilir. −1 < x < 1 iken x2 + 1 parabolux−ekseninin altında kaldıgından, y = |x2 − 1| nin grafigini x−ekseninegore yansıtarak buluruz.

and when . This tells us how to get the graph offrom the graph of : The part of the graph that lies above the -axis

remains the same; the part that lies below the -axis is reflected about the -axis.

EXAMPLE 5 Sketch the graph of the function .

SOLUTION We first graph the parabola in Figure 10(a) by shifting theparabola downward 1 unit. We see that the graph lies below the x-axis when

, so we reflect that part of the graph about the x-axis to obtain the graphof in Figure 10(b).

Combinations of Functions

Two functions and can be combined to form new functions , , , andin a manner similar to the way we add, subtract, multiply, and divide real numbers.

If we define the sum by the equation

then the right side of Equation 1 makes sense if both and are defined, that is,if x belongs to the domain of and also to the domain of . If the domain of is Aand the domain of is B, then the domain of is the intersection of thesedomains, that is, .

Notice that the sign on the left side of Equation 1 stands for the operation ofaddition of functions, but the sign on the right side of the equation stands for addi-tion of the numbers and .

Similarly, we can define the difference and the product , and their domainsare also . But in defining the quotient we must remember not to divide by 0.

Algebra of Functions Let and be functions with domains and . Then thefunctions , and are defined as follows:

f

t��x� �

f �x�t�x�

domain � �x � A � B � t�x� � 0�

� ft� �x� � f �x�t�x� domain � A � B

� f � t� �x� � f �x� � t�x� domain � A � B

� f � t� �x� � f �x� � t�x� domain � A � B

ftf � t, f � t, ftBAtf

ftA � Bftf � t

t�x�f �x��

�A � B

f � tt

ftft�x�f �x�

� f � t��x� � f �x� � t�x�1

f � t

ft

ftf � tf � ttf

0 x

y

_1 1 0 x

y

_1 1

FIGURE 10 (a) y=≈-1 (b) y=| ≈-1 |

y � � x 2 � 1��1 � x � 1

y � x 2y � x 2 � 1

y � � x 2 � 1 �

xxxy � f �x�y � � f �x��

f �x� � 0y � �f �x�f �x� � 0

42 � CHAPTER 1 FUNCTIONS AND MODELS

and when . This tells us how to get the graph offrom the graph of : The part of the graph that lies above the -axis

remains the same; the part that lies below the -axis is reflected about the -axis.

EXAMPLE 5 Sketch the graph of the function .

SOLUTION We first graph the parabola in Figure 10(a) by shifting theparabola downward 1 unit. We see that the graph lies below the x-axis when

, so we reflect that part of the graph about the x-axis to obtain the graphof in Figure 10(b).

Combinations of Functions

Two functions and can be combined to form new functions , , , andin a manner similar to the way we add, subtract, multiply, and divide real numbers.

If we define the sum by the equation

then the right side of Equation 1 makes sense if both and are defined, that is,if x belongs to the domain of and also to the domain of . If the domain of is Aand the domain of is B, then the domain of is the intersection of thesedomains, that is, .

Notice that the sign on the left side of Equation 1 stands for the operation ofaddition of functions, but the sign on the right side of the equation stands for addi-tion of the numbers and .

Similarly, we can define the difference and the product , and their domainsare also . But in defining the quotient we must remember not to divide by 0.

Algebra of Functions Let and be functions with domains and . Then thefunctions , and are defined as follows:

f

t��x� �

f �x�t�x�

domain � �x � A � B � t�x� � 0�

� ft� �x� � f �x�t�x� domain � A � B

� f � t� �x� � f �x� � t�x� domain � A � B

� f � t� �x� � f �x� � t�x� domain � A � B

ftf � t, f � t, ftBAtf

ftA � Bftf � t

t�x�f �x��

�A � B

f � tt

ftft�x�f �x�

� f � t��x� � f �x� � t�x�1

f � t

ft

ftf � tf � ttf

0 x

y

_1 1 0 x

y

_1 1

FIGURE 10 (a) y=≈-1 (b) y=| ≈-1 |

y � � x 2 � 1��1 � x � 1

y � x 2y � x 2 � 1

y � � x 2 � 1 �

xxxy � f �x�y � � f �x��

f �x� � 0y � �f �x�f �x� � 0

42 � CHAPTER 1 FUNCTIONS AND MODELS

MAT 1009 Kalkulus I 76 / 79

Eski Fonksiyonlardan Yenilerini Elde Etmek Fonksiyonların Cebiri

Fonksiyonların Cebiri

Fonksiyonların Cebiri

f ve g tanım kumeleri A ve B olan fonksiyonlar olsun.f + g, f − g, fg ve f/g fonksiyonları asagıdaki sekilde tanımlanır.

(f + g)(x) = f(x) + g(x) tanım kumesi = A ∩B(f − g)(x) = f(x)− g(x) tanım kumesi = A ∩B

(fg)(x) = f(x)g(x) tanım kumesi = A ∩B(f/g)(x) = f(x)/g(x) tanım kumesi = {x ∈ A ∩B : g(x) 6= 0}

MAT 1009 Kalkulus I 77 / 79

Eski Fonksiyonlardan Yenilerini Elde Etmek Fonksiyonların Cebiri

Ornek 21

f(x) =√x ve g(x) =

√4− x2 ise, f + g, f − g, fg ve f/g fonksiyonlarını

bulunuz.

Cozum.

f(x) =√x fonksiyonunun tanım kumesi [0,∞) dir.

g(x) =√4− x2 fonksiyonunun tanım kumesi, 4− x2 ≥ 0 esitsizligini

saglayan x degerlerinden olusur.

Her iki tarafın kare-kokunu alırsak, |x| ≤ 2 veya −2 ≤ x ≤ 2 elde ederiz.Dolayısıyla g fonksiyonunun tanım kumesi [−2, 2] aralıgıdır.

f ve g nin tanım kumelerinin kesisimi

[0,∞) ∩ [−2, 2] = [0, 2]

kumesidir.

MAT 1009 Kalkulus I 78 / 79

Eski Fonksiyonlardan Yenilerini Elde Etmek Fonksiyonların Cebiri

Cozum (devamı).

Boylece tanımlardan,

(f + g)(x) =√x+

√4− x2 0 ≤ x ≤ 2

(f − g)(x) =√x−

√4− x2 0 ≤ x ≤ 2

(fg)(x) =√x√

4− x2 =√4x− x3 0 ≤ x ≤ 2(

f

g

)(x) =

√x√

4− x2=

√x

4− x20 ≤ x < 2

buluruz. f/g nin tanım kumesinde g(x) = 0 veren x = ±2 noktalarınınolmaması gerektiginden, f/g nin tanım kumesi [0, 2) aralıgıdır.

MAT 1009 Kalkulus I 79 / 79