ap calculus ab semester i ago-dic 2009 ii

8/14/2019 AP Calculus Ab Semester i Ago-dic 2009 II

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AP CALCULUS ABAP CALCULUS AB

SEMESTER I AGO-DIC 2009SEMESTER I AGO-DIC 2009


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469 / 470 BC - 399 BC

Socratic Ignorance"I know that I know nothing"

SOCRATESSOCRATES


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CALCULUS HISTORYCALCULUS HISTORY

PITHAGORAS (600 BC) ZENO (500 BC) EUDOXUS (400 BC) EUCLID (300 BC)

ARCHIMIDES (200 BC)*V KEPLER (1500 AC) GALILEO (15

00 AC) FERMAT (1600 AC) CAVALIERI (1600 AC) DESCARTES (1600 AC) ISAAC BARROW (1600 AC) NEWTON (1700 AC)*V


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ArchimedesArchimedes

Was born and worked in Syracuse (Greek city inSicily) 287 BCE and died in 212 BCE

Friend of King Hieron II

Eureka! (discovery of hydrostatic law)

n Invented many mechanisms, some of which were used for the defenceof Syracuse

n Other achievements in mechanics usually attributed to Archimedes(the law of the lever, center of mass, equilibrium, hydrostatic pressure)

n

Used the method of exhaustions to show that the volume of sphereis 2/3 that of the enveloping cylinder

n According to a legend, his last words were Stay away from mydiagram!, address to a soldier who was about to kill him


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was designed to find areas and volumes ofcomplicated objects (circles, pyramids,spheres) using

approximations by simple objects(rectangles, triangles, prisms)having known areas (or volumes)

The Method of ExhaustionThe Method of Exhaustion


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Triangles1 ,2 ,3 ,4,

Note that

2 +3 = 1/4 1

Similarly4 +5 +6 +7

= 1/16 1and so on

4.4 The area of a Parabolic Segment4.4 The area of a Parabolic Segment[Archimedes (287 212[Archimedes (287 212 BCEBCE)])]

1

3

4 7

6

2

5

O

Y

Q

R

XP

S Z

Thus A = 1

(1+1/4 + (

1/4)

2+) = 4/3

1


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Calculus appeared in 17th century as a system ofshortcuts to results obtained by the method of

exhaustion Calculus derives rules for calculations Problems, solved by calculus include finding areas,

volumes (integralcalculus), tangents, normals andcurvatures (differentialcalculus) and summing of

infinite seriesThis makes calculus applicable in a wide variety of

areas inside and outside mathematics In traditional approach (method of exhaustions)

areas and volumes were computed using subtlegeometric arguments

In calculus this was replaced by the set of rules forcalculations

What is Calculus?What is Calculus?


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Cylon, a Crotoniate and leading citizen by birth,fame and riches, but otherwise a difficult,violent, disturbing and tyrannically disposedman, eagerly desired to participate in thePythagorean way of life. He approached

Pythagoras, then an old man, but was rejectedbecause of the character defects just described.When this happened Cylon and his friends vowedto make a strong attack on Pythagoras and hisfollowers. Thus a powerfully aggressive zealactivated Cylon and his followers to persecutethe Pythagoreans to the very last man. Becauseof this Pythagoras left for Metapontium andthere is said to have ended his days.

Pythagoras deathPythagoras death


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Los cylons son unacivilizacinciberntica queest en guerra con

las Doce Coloniasde la humanidaden la pelcula yseries de

BattlestarGalactica ...

CylonCylon


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"Imagination is more important than knowledge."


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ALBERT EINSTEIN WROTE THISRIDDLE EARLY DURING THE 19thCENTURY. HE SAID THAT 98% OF

THE WORLD POPULATION WOULDNOT BE ABLE TO SOLVE IT.ARE YOU IN THE TOP 2% OF

INTELLIGENT PEOPLE IN THEWORLD?

SOLVE THE RIDDLE AND FINDOUT.

Einsteins RiddleEinsteins Riddle


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There are no tricks, just purelogic, so good luck and don't giveup.

1. In a street there are five houses,painted five different colours.2. In each house lives a person ofdifferent nationality3. These five homeowners eachdrink a different kind of beverage,smoke different brand of cigar andkeep a different pet.

THE QUESTION:

WHO OWNS THE FISH?


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Einstein's Riddle - ANSWER

The German owns the fish.


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"Do not worry about your difficulties inMathematics.

I can assure you mine are still greater."


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CALCULUS: CALCULAE: STONES

TWO FUNDAMENTAL IDEAS OF CALCULUSDERIVATIVE-INTEGRAL

CALCULUS APPLICATIONS BOOK RESOURCES

TI 84 PLUS

CALCULUS INTROCALCULUS INTRO


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Calculus is deeply integrated in every branch of thephysical sciences, such as physics and biology. It isfound in computer science, statistics, andengineering; in economics, business, and medicine.Modern developments such as architecture,

aviation, and other technologies all make use ofwhat calculus can offer.

Finding the Slope of a Curve Calculating the Area of Any Shape Visualizing Graphs Finding the Average of a Function Calculating Optimal Values

CALCULUS APLICATIONSCALCULUS APLICATIONS


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HOW TO FIND:

INSTANTANEOUS RATE OF CHANGE

AREA UNDER A CURVE

THE TWO BIG QUESTIONS OFTHE TWO BIG QUESTIONS OFCALCULUSCALCULUS

A B


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R= D / TRATE = CHANGE IN DISTANCE/ CHANGE IN

TIME

INSTANTANEOUS RATE OFINSTANTANEOUS RATE OFCHANGECHANGE

TIME

DISTANC THE AVERAGE RATE OF CHANGE

BETWEEN TWO POINTS =THE SLOPE OF THE SECANT LINECONNECTING THE TWO POINTS

THE INSTANTANEOUSRATE OFCHANGE =

THE SLOPE OF THETANGENT LINE

R = CHANGE IN D / CHANGE IN TR = O / O = UNDEFINED

BIG PROBLEM


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BLACKBOARD EXAMPLE:From home to school.

SKETCHPADRate of change


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LIMITSLIMITS

TIME

DISTANC

THE INSTANTANEOUS RATE OF CHANGE

THE DEFINITION OF THEDERIVATIVE

f

THE DERIVATIVE OF f(x) AT x REPRESENTSTHE SLOPE OF THE TANGENT LINE AT A POINT x


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The DerivativeThe Derivative

THE DERIVATIVE OF f(x) AT x REPRESENTSTHE SLOPE OF THE TANGENT LINE AT A POINT x

THE INSTANTANEOUS RATE OF CHANGE

UNDERSTANDING LIMITSUNDERSTANDING LIMITS


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Given the graph of below, evaluate the followinglimits.

(a) (b) (c)(d) (e) (f)

(g) (h) (i)



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1st Direct Substitution If it fails

2nd Factoring If it fails

3rd The Conjugate Method

Evaluating LimitsEvaluating Limits


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Algebraic Limits:

(a) (b) (c)

(d) (e) (f)


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Workout the MAGIC (Algebra)

Review: ALGEBRA

ECUATIONS, RELATIONS, AND FUNCTIONS

TRIGONOMETRY

LimitsLimits


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F 1998 AB4From 1998 AB4


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Let fbe a function with f(1) = 4 such that for all

points (x, y) on the graph of f. The slope is givenby

(a)Find the slope of the graph of f at the pointwherex= 1.

(b)Write an equation for the line tangent to the

graph offat x= 1 and use it to approximatef(1.2)

From 1998 AB4From 1998 AB4


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The graph of the velocity v(t), in ft/sec, of a car traveling on astraight road, for is shown above.

A table of values for v(t), at 5 second intervals of time t, is shownto the right of the graph.(a)During what intervals of time is the acceleration of the car

positive? Give a reason for your answer

(b)Find the average acceleration of the car, in ft/sec2, over theinterval

(c)Find one approximation for the acceleration of the car, in ft/sec2,at t= 40. Show the computations you used to arrive

at your answer.

From 1998From 1998AB3AB3


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An equation for the line tangent to the graphof at is:

(a) (b)

(c) (d)

(e)

1997MC AB101997MC AB10


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At what point on the graph ofis the tangent line parallel to the line ?

a (0.5, -0.5) b (0.5, 0.125) c (1, -0.25)d (1, 0.5) e (2, 2)

The following table gives US populations at timet:

Estimate and interpret P(1996).

1997MC AB121997MC AB12


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ILS AP CALCULUS ABILS AP CALCULUS ABWORKSHEET 3WORKSHEET 3


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ken s. . . It also helps you to practice and develop yourlogic/reasoning skills. Calculus throws you challengingproblems your way which make you think.

Life after school and college will likewise undoubtedlythrow you problems which you will have to learn to solve.Although you may never use calculus ever again in yourlifetime or career, you will definitely hold on to the lessonsthat calculus taught you.

Things like time management, how to be organized and

neat, how to hand in things on time, how to perform underpressure when tested, how to be responsible for yourfuture boss, how to be amongst people in your class (whoare analogous to your future clients and co-workers).Calculus on face-value may not seem important to you andmay seem useless, but the lessons and skills you are

learning will be with you your whole lifetime.

secretussecretus......What is theWhat is theimportance ofimportance of
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Olivia J: learning advanced math helps youstrengthen your mind overall. Think of yourmind as a muscle. When you lift heavythings for a while, the lighter things seem

really easy.

whats my name again: If you want to be amath teacher you can use it to torture a

whole other generation of kids.
http://answers.yahoo.com/my/profile;_ylt=AqSdvERHfO.uWn5K4AS6TSjExQt.;_ylv=3?show=vGxipYJnaahttp://answers.yahoo.com/my/profile;_ylt=AqSdvERHfO.uWn5K4AS6TSjExQt.;_ylv=3?show=WbGt7t8Iaahttp://answers.yahoo.com/my/profile;_ylt=AqSdvERHfO.uWn5K4AS6TSjExQt.;_ylv=3?show=WbGt7t8Iaahttp://answers.yahoo.com/my/profile;_ylt=AqSdvERHfO.uWn5K4AS6TSjExQt.;_ylv=3?show=WbGt7t8Iaahttp://answers.yahoo.com/my/profile;_ylt=AqSdvERHfO.uWn5K4AS6TSjExQt.;_ylv=3?show=WbGt7t8Iaahttp://answers.yahoo.com/my/profile;_ylt=AqSdvERHfO.uWn5K4AS6TSjExQt.;_ylv=3?show=WbGt7t8Iaahttp://answers.yahoo.com/my/profile;_ylt=AqSdvERHfO.uWn5K4AS6TSjExQt.;_ylv=3?show=WbGt7t8Iaahttp://answers.yahoo.com/my/profile;_ylt=AqSdvERHfO.uWn5K4AS6TSjExQt.;_ylv=3?show=vGxipYJnaa


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KillerLi...You may not use Calculus, but much ofour society relies on it.The financial operation of our economy relies onforecastings and predictions that only Calculuscan provide. Electrical Engineers use Calculus to

optimize the processing power of the CPU thatruns your computer. City planners and surveyorsuse Calculus to find the exact areas of irregularregions of land. So calculus is very important inlife.As for the meaning of life, Calculus gives noanswers, as it is strictly analytical, and notinterpretational.
http://answers.yahoo.com/my/profile;_ylt=AqSdvERHfO.uWn5K4AS6TSjExQt.;_ylv=3?show=8dHuLVU1aahttp://answers.yahoo.com/my/profile;_ylt=AqSdvERHfO.uWn5K4AS6TSjExQt.;_ylv=3?show=8dHuLVU1aahttp://answers.yahoo.com/my/profile;_ylt=AqSdvERHfO.uWn5K4AS6TSjExQt.;_ylv=3?show=8dHuLVU1aa


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ILS AP CALCULUS AB

If you multiply two terms with the same base(here its x), add the powers and keep the base.

If you divide two terms with the same base,subtract the powers and keep the base.

A negative exponent indicates that a variable is inthe wrong spot, and belongs in the opposite part ofthe fraction, but it only affects the variable itstouching. Note that the exponent becomes positivewhen it moves to the right place.

EXPONENTIAL RULES ANDEXPONENTIAL RULES AND


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ILS AP CALCULUS AB

If an exponential expression is raised to apower, you should multiply the exponents andkeep the base.

The numerator of the fractional power remains

the exponent. The denominator of the powertells you what sort of radical (square root, cuberoot, etc.).


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ILS AP CALCULUS AB

Example 4:

Simplify

Solution:

First raise to the third power.

Then

Multiply the xs and ys together


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ILS AP CALCULUS AB

Problem 4:Simplify the expressionusing exponential rules.

LOGARITHMIC RULES AND


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LOGARITHMIC RULES ANDLOGARITHMIC RULES ANDPROPERTIESPROPERTIES


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ss worksheet 3


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Complete the tableComplete the table


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AnswersAnswers

FACTORING POLYNOMIALSFACTORING POLYNOMIALS


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Greatest Common FactorsFactoring using the greatest common factor is the easiestmethod of factoring and is used whenever you see terms thathave pieces in common.Take, for example, the expression 4x + 8. Notice that bothterms can be divided by 4, making 4 a common factor.Therefore, you can write the expression in the factored form of

4(x + 2).In effect, I have pulled out the common factor of 4, andwhats left behind are the terms once 4 has been divided out ofeach.In these type of problems, you should ask yourself, What doeach of the terms have in common? and then pull that greatestcommon factor out of each to write your answer in factored

form.

Problem 5: Factor the expression

FACTORING POLYNOMIALSFACTORING POLYNOMIALS

S i l i


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You should feel comfortable factoringtrinomials such as x + 5x + 4 = 0using whatever method suits you.

Most people play with binomial pairs untilthey stumble across some-thing that works,in this case

(x + 4)(x + 1)

Special Factoring PatternsSpecial Factoring Patterns

S i l F i P


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There are some patterns that you should have memorized: Difference of perfect squares: a b = (a + b)(a b)

Explanation: A perfect square is a number like 16, whichcan be created by multiplying something times itself. Inthe case of 16, that something is 4, since 4 times itself is

16. If you see one perfect square being subtracted fromanother, you can automatically factor it using the patternabove.

For example, x 25 is a difference of x and 25, and bothare perfect squares. Thus, it can be factored as

(x + 5)(x 5).

You cannot factor the sum of perfect squaresso whereas x 4 is factorable, x + 4 is not!

Special Factoring PatternsSpecial Factoring Patterns


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Sum of perfect cubes:a + b = (a + b)(a ab + b)

Explanation: Perfect cubes are similar to

perfect squares. The number 125 is a perfectcube because5 5 5 = 125. This formula can be altered

just slightly to factor the difference of perfectcubes, as illustrated in the next bullet. Otherthan a couple of sign changes, the process isthe same.


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Difference of perfect cubes:a b = (a b)(a2 + ab + b2)Example 5: Factor x 27 using the difference of

perfect cubes factoring pattern. Solution: Note that xis a perfect cube since x x x = x, and 27 is also,since 3 3 3 = 27. Therefore, x 27 corresponds to

a b in the formula, making a = x and b = 3. Now,all thats left to do is plug a and b into the formula:

You cannot factor (x + 3x + 9) any further,so you are finished.


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Problem 6:Factor the expression 8x + 343

Solving QuadraticSolving Quadratic


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Method One: Factoring

Method Two: Completing the Square

Method Three: The Quadratic Formula

Solving QuadraticSolving QuadraticE uationsE uations

Method One: FactoringMethod One: Factoring


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To begin, set your quadratic equation equal to 0; If the resulting equation is factorable, factor it and set each

individual term equal to 0. These equations will give you thesolutions to the equation. Thats all there is to it.

Example 6: Solve the equation 3 x + 4x = 1 by factoring

Solution: Always start the factoring method by setting the

equation equal to 0. 3x + 4x + 1 = 0.Now, factor the equation and set each factor equal to 0.

(3x + 1)(x + 1) = 03x + 1 = 0 x + 1 = 0x = - 1/3 x = - 1

This equation has two solutions: x = -1/3 or x = 1You can check them by plugging each separately into theoriginal equation, and youll find that the result is true.

Method One: FactoringMethod One: Factoring


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Method Two: Completing theMethod Two: Completing the


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Example 7: Solve the equation 2x + 12x 18 = 0by completing the square.

Solution:Move the constant to the right side of the equation:

2x + 12x = 18

This is important: For completing the square to work, thecoefficient of x2 must be 1. Divide every term in the equation by2:

x + 6x = 9Heres the key to completing the square: Take half of thecoefficient of the x term, square it, and add it to both sides.In this problem, the x coefficient is 6, so take half of it (3) andsquare that (3 = 9). Add the result (9) to both sides of theequation:

x + 6x + 9 = 9 + 9x + 6x + 9 = 18

Method Two: Completing theMethod Two: Completing theSquareSquare


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At this point, if youve done everything correctly, theleft side of the equation will be factorable. In fact, itwill be a perfect square!

(x + 3)(x + 3) = 18(x + 3) = 18

To solve the equation, take the square root of bothsides. That will cancel out the exponent. Wheneveryou do this, you have to add a sign in front of theright side of the equation. This is always done whensquare rooting both sides of any equation:

(x + 3) = 18

x + 3 = 18 Solve forx, and thats it. It would also be good form

to simplify into : x= -3 18x = -3 3 2

x = - 3 + 3 2 x = - 3 - 3 2

Method Three:Method Three:


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The quadratic formula

Set the equation equal to 0, and youre halfwaythere. Your equation will then look like this:

ax + bx + c = 0

where a, b, and c are the coefficients asindicated.Take those numbers and plug them straight intothis formula :

Youll get the same answer you would achieve bycompleting the square.

Method Three:The QuadraticThe Quadratic

Solve the equation 2x + 12x 18 = 0


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Solve the equation 2x + 12x 18 0using the quadratic formula.

Solution: The equation is already set equal to 0,

in form

ax + bx + c = 0, and a = 2, b = 12, and c = 18

Plug these values intothe quadratic formula

and simplify:


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Problem 7: Solve the equation3x + 12x = 0

three times, using all the methods you havelearned for solving quadratic equations.

The Least You Need to KnowThe Least You Need to Know


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Basic equation solving is an important skill in

calculus.

Reviewing the five exponential rules willprevent arithmetic mistakes in the long run.

You can create the equation of a line with justa little information using point-slope form.

There are three major ways to solve quadraticequations, each important for different reasons.

ECUATIONS, RELATIONS, ANDECUATIONS, RELATIONS, AND


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WHEN IS AN ECUATION A FUNCTION? IMPORTANT FUNCTION PROPERTIES FUNCTION SKILLSTHE BASIC PARAMETRIC ECUATIONS

, ,, ,FUNCTIONSFUNCTIONS

GO TO THE

BOOK: 1 A LIBRARY OFBOOK: 1 A LIBRARY OF


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FUNCTIONSFUNCTIONS

FUNCTIONSFUNCTIONS


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The Rule of four:

Tables, Graphs, Formulas, and Words.

C=4T - 160

The Chirp Rate is a Functionof Temperature C(T)=4T-160

T

FUNCTIONSFUNCTIONS C=4T - 160


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Domain (inputs)

=All T values between 40F and 136F =All T values with 40x136

=[40,136]

Range (outputs) =All C values from 0 to 384

=All C value with 0C384

T


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This function, called g, accepts any realnumber input. To find out the output g gives,you plug the input into the x slot.

Real life examples A persons height is a function of time

Other examples (by ss)


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Piecewise-defined function

Evaluate f(1)=

f(2)=

f(3)=

f(10)= f(0)=

Vertical line testVertical line test


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The last important thing you should knowabout functions is the vertical line test.This test is a way to tell whether or not agiven graph is the graph of a function or not.

Vertical line testVertical line test


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1.1 FUNCTIONS AND1.1 FUNCTIONS AND


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Linear functionsy=f(x)=b +mx

y-y=m(x-x)

CHANGECHANGE

1.2 EXPONENTIAL1.2 EXPONENTIAL


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Nmero de habitantes En el II Conteo de Poblacin y Vivienda 2005,

realizado por el INEGI, se contaron 103 263 388habitantes en Mxico.

Por ello, Mxico est entre los once pases mspoblados del mundo, despus de:China, India, Estados Unidos de Amrica,Indonesia, Brasil, Pakistn, Rusia, Bangladesh,Ni eria a n.

FUNCTIONSFUNCTIONS

1.2 EXPONENTIAL1.2 EXPONENTIAL


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THE GENERAL EXPONENTIAL FUNCTION Pis an exponential funtion oftwith base a if

Where P is the initial quantity (when t=0) and a is

the factor by which P changes when t increases by1.

If a>1, we have exponential growth If 0


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Exponential

Calculate the Exponential

t (years since 1980)

What is the initial quantity?

What is the Growth Rate?

Evaluate and Interpret P(2005):P(2009):

For what year was thePopulation


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t (hours)

Exponential

Calculate the Exponential

What is the initial quantity?

What is the Growth Rate?Evaluate and Interpret Q(10):

How many hours does it takefor the drug to decrease to 0.001mg?

Example 1


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Suppose that Q=f(t) is an exponential function of t.

If f(20)=88.2 and f(23)=91.4

a. Find the base b. Find the growth rate c. Evaluate f(25)

Exponential FunctionsExponential Functions


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Any exponential Growth function can be written, for somea>1 and k>0, in the form

or

And any exponential Decay function can be written, forsome 0


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The graph of a function is concave up if it bendsupward as we move from left to right; It is concave down if it bends downward.

ConcavityConcavity

Exercisespg. 14: 1,2,3,4,5,6,7,8,9,10,1112,17,23,24,25,26,27,37,39

1.3 NEW FUNCTIONS FROM1.3 NEW FUNCTIONS FROM


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Shifts and StretchesOLDOLD

Multiplying a function by a constant, c, stretches the graph vertically(if c>1). Or shrinks the graph vertically (if 0


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Composite Functions A Function of a Function Example 1. If f(x)=x and g(x)=x+1, find:

a. f(g(2))

b. g(f(2))

c. f(g(x))

d. g(f(x))

Exmp 2. Express the following function as acomposition.

h(t)=(1+t)

Odd and Even Functions: Symmetry


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The graph of any polynomial involving only even powersof x has symmetry about the x-axis.

(Even functions. E.g. f(x)=x)

Polynomials with only odd powers of x are symmetricabout the origin.(Odd functions. E.g. g(x)=x)

f(x)=x g(x)=x

For any function f,f is an Even function iff(-x)=f(x) for all x.f is an Odd function if f(-x)=-f(x) for all x.

Even Odd function


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1.4 LOGARITHMIC FUNCTIONS1.4 LOGARITHMIC FUNCTIONSTh l ith t b 10 f itt l


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The logarithm to base 10 of x, written logx,is the power of 10 we need to get x.

The naturallogarithm of x, written lnx,is the powerof e needed to get x.

Properties of Logarithms

3. Log (AB)=log A + log B4. Log (A/B)=log A log B5. Log A^p= p log A

6. Log 10^x= x7. 10 ^ log x= x

Log x and Lnx are not defined when x is negative or0.

Log 1=0 Ln 1=0

Solving Equations usingSolving Equations using


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EX 1. Find t such that

EX 2. Find when the population of Mexicoreaches 200 million by solving

EX 3 What is the half life of ozone?


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EX 3. What is the half life of ozone?(Decaying exponentially at a continuous rateof 0.25% per

year)

EX 4 The population of Kenya was 19 5 million in


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EX 4. The population of Kenya was 19.5 million in1984, and 21.2 million in 1986. Assuming it

increases exponentially, find a formula for thepopulation of Kenya as a function of time.

Give a formula for the inverse of the following


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gfunction. (Solve for tin terms ofP )

Exercises pg 27:

1.5 TRIGONOMETRIC1.5 TRIGONOMETRICFUNCTIONSFUNCTIONS


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FUNCTIONSFUNCTIONS

The Unit

= 1

An angle of 1 radian is defined to be theangleat the center of a unit circle which cuts off an

Arc length= 180 = radians 1 radian = 180 /

Equation of the unit circle: x + y =1Fundamental Identity: cos t + sin t = 1

Amplitude, Period, and Phase

F P i di f i f i


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For any Periodic function of timeAmplitude is half the distance between the maximum and the minimumvalues. (if it exists)

Period is the smallest time needed for the function to execute onecomplete cycle.

Phase is the difference a periodic function is shifted with respect to

Amplitude

Period = 2

Sine and Cosine graphsare shifted horizontally /2

cos t = sin(t+ /2)sin t = cos(t /2)

The phase difference orphase shift between

sin t and cos t is /2Phase = /2

To describe arbitrary amplitudes and periods of Sinusoidalfunctions:


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f(t)=A sin( B t ) and g(t)=A cos( B t )

The graph of a sinusoidal function is shifted horizontally by a distance |h|when t is replaced by t-h or t+h.

Functions of the form f(t)=A sin (Bt) + C and g(t)=A cos( Bt) +

Ex 1. Find and show on the graph the Amplitude and Period of thefunctions.

EX 2 Fi d ibl f l f th f ll i i id l


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EX 2. Find possible formulas for the following sinusoidal

-6 6t

g

3

-3

-1 3t

f

2

-2

-5 7t

h

3

-3

EX 3. The High tide was 9.9 feet at midnight. Later at Low tide, it was 0.1feet.


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the next High tide is at exactly 12 noon and the height of the water isgiven by a sine or cosine curve.

Ex 4. The interval between high tides actually averages 12 hours 24

minutes.

Ex 5. Using the info from Ex 4.Write a formula for the water level, when the high tide is at 2

Exercises 35. 13,14,15,16,17,19,20,24,25,38


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1.6 POWERS, POLYNOMIALS, AND RATIONAL1.6 POWERS, POLYNOMIALS, AND RATIONALFUNCTIONSFUNCTIONS


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A power function has the

form

Where k and p are constant.

Ex: the volume, V, of a sphere of radius r is givenby

V= g(r)=4/3 r

Ex2: Newtons Law of Gravitation

Polynomialsare the sums of power functions with nonnegative integer


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exponents

n is a nonnegative integer called the degree of the polynomial.

degree of the function=_____

The shape of the graph of a polynomial depends on its degree.A leading negative coefficient turns the graph upside down.The quadratic (n=2) turns around once.The cubic (n=3) turns around twice.The quartic (n=4) turns around three times.An degree polynomial turns around at most n-1 times.**There ma be fewer turns**

n= n=n=

n=

EX1: Find possible formulas for the


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-2 2x

f4

-3 2x

g

1 -3 2x

h

EX1: Find possible formulas for the

-12

Rational functions

i f l i l d


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are ratios of polynomials, p and q:

x

y y=0 is a Horizontal Asymptoteor

y0 as x and y0 as x-

x=K is a Vertical Asymptote

ify or y- as x K

x

y

K

The graphs in Rationalfunctions

may have vertical asymptotes

Rational functionshave horizontal asymptotesif f(x) approaches a finitenumber

Exercisesp

g42:5,7,8,

9,1

1.7 INTRODUCTION TO CONTINUITY1.7 INTRODUCTION TO CONTINUITYA continuous function has a graph which can bedrawn


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A function is said to be continuous on an interval

if its graph has no breaks, jumps or holes in that

To be certain that a function has a zero in an intervalon which it changes sign, we need to know that thefunction

-1 1x

f(x)=3x-x+2x-15

-5

-1

1x

f(x)=1/x

No zero for -1x1although f(-1) and f(1)

have opposite signs

Zero for 0x1F(0)=-1 and f(1)

=3

drawn


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A continuous function cannot skip

The function f(x)=cos x -2x must have a zerobecause its graph cannot skip over the x-axis.

f(x) has at least one zero in the interval0.6x0.8

since f(x) changes from positive to negative onx

f(x)=cos x -2x1

-1

0.4 0.6 0.8 1

The Intermediate Value TheoremSuppose f is a continuous function on a closed interval [a,b].If k is any number between f(a) and f(b), then there is at

EX I ti t th ti it f f( ) t


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EX: Investigate the continuity of f(x)=x at

The values of f(x)=x approach f(2)=4 as x approaches 2.Thus f appears to be continuous at x=2

ContinuityThe function f is continuous atx=c iffis defined atx=c andif

Exercises pg 47: 15, 17,

15. An electrical circuit switches instantaneously from a 6 volt batteryto a 12 volt battery 7 seconds after being turned on. Graph the


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to a 12 volt battery 7 seconds after being turned on. Graph thebatteryvoltage against time. Give formulas for the function represented by

17. Find k so that the following function is continuous on any

t

f


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Given the graph of below, evaluate the followinglimits.



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(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Evaluating LimitsEvaluating Limits


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1st

Direct Substitution If it fails (0/0 Indeterminate form)

2nd Factoring

If it fails

3rd The Conjugate Method


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Algebraic Limits:

(a) (b) (c)

(d) (e) (f)

(g)

Chapter1 REVIEW EXERCISES ANDChapter1 REVIEW EXERCISES ANDPROBLEMSPROBLEMS


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PROBLEMSPROBLEMS

1st Period Exam Review1st Period Exam Review


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Concepts are key to AP Exams

Functions

Linear functions Exponential functions

New from old functions

Logarithmic functions

Trigonometric functions

Powers, Polynomials, and Rational functions

Continuity Limits

A derivative is Continuity lim f(x)=f(x)

ap calculus ab semester i ago-dic 2009 ii

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