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A Jet Tour Of Calculus in Five Days

Introduce your students the

main concepts of Calculus

in the first 5 days.

Five Day Lesson James Rahn

Each lesson includes a one or two page lesson

and a two question assignment

t

h

Figure 1

t

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Figure 2

A Jet Tour of Calculus Day 1: Instantaneous Rate of Change

A homemade rocket is fired initially from a platform 400 feet above ground at a velocity of 300 ft/sec. After 25 seconds, the rocket hits the ground. Make a sketch of the height of the rocket as a function oftime in Figure 1.

The rocket’s height can be modeled by the function at any time t, . 216 300 400( )h t t t 0t Make a sketch of the model in Figure 2. How is your graph similar to Figure 1? How is it different?

How long was the rocket in the air? How can you tell?

What was the maximum height reached by the rocket? At what time did it reach that height?

At what height was the rocket at 1 second? Based upon the model, was the rocket on its way up or downat t=1 second? How can you tell?

What was the average rate at which the rocket was traveling during the first second of flight? Explain howyou found this rate? Draw a line segment in Figure 2, whose slope represents this average rate of change.

Approximate the rate of change of the height of the rocket at 1 second, by first calculating the height ofthe rocket at various times:

t 0.9 0.99 0.999 0.9999 0.99999 0.999999 1.0

h

Use these heights to calculate the average rate of change of the height over the following intervals? Whatis the unit of measure on this rate of change?

Time Interval [0.9,1] [0.99,1] [0.999,1] [0.9999,1] [0.99999,1] [0.999999,1]

Rate of Change

Rahn (c) 2014

What do you notice about the rates of change of the height as the width of the intervals are decreased?

Use the table above to approximate the rate of change of the height of rocket at 1 second.

When is the rocket at the same height it was at 1 second?

Create a table of values to approximate the rate of change of the height at time when the rocket is at thesame height it was at 1 second?

Time Interval [17.749,17.75] [17.7499,17.75] [17.74999,17.75]

Rate of Change

Time Interval [17.749999,17.75] [17.7499999,17.75] [17.74999999,17.75]

Rate of Change

Use the table above to approximate the rate of change of the height of rocket at the moment it reachedthe same height as it did at 1 second.

What do you notice about the average rate of change of the height at this time?

At what time did the rocket reach its maximum height? What do you believe the rate of change of theheight was at that time? Complete a chart to estimate the instantaneous rate of change of the height atthat time.

Time Interval

Rate of Change

Time Interval

Rate of Change

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h


ANSWERS

A homemade rocket is fired initially from a platform 400 feet above ground at a velocity of 300 ft/sec. After 25 seconds, the rocket hits the ground. Make a sketch of the height of the rocket as a function oftime in Figure 1.

Figure 1 sketches should be close to the model graphed in Figure 2.

The rocket’s height is modeled by the function at any time t, . 216 300 400( )h t t t 0t Make a sketch of the model in Figure 2. How is your graph similar to Figure 1? How is it different? How long was the rocket in the air? How can you tell? They should both resemble parabolas. They maynot begin and end at the same locations. They should both illustrate a maximum height at approximatehalfway through the trip. What was the maximum height reached by the rocket? At what time did it reach that height? Themaximum height is 1806.25 feet at 9.375 minutes.At what height was the rocket at 1 second? Based upon the model, was the rocket on its way up or downat t=1 second? How can you tell? The rocket is at a height of 684 feet. The rocket is on its way upbecause the heights are increasing when time is near t = 1 second. What was the average rate at which the rocket was traveling during the first second of flight? Explain howyou found this rate? Draw a line segment in Figure 2, whose slope represents this average rate of change.

684 400

2841 sec

ft

Approximate the rate of change of the height of the rocket at 1 second, by first calculating the height ofthe rocket at various times:

t 0.9 0.99 0.999 0.9999 0.99999 0.999999 1.0

h 657.04 681.318 683.731 683.973 683.997 683.9997 684

What was the average rate of change of the height over the following intervals? What is the unit ofmeasure on this rate of change?

Rahn (c) 2014

Time Interval [0.9,1] [0.99,1] [0.999,1] [0.9999,1] [0.99999,1] [0.999999,1]

Rate of Change 269.6 268.16 268.016 268.0016 268.00016 268.000016

What do you notice about the rates of change of the height as the width of the intervals are decreased? The average rates of change appear to be approaching the number 268 ft/sec.

Use the table above to approximate the rate of change of the height of rocket at 1 second.It appears that the rocket is traveling at 268 ft/sec when at the time 1 second.When is the rocket at the same height it was at 1 second? The rocket is at the height of 684 feet whent=17.75 seconds.

Create a table of values to approximate the rate of change of the height at time when the rocket is at thesame height it was at 1 second?

Time Interval [17.749,17.75] [17.7499,17.75] [17.74999,17.75]

Rate of Change -267.984 -267.9984 -267.99984

Time Interval [17.749999,17.75] [17.7499999,17.75] [17.74999999,17.75]

Rate of Change -267.999984 -267.9999984 -267.99999984

Use the table above to approximate the rate of change of the height of rocket at the moment it reachedthe same height as it did at 1 second. It appears that the rate of change of the height is approaching -268ft/sec when the rocket is at 684 ft from the ground on its return trip. This is the opposite of the rate attime 1 second because the rocket is approaching the ground rather than moving away from the ground.

What do you notice about the average rate of change of the height at this time? It is opposite the rate attime 1 second.

At what time did the rocket reach its maximum height? What do you believe the rate of change of theheight was at that time? Complete a chart to estimate the instantaneous rate of change of the height atthat time. The rocket reaches it maximum height of 1806.25 feet at 9.375 seconds. Intervals may vary.

Time Interval [9.3749,9.375] [9.37499,9.375] [9.374999,9.375]

Rate of Change 0.0016 0.00016 0.000016

Time Interval [9.3749999,9.375] [9.37499999,9.375] [9.374999999,9.375]

Rate of Change 0.0000016 0.00000016 0.000000016

The rocket’s rate of change of height at 9.375 seconds appears to be approaching 0 ft/sec.

Rahn (c) 2014


Assignment

1. A leaf is dropped from the observation deck of the city tower. The distance the leaf falls is given by

the formula , where g is the force of gravity working on the leaf or g= 9.8 meter/sec221

2d gt

and t is measured in seconds. Estimate the rate of change of the distance with respect to time theleaf is falling at time 1 second and 2 seconds. Show work that supports your answer. Explainwhy the rate of change of the distance with respect to time at 2 seconds makes sense based uponthe rate of change of distance with respect to time at 1 second.

2. A cylindrical shaped oil spill has a constant height of .25 feet and a radius that is changing at the

rate of 6t +1 feet every hour. The volume of the oil spill is given by where t is2.25 (6 1)V t measured in hours. Estimate the rate of change of the volume when t is 1 second.

Rahn (c) 2014


Assignment Answers

1. A leaf is dropped from the observation deck of the city tower. The distance the leaf falls is given by

the formula , where g is the force of gravity working on the leaf or g= 9.8 meter/sec221

2d gt

and t is measured in seconds. Estimate the rate of change of the distance with respect to time theleaf is falling at time 1 second and 2 seconds. Show work that supports your answer. Explainwhy the rate of change of the distance with respect to time at 2 seconds makes sense based uponthe rate of change of distance with respect to time at 1 second.

Time Interval [0.9,1] [0.99,1] [0.999,1] [0.9999,1] [0.99999,1] [0.999999,1]

Rate of Change 9.31 9.751 9.795 9.7995 9.7999 9.7999

Time Interval [1.9,2] [1.99,2] [1.999,2] [1.9999,2] [1.99999,2] [1.999999,2]

Rate of Change 19.11 19.55 19.595 19.59951 19.599951 19.5999951

The rate of change of the distance with respect to time should increase as the leaf falls. Therefore, therate of change of the distance should be larger at t=2 than t=1. The rate of change of distance withrespect to time is about 9.8 meters per second at time t=1 second and 19.6 meters per second at time t=2seconds.

2. A cylindrical shaped oil spill has a constant height of .25 feet and a radius that is changing at the

rate of 6t +1 feet every hour. The volume of the oil spill is given by where t is2.25 (6 1)V t measured in hours. Estimate the rate of change of the volume when t is 1 second.

Time Interval [0.9,1] [0.99,1] [0.999,1] [0.9999,1] [0.99999,1] [0.999999,1]

Rate of Change 63.146 65.690 65.945 65.970 65.973 65.973

The rate of change of the volume with respect to time at time t = 1 second is about 65.97 feet per second.

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x

y

1xy e

x

y

2( 2) 1y x

x

y

3( 1) 1y x

x

y

4( 2) 2y x

A Jet Tour of Calculus Day 2: Behavior of Functions

Make a sketch of each function in the given window. At x = 1 draw a tangent line that approximates thesteepness of the function at x = 1. Approximate the slope of the tangent line. Describe how this tangentline describes the behavior of the graph at x = 1.

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x

y

2( 2) 1y x The tangent line to the graph at x=1appears to have a slope of -2. Thenegative slope on the tangent line

indicates that the graph is decreasingat x = 1.

x

y

1xy e The tangent line to the graph at x = 1

appears to have a slope of 1. Thepositive slope on the tangent line

indicates that the graph is increasingat x = 1.

x

y

4( 2) 2y x The tangent to graph at x = 1 appears

to have a slope of about 4. Thepositive slope on the tangent line

indicates the graph is increasing at x= 1.

x

y

3( 1) 1y x The tangent line to the graph at x = 1appears to have a slope of zero. The

zero slope on the tangent lineindicates that the graph has stopped

decreasing at x = 1.


ANSWERS

Make a sketch of each function in the given window. At x = 1 draw a tangent line that approximates thesteepness of the function at x = 1. Approximate the slope of the tangent line. Describe how this tangentline describes the behavior of the graph at x = 1.

Rahn (c) 2014

x

y

x

y


Assignment1. Create a sketch of a function f that has a tangent line whose slope is positive at x = 1 and negative at x= 3. What is the behavior of the function at each of these two x values?

2. Create a sketch of a function g that has a tangent line whose slope is zero at x = 1 and negative at x =2. What is the behavior of the function at each of these two x values?

Rahn (c) 2014

x

y

Answers will vary, but the graphabove has a positive slope at x = 1and therefore, f is increasing. Thegraph has a negative slope at x = 3and therefore, is f is decreasing.

x

y

Answers will vary, but the graphabove has a zero slope at x = 1 andtherefore, f is leveling off. The graph

has a negative slope at x = 3 andtherefore, is f is decreasing.


Assignment Answers

1. Create a sketch of a function f that has a tangent line whose slope is positive at x = 1 and negative at x= 3. What is the behavior of the function at each of these two x values?

2. Create a sketch of a function g that has a tangent line whose slope is zero at x = 1 and negative at x =2. What is the behavior of the function at each of these two x values?

Rahn (c) 2014

minutes

feet per minute

minutes

feet per minute

A Jet Tour of Calculus Day 3: What Can Area Represent?

As you pull out on the highway on your road bike you gradually increase your speed according the graphbelow. Then you notice your speedometer approaching 465 ft per minute so you tap hand brake to slowdown your speed to a constant rate of 465 feet per minute.

Notice that the portion of the velocity graph between time t=60 minutes and t=100 minutes is constant. The distance traveled during this time can be represented by what geometric shape? What are the unitsof measure for the height of region? What are the units of measure for the length (or base) of this region?Using correct units, what is the area of this region? Explain how you determined this unit of measure.

Each rectangular region on the graph represent what distance? Explain how you found your answer.

If the rectangle at the right has the same dimensions at those in the graph above, how many feetdoes the shaded part of the rectangle represent?

Find an estimate for the distance traveled by the cyclist from t=0 to t = 60 minutes.

Find out how far your bicycle traveled in the first 100 minutes of the trip.

The distance you traveled on your road bike is represented by the bounded area under the velocity graphand above the time axis and the two vertical lines t = 0 and t = 100. This area is called the definiteintegral of the velocity from time t=0 to t= 100 minutes.

You have just found a geometric method to find an approximate value for the definite integral of thevelocity from t=0 to t= 60 minutes. It involved both estimating bounded area and using geometric areaformulas.

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Centimeters

Area (sq. cm.)

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.

4

8

12

16

20

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.

4

8

12

16

20

The picture below illustrates a right circular cone sitting on its circular base. Suppose we slice the coneparallel to the base at a point x centimeters from the base. What shape is each cross section?

The graph below shows the area of each cross-section as a function of height where we took the crosssection.

What is the largest cross-sectional area?

What is the cross-sectional area created at a point 1.75 cm from the base?

At what distance from the base was a 4 square centimeter circle cut?

What does each rectangle in this graph represent? Explain your answer.

The definite integral of the area from time x=0 to x=5 centimeters will determine the volume of thecone.

Estimate the definite integral of y, the area, with respect to x for 0#x#5. Show work that leads to youranswer.

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minutes

feet per minute


ANSWERS

As you pull out on the highway on your road bike you gradually increase your speed according the graphbelow. Then you notice your speedometer approaching 465 ft per minute so you tap hand brake to slowdown your speed to a constant rate of 465 feet per minute.

Notice that the portion of the velocity graph between time t=60 minutes and t=100 minutes is constant. The distance traveled during this time can be represented by what geometric shape? What are the unitsof measure for the height of region? What are the units of measure for the length (or base) of this region?Using correct units, what is the area of this region? Explain how you determined this unit of measure. Theregion is a rectangle. The units for the height are feet per minute. The units for the length are minutes.

Therefore the units for each rectangle are .feet

•minute=feetminute

Each rectangular region on the graph represent what distance? Explain how you found your

answer. Each region represent .feet

100 •10minutes = 1000 feetminute

If the rectangle at the right has the same dimensions at those in the graph above, how many feet does theshaded part of the rectangle represent? It appears to represent about 0.4 x 1000 feet or 400 feet.

Find an estimate for the distance traveled by the cyclist from t=0 to t = 60 minutes. about 19,000 feet

Find out how far your bicycle traveled in the first 100 minutes of the trip. About 37,600 feet

The distance you traveled on your road bike is represented by the bounded area under the velocity graphand above the time axis and the two vertical lines t = 0 and t = 100. This area is called the definiteintegral of the velocity from time t=0 to t= 100 minutes.

You have just found a geometric method to find an approximate value for the definite integral of thevelocity from t=0 to t= 60 minutes. It involved both estimating bounded area and using geometric areaformulas.

Rahn (c) 2014

Centimeters

Area (sq. cm.)

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.

4

8

12

16

20

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.

4

8

12

16

20

The picture below illustrates a right circular cone sitting on its circular base. Suppose we slice the coneparallel to the base at a point x centimeters from the base. What shape is each cross section? Each shapeis a circle.

The graph below shows the area of each cross-section as a function of height where we took the crosssection.

What is the largest cross-sectional area? The largest area is about 20 square centimeters.

What is the cross-sectional area created at a point 1.75 cm from the base? About 8 square centimeters.

At what distance from the base was a 4 square centimeter circle cut? About 2.75 centimeters.

What does each rectangle in this graph represent? Explain your answer. Each rectangle represents.2squarecentimeters•0.25centimeters = 0.50 cubic centimeters

The definite integral of the area from time x=0 to x=5 centimeters will determine the volume of thecone.

Estimate the definite integral of y, the area, with respect to x for 0#x#5. Show work that leads to youranswer. About 32.5 cubic centimeters (Each square in the graph represents 0.5 cubic centimeters.)

Rahn (c) 2014

inches

square inches

Figure 1Area of each cross-section of a hemisphere

for 0 5x

min

in/min

Figure 2Velocity of a caterpillar


Assignment1. A hemisphere is placed on the table so it is sitting on a circular base. Each cross-section, parallel to

the base, is also a circle. The graph in Figure 1 represents the area of each cross-section at adistance x units from the base. The definite integral from 0 to 5 represents the volume of thehemisphere by calculating the area bounded by the graph. What does each rectangle in the graphrepresent? Explain your answer. Estimate the volume of the hemisphere. Include units on youranswer.

2. The velocity of a caterpillar, traveling along a branch, is given in Figure 2. A definite integral from t= 0 to t = 5 would represent the distance the caterpillar travels during the first five seconds. Whatdoes each square in the graph represent. Support your reasoning. Estimate the distance thecaterpillar travels during the five seconds. If the velocity of the caterpillar is given by

, estimate the rate of change of the velocity (acceleration) of the caterpillar at 231( ) 32

2v t t

minutes. Draw a tangent line that represents thisrate of change.

Rahn (c) 2014

inches

square inches

Figure 1Area of each cross-section of a hemisphere

for 0 5x

min

in/min

Figure 2Velocity of a caterpillar


Assignment Answers

1. A hemisphere is placed on the table so it is sitting on a circular base. Each cross-section, parallel tothe base, is also a circle. The graph in Figure 1 represents the area of each cross-section at adistance x units from the base. The definite integral from 0 to 5 represents the volume of thehemisphere by calculating the area bounded by the graph. What does each rectangle in the graphrepresent? Explain your answer. Estimate the volume of the hemisphere. Include units on youranswer.

Each rectangle in the graph represents (10 square inches)(0.25 inches) or 2.5 cubic inches. Thehemisphere has a volume of about 260 cubic inches.

2. The velocity of a caterpillar, traveling along a branch, is given in Figure 2. A definite integral from t= 0 to t = 5 would represent the distance the caterpillar travels during the first five seconds. Whatdoes each square in the graph represent. Support your reasoning. Estimate the distance thecaterpillar travels during the five seconds. If the velocity of the caterpillar is given by

, estimate the rate of change of the velocity (acceleration) of the caterpillar at 231( ) 32

2v t t

minutes. Draw a tangent line that represents this rate of change. Estimate the rate of change ofthe velocity (acceleration) of the caterpillar at 2 minutes. Show all work that leads to your answer. Draw a tangent line that represents this rate of change.

Each square in the graph represents (4 in/min)(0.5 min) or 2inches. At the end of 5 minutes the caterpillar will have travelabout 110 inches.

Time Interval [1.9,2] [1.99,2] [1.999,2]

Rate of Change -5.705 -5.97005 -5.9970005

Time Interval [1.9999,2] [1.99999,2] [1.999999,2]

Rate of Change -5.9997 -5.99997 -5.999997

The rate of change of the velocity at 2 is about -6 in/min/min.

Rahn (c) 2014

day

1000 gallons/day

Figure 1

A Jet Tour of Calculus Day 4: Determining a Definite Integral with Formulas

1. Water is being pumped into a large storage tank at a rate, thousands of3( ) ( 2) 12R t x gallons/day. Draw a sketch of R(t) in Figure 1 for time . The definite integral of R(t)0 4t days from t = 0 to t = 4 represents the thousands of gallons pumped into the tank during the four days.

What does the area of each rectangular region represent in this problem?

Draw five vertical line segments to separate the time interval into four equal regions. Use0 4t days these five line segments to create rectangles that will approximate the area under the graph. Find anestimate for the number of gallons pumped into the tank during the four days.

Draw four additional vertical line segments to separate the time interval into eight equal0 4t days regions. Find a second estimate for the number of gallons pumped into the tank during the four days.

Explain how increasing the number of line segments will change your estimate for the number of gallonsbeing pumped into the tank.

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minutes

mm/minutes

2. The velocity of an inch worm is given by the function . Draw a sketch of this function2( ) 8v t x x in figure 2.

What does the area of each rectangle in the graph represent?

Draw vertical line segments to separate the time interval into four equal regions. Use0 8 secondst these line segments, and points along the graph of v(t) to create triangles or trapezoids that willapproximate the area under the graph. Find an estimate for the distance traveled by the inch worm in 8seconds.

Draw additional vertical line segments to separate the time interval into more equal0 8 secondst regions. Use these new intervals and points along the graph of v(t) to find a second estimate for thedistance traveled by the inch worm in 8 seconds.

Explain how additional number of line segments will change your estimate for the distance traveled by theinch worm in 8 seconds.

Rahn (c) 2014

Figure 1


ANSWERS

1. Water is being pumped into a large storage tank at a rate, thousands of3( ) ( 2) 12R t x gallons/day. Draw a sketch of R(t) in Figure 1 for time . The definite integral of R(t)0 4t days from t = 0 to t = 4 represents the thousands of gallons pumped into the tank during the four days.

What does the area of each rectangular region represent in this problem? Each rectangle represent 2000gallons because the dimensions are 4000 gallons/day by 0.5 days.

Draw five vertical line segments to separate the time interval into four equal regions. Use0 4t days these five line segments to create rectangles that will approximate the area under the graph. Find anestimate for the number of gallons pumped into the tank during the four days. Answers will vary. The figure shows four rectangles, whose height is drawn at the left hand endpoint ofeach interval. The numbers in each rectangle are in 1000's of gallons. These rectangles have areas thatadd up to 40,000 gallons. Students may use other types of rectangles andhave answers between 40,000 and 56,000 gallons.

Draw four additional vertical line segments to separate the time interval into eight equal regions. Find a second estimate for the0 4t days

number of gallons pumped into the tank during the four days. Answers willvary. As four additional line segments are added the area can rangebetween 44,000 and 56,000 gallons.

Explain how increasing the number of line segments will change your estimatefor the number of gallons being pumped into the tank.

As the number of rectangles are increased the answer for the number of gallons pumped into the tankapproach 48,000 gallons, the actual area.

Rahn (c) 2014

Figure 2

2. The velocity of an inch worm is given by the function . Draw a sketch of this function2( ) 8v t x x in figure 2.

What does the area of each rectangle in the graph represent? The area represents 2 mm because thedimensions are 2 mm/minute by 1 minute.

Draw vertical line segments to separate the time interval into four equal regions. Use0 8 secondst these line segments, and points along the graph of v(t) to create triangles or trapezoids that willapproximate the area under the graph. Find an estimate for the distance traveled by the inch worm in 8minutes. The area of the two triangles and two trapezoids adds up to 80 mm. This is how far the inchworm crawls in 8 minutes.

Draw additional vertical line segments to separate the time interval into more equal regions. Use these new intervals and0 8 secondst

points along the graph of v(t) to find a second estimate for the distancetraveled by the inch worm in 8 seconds. Using eight trapezoids and/ortriangles the area will be 84 mm.

Explain how additional number of line segments will change your estimate forthe distance traveled by the inch worm in 8 seconds. Will these estimatesbe an over or under estimate for the distance?

As additional trapezoids are added the area of all the triangles and trapezoids will approach the exact areabounded under the velocity graph because the slanted sides better approximate the curvature of thevelocity graph. The area will approach the value of 85 1/3 mm. These estimates will be under estimatessince the graph of v(t) is concave down. The straight segments will be below the graph

Rahn (c) 2014

x

y


Assignment1. A region R is defined by the graph of and the x-axis is graphed in Figure 1. What2( ) 81f x x

does the area of each of the squares on the graph represent?

Estimate the bounded area by thinking about the full and partial squares contained in the region R.

Use 6 rectangles to approximate the area of region R.

Approximate area of the bounded region R using 6 trapezoids and/or triangles.

Explain why the three estimates differ from each other. Explain why one of the answer is a betterapproximation.

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inches

square inches

Figure 2

2. A solid is sliced into cross sections whose area, A(x), is represented by the graph in Figure 2. Thedefinite integral of A(x) from x=0 to x = 4 will find the volume of the solid.

What does the area of each of the rectangles on the graph represent in the context of this problem?

Use two different estimation techniques to approximate volume of the solid. Compare the twoestimates to the actual volume of the solid.

Rahn (c) 2014

x

y


Assignment Answers

1. A region R is defined by the graph of and the x-axis is graphed in Figure 1. What2( ) 81f x x does the area of each of the squares on the graph represent? Each rectangle represents 1 squareunit.

Estimate the bounded area by thinking about the full and partial squares contained in the region R. The area of the bounded region is about 108 full and/or partial squares or about 108 cubic units.

Use 6 rectangles to approximate the area of region R. Using 6 rectangles the area can be between92 square units and 145 square units.

Approximate area of the bounded region R using 6 trapezoids and/or triangles. Using 6 trapezoidsand/or triangles the area will be about 118.16 square units.

Explain why the three estimates differ from each other. Explain why one of the answer is a betterapproximation. The estimate using trapezoids and/or triangles will be closer to the actual areasince the shapes fit closer to the actual shape, but less than the actual area. The actual region is a

semicircle with a radius of 9 so its area is or 127.234 square units. The rectangles extend812

over the graph or under the graph, but the trapezoids fit tighter to the graph and all remain underthe graph.

Rahn (c) 2014

inches

square inches

Figure 2

2. A solid is sliced into cross sections whose area, A(x), is represented by the graph in Figure 2. Thedefinite integral of A(x) from x=0 to x = 4 will find the volume of the solid.

What does the area of each of the rectangles on the graph represent in the context of this problem? Each rectangle represent 0.025 cubic inches of volume since the dimension are 0.1 square inchesby 0.25 inches.

Use two different estimation techniques to approximate volume of the solid. Compare the twoestimates to the actual volume of the solid.

Using 6 rectangles the approximate volume of the solid will be between 1.59 cubic inches and 2.647cubic inches. Using 6 trapezoids the approximate volume is 2.123 cubic inches. The estimateusing 6 trapezoids should be a better approximation than some of the rectangular approximationsince the trapezoids fit the shape better than the rectangles. Rectangles that fit below the graphwill give an underestimate. Rectangles that fit above the graph will be an overestimate. Trapezoidswill give an overestimater.

Rahn (c) 2014

x

y

Figure 1

Graph of f(x) in a smallerwindow

A Jet Tour of Calculus Day 5: What is a Limit?

Consider the graph of the function . Make a sketch of this function in Figure 1.22 5 2

( )2

x x

f xx

How does the graph differ from what youexpected to see?

What does your graph indicate about the value off(2)? Why is this? Algebraically calculate f(2). Why is there no value for f(2)?

Let’s investigate how large the hole is in the function at x = 2 by looking at function values when x is closeto 2.

Complete this table of values for f(x) and thensketch a new graph of f(x) in a window with thosedimensions:

x 1.9

f(x)

x 2.1

f(x)

What does this new graph illustrate about f(x)when you are even closer to x = 2?

Complete this new table of values for f(x) and then sketch a new graph of f(x) in a window with thosedimensions. What does this new graph illustrate about f(x) when you are even closer to x = 2?

x 1.99

f(x)

x 2.01

f(x)

Rahn (c) 2014

Graph of g(x) in a smallerwindow

Graph of g(x) in a smallerwindow

What value is f(x) getting close to as x approaches 2? Explain why you chose this value.

We call this function value the LIMIT of the function f(x) as x approaches 2. It is the number that allthe y values stay close to as x is very close to 2.

What is the limit of the function f(x) as x approaches 2?

Notice that as we got closer to x=2 the function values stayed very closed to your limit value. From thelast set of table values and the graph we can notice that all the function values for f(x) were within 0.02 of-3 when the x values were within 0.01 of x=2.

Consider a new function . Consider the limit of this function as x approaches 3. 2 6( )2 6

x xg xx

Create table values for g(x) near x = 3. Complete the table and build a window to represent the numbersin the shaded column:

x 2.9 2.99

g(x)

x 3.1 3.01

g(x)

As x got closer to x=3 the function values appears to be approaching what LIMIT value? Fill in thefollowing statement based on your table values and graphing window:

All the function values for f(x) were within ______ of g(x)=_____ when the x values were within 0.01 ofx=3. Predict how close all function values for f(x) will be from the limit of g(x)=_____ when x values are keptwithin 0.001 of x = 3. Show this on a new graphing window:

Rahn (c) 2014

x

y

Figure 1

Graph of f(x) in a smallerwindow


ANSWERSConsider the graph of the function . Make a sketch of this function in Figure 1.

22 5 2( )

2

x xg x

x

How does the graph diffe r from what youexpected to see? This graph is a linear functionwith a hole in it. Some students might expect tosee an asymptotic function.

What does your graph indicate about the value off(2)? Why is this? Algebraically calculate f(2). Why is there no value for f(2)? There is no valueat x = 2. . There is a(1 2 )( 2)( ) (1 2 ), x 2

( 2)x xf x x

x

hole at x = 2. There is no value at x = 2 since

is not defined at x =2.

22

xx

Let’s investigate how large the hole is in the function at x = 2 by looking at function values when x is closeto 2.

Complete this table of values for f(x) and then sketch a new graph of f(x) in a window with thosedimensions:

x 1.9

f(x) -2.8

x 2.1

f(x) -3.2

What does this new graph illustrate about f(x)when you are even closer to x = 2? This graphshows that all function values are staying veryclose to f(x)=-3.

Complete this new table of values for f(x) and then sketch a new graph of f(x) in a window with thosedimensions. What does this new graph illustrate about f(x) when you are even closer to x = 2? Thefunction values are all staying very close to -3.

x 1.99

f(x) -2.98

x 2.01

f(x) -3.02

Rahn (c) 2014

The graph of g(x) in a smallerwindow

The graph of f(x) in a smallerwindow.

What value is f(x) getting close to as x approaches 2? Explain why you chose this value. f(x) is gettingvery close to -3 as x approaches 2. This is visible from the table values and the graph.

We call this function value the LIMIT of the function f(x) as x approaches 2. It is the number that allthe y values stay close to as x is very close to 2.

What is the limit of the function f(x) as x approaches 2? The limit of f(x) as x approaches 2 is -3.

Notice that as we got closer to x=2 the function values stayed very closed to your limit value. From thelast set of table values and the graph we can notice that all the function values for f(x) were within 0.02 of-3 when the x values were within 0.01 of x=2.

Consider a new function . Consider the limit of this function as x approaches 3. 2 6( )2 6

x xg xx

Create table values for g(x) near x = 3. Complete the table and build a window to represent the numbersin the shaded column:

x 2.9 2.99

g(x) 2.45 2.495

x 3.1 3.01

g(x) 2.55 2.505

As x got closer to x=3 the function values appears to be approaching what LIMIT value? The functionappears to be approaching 2.5. Fill in the following statement based on your table values and graphingwindow:

All the function values for f(x) were within 0.005 of g(x)=2.5 when the x values were within 0.01 of x=3. Predict how close all function values for f(x) will be from the limit of g(x)=2.5 when x values are keptwithin 0.001 of x = 3. f(x) should be within 0.0005 of 2.5. Show this on a new graphing window:

Rahn (c) 2014


Assignment

1. Consider the function .

23 11 6( )( 3)

x xh xx

What is the value of h(0)? Support your answer algebraically.

Determine the limit of this function as h approaches zero by considering the following table values.

x 2.9 2.99 2.999 2.9999 2.99999

h(x)

x 3.1 3.01 3.001 3.0001 3.00001

h(x)

If you chose a number in between , complete the following statement about all function 2.95 3.05xvalues: . ___ ( ) _____h x

Show this visually on a graph of h(x) in a small window around (3,7)


2 4, x 3( ) 1 1 , 3

2 2

xk x

x x

Notice this function is not defined at x = 3.

Determine if it appears that a limit as x approaches 3 by considering the following table values.

x 2.9 2.99 2.999 2.9999 2.99999

k(x)

x 3.1 3.01 3.001 3.0001 3.00001

k(x)

Show visually through a graph what is happening when and . 2.9 3.1x .9 ( ) 2.1k x

Rahn (c) 2014


Assignment Answers


23 11 6( )( 3)

x xh xx

What is the value of h(0)? Support your answer algebraically. There is no value for h(0)

since

23 11 6(3 2)( 3)() 3 2,x3( 3) ( 3)

x x x xhx xx x

Determine the limit of this function as h approaches zero by considering the following table values.

x 2.9 2.99 2.999 2.9999 2.99999

h(x) 6.7 6.97 6.997 6.9997 6.99997

x 3.1 3.01 3.001 3.0001 3.00001

h(x) 7.3 7.03 7.003 7.0003 7.00003

If you chose a number in between , complete the following 2.95 3.05xstatement about all function values: . ( )6.85 7.15h x

Show this visually on a graph of h(x) in a small window around (3,7)


2 4, x 3( ) 1 1 , 3

2 2

xk x

x x

Notice this function is not defined at x = 3.

Determine if it appears that a limit as x approaches 3 by considering the following table values.

x 2.9 2.99 2.999 2.9999 2.99999

k(x) 1.8 1.98 1.998 1.9998 1.99998

x 3.1 3.01 3.001 3.0001 3.00001

k(x) 1.05 1.005 1.0005 1.00005 1.000005

It appears that the function k approaches a limit of 2 from the left side of 3and 1 from the right side of 3. Therefore, a limit does not exist at x = 3.

Show visually through a graph what is happening when and 2.9 3.1x. .9 ( ) 2.1k x

Rahn (c) 2014

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