explorations chapter 1

Calculus: Concepts and Applications Instructor’s Resource Book Exploration Masters / 51©2005 Key Curriculum Press

The diagram shows a door with an automatic closer. Attime t H 0 s, someone pushes the door. It swings open,slows down, stops, starts closing, then slams shut at timet H 7 s. As the door is in motion, the number of degrees, d,it is from its closed position depends on t.

1. Sketch a reasonable graph of d versus t.

2. Suppose that d is given by the equation

d H 200t • 2Dt

Plot this graph on your grapher. Sketch the resultshere.

3. Make a table of values of d for each second fromt H 0 through t H 10. Round to the nearest 0.1−.

t d

0

1

2

3

4

5

6

7

8

9

10

Door

d

4. At time t H 1 s, does the door appear to be openingor closing? How do you tell?

5. What is the average rate at which the door is movingfor the time interval [1, 1.1]? Based on your answer,does the door seem to be opening or closing at timet H 1? Explain.

6. By finding average rates using the time intervals[1, 1.01], [1, 1.001], and so on, make a conjectureabout the instantaneous rate at which the door ismoving at time t H 1 s.

7. In calculus you will learn by four methods:

• algebraically,

• numerically,

• graphically,

• verbally (talking and writing).

What did you learn as a result of doing thisExploration that you did not know before?

8. Read Section 1-1. What do you notice?

Name: Group Members:

Date: Exploration 1-1a: Instantaneous Rateof Change of a Function

Objective: Explore the instantaneous rate of change of a function.

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For each function:

a. Without using your grapher, sketch the graph on theaxes provided.

b. Confirm by grapher that your sketch is correct.

c. Tell whether the function is increasing, decreasing,or not changing when x H 1. If it is increasing ordecreasing, tell whether the rate of change is slowor fast.

1. f (x) H 3Dx

2. f (x) H sin �π2�x

f (x)

x

5

5

–5

–5

f (x)

x

5

5

–5

–5

3. f (x) H x2 C 2x D 2

4. f (x) H sec x

5. f (x) H �1x�

f (x)

x

5

5

–5

–5

f (x)

x

5

5

–5

–5

f (x)

x

5

5

–5

–5


Date: Exploration 1-2a: Graphs of Familiar Functions

Objective: Recall the graphs of familiar functions, and tell how fast the function ischanging at a particular value of x.


Calculus: Concepts and Applications Instructor’s Resource Book Exploration Masters / 53©2005 Key Curriculum Press

As you drive on the highway you accelerate to 100 ft/s topass a truck. After you have passed, you slow down to amore moderate 60 ft/s. The diagram shows the graph ofyour velocity, v(t), as a function of the number of seconds,t, since you started slowing.

1. What does your velocity seem to be between t H 30and t H 50 s? How far do you travel in the timeinterval [30, 50]?

2. Explain why the answer to Problem 1 can berepresented as the area of a rectangular region ofthe graph. Shade this region.

3. The distance you travel between t H 0 and t H 20 canalso be represented as the area of a region boundedby the (curved) graph. Count the number of squaresin this region. Estimate the area of parts of squaresto the nearest 0.1 square space. For instance, howwould you count this partial square?

100

10 30 500

60

t

v (t )

20 40

4. How many feet does each small square on the graphrepresent? How far, therefore, did you go in the timeinterval [0, 20]?

5. Problems 3 and 4 involve finding the product of thex-value and the y-value for a function where y mayvary with x. Such a product is called the definite

integral of y with respect to x. Based on the units oft and v (t), explain why the definite integral of v (t)with respect to t in Problem 4 has feet for its units.

6. The graph shows the cross-sectional area, y, in in.2,of a football as a function of the distance, x, in in.,from one of its ends. Estimate the definite integralof y with respect to x.

7. What are the units of the definite integral inProblem 6? What, therefore, do you suppose thedefinite integral represents?

8. What did you learn as a result of doing thisExploration that you did not know before?

6 12

x

y

10

20

30

0


Date: Exploration 1-3a: Introduction to Definite Integrals

Objective: Find out what a definite integral is by working a real-world problemthat involves the speed of a car.


54 / Exploration Masters Calculus: Concepts and Applications Instructor’s Resource Book©2005 Key Curriculum Press

Rocket Problem: Ella Vader (Darth’s daughter) is driving inher rocket ship. At time t H 0 min, she fires her rocketengine. The ship speeds up for a while, then slows downas Alderaan’s gravity takes its effect. The graph of hervelocity, v (t), in miles per minute, is shown below.

1. What mathematical concept would you use toestimate the distance Ella goes between t H 0 andt H 8?

2. Estimate the distance in Problem 1 graphically.

3. Ella figures that her velocity is given by

v(t) H t3 D 21t2 C 100t C 110

Plot this graph on your grapher. Does the graphconfirm or refute what Ella figures? Tell how youarrive at your conclusion.

100

200

5 10

t

v(t)

4. Divide the region under the graph from t H 0 tot H 8, which represents the distance, into fourvertical strips of equal width. Draw four trapezoidswhose areas approximate the areas of these stripsand whose parallel sides extend from the x-axis tothe graph. By finding the areas of these trapezoids,estimate the distance Ella goes. Does the answeragree with the answer to Problem 2?

5. The technique in Problem 4 is the trapezoidal rule.

Put a program into your grapher to use this rule. Thefunction equation may be stored as y1. The inputshould be the starting time, the ending time, and thenumber of trapezoids. The output should be thevalue of the definite integral. Test your program byusing it to answer Problem 4.

6. Use the program from Problem 5 to estimate thedefinite integral using 20 trapezoids.

7. The exact value of the definite integral is the limitof the estimates by trapezoids as the width of eachtrapezoid approaches zero. By using the programfrom Problem 5, make a conjecture about the exactvalue of the definite integral.

8. What is the fastest Ella went? At what time was that?

9. Approximately what was Ella’s rate of change ofvelocity when t H 5? Was she speeding up or slowingdown at that time?

10. Based on the equation in Problem 3, there arepositive values of time t at which Ella is stopped.What is the first such time? How did you find youranswer?

11. What did you learn as a result of doing thisExploration that you did not know before?


Date: Exploration 1-4a: Definite Integrals by Trapezoidal Rule

Objective: Estimate the definite integral of a function numerically rather than graphicallyby counting squares.


Solutions for the ExplorationsChapter 1

Exploration 1-1a

1. Such a graph might look like this:

2. d H 200t • 2Dt:

3. t d

0 0.0−1 100.0−2 100.0−3 75.0−4 50.0−5 31.3−6 18.8−7 10.9−8 6.3−9 3.5−

10 2.0−

4. Door appears to be opening. The graph of d shows that dwas less than 100− before t H 1 s and greater than 100−after t H 1 s.

5. Average Rate H (change in value)/(Time)H (200(1.1) • 2D1.1 D 200(1) • 2D1)/(1.1 D 1)≈ (102.6− D 100−)/0.1 sH 26−/s

This number is greater than zero, which shows that the dooris still opening because d is increasing.

90

180

7

t

d

t

d

1 2 3 4 5 6 7 8 9 10–1

90

180

6. Average rate for time interval [1, 1.01] ≈ 30−/s.Average rate for time interval [1, 1.001] ≈ 31−/s.Average rate for time interval [1, 1.0000001] ≈ 31−/s.The average rate seems to be approaching 30.68−/s ≈ 31−/s!

7. Answers will vary.

8. The example in Section 1-1 is the same as this Exploration!

Exploration 1-2a

1. a. f (x) H 3Dx:

b. Grapher confirms sketch.

c. Decreasing slowly

2. a. f (x) H sin �π2�x :


c. Not changing

f (x)

x5–5

5

–5

x

5–5

5

–5

f(x)

Calculus: Concepts and Applications Instructor’s Resource Book Solutions for the Explorations / 169©2005 Key Curriculum Press

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170 / Solutions for the Explorations Calculus: Concepts and Applications Instructor’s Resource Book©2005 Key Curriculum Press

3. a. f (x) H x2 C 2x D 2:


c. Increasing quickly

4. a. f (x) H sec x :


c. Increasing quickly

5. a. f (x) H �1x�:

x

5–5

5

–5

f(x)

x

5–5

5

–5

f(x)

x

5–5

5

–5

f(x)


c. Decreasing slowly

Exploration 1-3a

1. From t H 30 and t H 50 s, the velocity seems to be about 60 ft/s.Distance H rate • time, so the distance traveled is about60 ft/s • (50 s D 30 s) H 1200 ft.

2. The rectangle on the graph with height H 60 and base fromt H 30 to t H 50 has area H base • height H 1200.

3. The sample partial square has about 0.6 square space underthe curve. All the partial squares under the graph from t H 0to t H 20 have area about 28.6 square spaces.

4. Each small space has base representing 5 s and heightrepresenting 10 ft/s. So the area of each small square Hbase • height H 5 s • 10 ft/s represents 50 ft. Therefore, thedistance was about 28.6 • 50 H 1430 ft. (Exact answer is1431.3207... ft.)

5. The x-value is in seconds, and the y-value is in feet/second,so their product (i.e., the definite integral) is inseconds • feet/second H feet.

6. The squares and partial squares under the curve have about45.2 square spaces of area. Each square space has baserepresenting 1 and height representing 5. So one squarespace represents 5 units of definite integral, and the totaldefinite integral is about 226 units. (Exact answer is226.1946....)

7. The x-units are in inches, and the y-units are in squareinches. So their product, the definite integral, is in cubicinches. The definite integral seems to represent the volumeof the football.


Exploration 1-4a

1. The graph shows time on the x-axis and velocity onthe y-axis. A definite integral can be used to findtime • velocity H distance.

2. There are about 60.8 square spaces under the graph. Eachsquare represents about 25 miles, so the total distance isabout 1520 miles.

3. v (t) H t3 D 21t2 C 100t C 110The graph confirms Ella’s conclusion. Tracing to integervalues of t shows the same v (t) values as on the graph.


Calculus: Concepts and Applications Instructor’s Resource Book Solutions for the Explorations / 171©2005 Key Curriculum Press

4. Divide the graph into “trapezoids” of width H 2.

Areas of trapezoids are

�12�(110 C 234) • 2 H 344

�12�(234 C 238) • 2 H 472

�12�(238 C 170) • 2 H 408

�12�(170 C 78) • 2 H 248

Integral ≈ sum H 1472, which is reasonably close to the areafound in Problem 2.

5. See program TRAPRULE in the Programs for GraphingCalculators section of the Instructor’s Resource Book.

6. Using 20 trapezoids, the definite integral is about 1518.08 mi.

7. The approximate definite integral is1519.52 miles for 40 trapezoids1519.9232 miles for 100 trapezoids1519.999232 miles for 1000 trapezoids

The exact definite integral appears to be 1520 mi!

8. According to the graph, Ella’s greatest velocity was about248 mi per minute at about t H 3 min. (Actual maximum was

124 C �934� • ��

437�� mi, at t H 7 D ��

437�� min.)

9. v (4.9) H 213.439, and v (5.1) H 206.441.Velocity changed D6.998 ft/s in 0.2 s.Rate of change ≈ D6.998/0.02 H D34.99.So, Ella was slowing down at about 35 miles/minute perminute.

10. Ella is first stopped at t H 11, because v (11) H 0. She stopsgradually at t H 11 because the graph gently levels off tov (t) H 0 at that point.


100

200

5 10

t

v (t )

110

234 238

170

78

Chapter 2

Exploration 2-1a

1.

2. f (3) H�27 D 633

DC

351 D 15� H �

00�, an indeterminate form.

3. f (x) is very close to 2 when x is close to 3.The limit is 2.

4. x f (x)

2.5 1.25

2.6 1.36

2.7 1.49

2.8 1.64

2.9 1.81

3.0 ?.??

3.1 2.21

3.2 2.44

3.3 2.69

3.4 2.96

3.5 3.25

5. f (x) stays between 1.25 and 3.25 if x is between 2.5 and 3.5(and x ≠ 3).

6. f (x) H H x2 D 4x C 5, x ≠ 3.

Set x2 D 4x C 5 H 1.99 and solve algebraically or numerically.Repeat with x2 D 4x C 5 H 2.01.If 2.99498743... < x < 3.00498756..., then f (x) will be between1.99 and 2.01. (Also if x is between 0.9950... and 1.0050..., butthis is not the interval of interest!)

7. On the left, keep x within 0.0050125... unit of 3.On the right, keep x within 0.0049875... unit of 3.

8. 0.0049875...

9. L H 2, c H 3, ε H 0.01, δ H 0.0049....


x3 D 7x2 C 17x D 15��x D 3

5

5

x

3

2

f(x)


explorations chapter 1

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