Active Calculus - Gvsu - Grand Valley State University

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18 1.5. INTERPRETING THE DERIVATIVE AND ITS UNITSActivity 1.12.A potato is placed in an oven, and the potato’s temperature F (in degrees Fahrenheit) at variouspoints in time is taken and recorded in the following table. Time t is measured in minutes.tF (t)0 7015 180.530 25145 29660 324.575 342.890 354.5(a) Use a central difference to estimate the instantaneous rate of change of the temperatureof the potato at t = 30. Include units on your answer.(b) Use a central difference to estimate the instantaneous rate of change of the temperatureof the potato at t = 60. Include units on your answer.(c) Without doing any calculation, which do you expect to be greater: F ′ (75) or F ′ (90)?Why?(d) Suppose it is given that F (64) = 330.28 and F ′ (64) = 1.341. What are the units on thesetwo quantities? What do you expect the temperature of the potato to be when t = 65?when t = 66? Why?(e) Write a couple of careful sentences that describe the behavior of the temperature of thepotato on the time interval [0, 90], as well as the behavior of the instantaneous rate ofchange of the temperature of the potato on the same time interval.⊳
1.5. INTERPRETING THE DERIVATIVE AND ITS UNITS 19Activity 1.13.A company manufactures rope, and the total cost of producing r feet of rope is C(r) dollars.(a) What does it mean to say that C(2000) = 800?(b) What are the units of C ′ (r)?(c) Suppose that C(2000) = 800 and C ′ (2000) = 0.35. Estimate C(2100), and justify yourestimate by writing at least one sentence that explains your thinking.(d) Which of the following statements do you think is true, and why?• C ′ (2000) < C ′ (3000)• C ′ (2000) = C ′ (3000)• C ′ (2000) > C ′ (3000)(e) Suppose someone claims that C ′ (5000) = −0.1. What would the practical meaning ofthis derivative value tell you about the approximate cost of the next foot of rope? Is thispossible? Why or why not?⊳
Page 1: Active CalculusMatt Boelkins, Lead
Page 5 and 6: Chapter 1Understanding the Derivati
Page 7 and 8: 1.1. HOW DO WE MEASURE VELOCITY? 3A
Page 9 and 10: 1.2. THE NOTION OF LIMIT 51.2 The n
Page 11 and 12: 1.2. THE NOTION OF LIMIT 7Activity
Page 13 and 14: 1.3. THE DERIVATIVE OF A FUNCTION A
Page 15 and 16: 1.3. THE DERIVATIVE OF A FUNCTION A
Page 17 and 18: 1.4. THE DERIVATIVE FUNCTION 131.4
Page 19 and 20: 1.4. THE DERIVATIVE FUNCTION 15rsxx
Page 21: 1.5. INTERPRETING THE DERIVATIVE AN
Page 25 and 26: 1.6. THE SECOND DERIVATIVE 21(a) In
Page 27 and 28: 1.6. THE SECOND DERIVATIVE 23• on
Page 29 and 30: 1.6. THE SECOND DERIVATIVE 25ffxxf
Page 31 and 32: 1.7. LIMITS, CONTINUITY, AND DIFFER
Page 33 and 34: 1.7. LIMITS, CONTINUITY, AND DIFFER
Page 35 and 36: 1.8. THE TANGENT LINE APPROXIMATION
Page 37 and 38: Chapter 2Computing Derivatives2.1 E
Page 39 and 40: 2.1. ELEMENTARY DERIVATIVE RULES 35
Page 41 and 42: 2.2. THE SINE AND COSINE FUNCTIONS
Page 43 and 44: 2.2. THE SINE AND COSINE FUNCTIONS
Page 45 and 46: 2.3. THE PRODUCT AND QUOTIENT RULES
Page 47 and 48: 2.3. THE PRODUCT AND QUOTIENT RULES
Page 49 and 50: 2.4. DERIVATIVES OF OTHER TRIGONOME
Page 51 and 52: 2.4. DERIVATIVES OF OTHER TRIGONOME
Page 53 and 54: 2.5. THE CHAIN RULE 492.5 The chain
Page 55 and 56: 2.5. THE CHAIN RULE 51Activity 2.14
Page 57 and 58: 2.6. DERIVATIVES OF INVERSE FUNCTIO
Page 59 and 60: 2.6. DERIVATIVES OF INVERSE FUNCTIO
Page 61 and 62: 2.7. DERIVATIVES OF FUNCTIONS GIVEN
Page 63 and 64: 2.7. DERIVATIVES OF FUNCTIONS GIVEN
Page 65 and 66: 2.8. USING DERIVATIVES TO EVALUATE
Page 67 and 68: 2.8. USING DERIVATIVES TO EVALUATE
Page 69 and 70: Chapter 3Using Derivatives3.1 Using
Page 71 and 72: 3.1. USING DERIVATIVES TO IDENTIFY
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3.1. USING DERIVATIVES TO IDENTIFY
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3.2. USING DERIVATIVES TO DESCRIBE
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3.2. USING DERIVATIVES TO DESCRIBE
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3.3. GLOBAL OPTIMIZATION 75Activity
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3.3. GLOBAL OPTIMIZATION 77Activity
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3.4. APPLIED OPTIMIZATION 79Activit
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3.4. APPLIED OPTIMIZATION 81Activit
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3.5. RELATED RATES 833.5 Related Ra
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3.5. RELATED RATES 85Activity 3.15.
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3.5. RELATED RATES 87Activity 3.17.
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Chapter 4The Definite Integral4.1 D
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4.1. DETERMINING DISTANCE TRAVELED
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4.1. DETERMINING DISTANCE TRAVELED
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4.2. RIEMANN SUMS 95⊲⊳
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4.2. RIEMANN SUMS 97Activity 4.5.Su
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4.3. THE DEFINITE INTEGRAL 994.3 Th
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4.3. THE DEFINITE INTEGRAL 101Activ
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4.3. THE DEFINITE INTEGRAL 103Activ
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4.4. THE FUNDAMENTAL THEOREM OF CAL
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4.4. THE FUNDAMENTAL THEOREM OF CAL
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Active Calculus - Gvsu - Grand Valley State University

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