Active Calculus - Gvsu - Grand Valley State University

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26 1.6. THE SECOND DERIVATIVEtF (t)0 7015 180.530 25145 29660 324.575 342.890 354.5tF ′ (t)0 NA15 6.0330 3.8545 2.4560 1.5675 1.0090 NA(a) What are the units on the values of F ′ (t)?(b) Use a central difference to estimate the value of F ′′ (30).(c) What is the meaning of the value of F ′′ (30) that you have computed in (c) in terms ofthe potato’s temperature? Write several careful sentences that discuss, with appropriateunits, the values of F (30), F ′ (30), and F ′′ (30), and explain the overall behavior of thepotato’s temperature at this point in time.(d) Overall, is the potato’s temperature increasing at an increasing rate, increasing at a constantrate, or increasing at a decreasing rate? Why?⊳
1.7. LIMITS, CONTINUITY, AND DIFFERENTIABILITY 271.7 Limits, Continuity, and DifferentiabilityPreview Activity 1.7. A function f defined on −4 < x < 4 is given by the graph in Figure 1.12.Use the graph to answer each of the following questions. Note: to the right of x = 2, the graph off is exhibiting infinite oscillatory behavior similar to the function sin( π x) that we encountered inFigure ?? in Example ??.f321-3 -2 -1-11 2 3-2-3Figure 1.12: The graph of y = f(x).(a) For each of the values a = −3, −2, −1, 0, 1, 2, 3, determine whether or not lim f(x) exists.x→aIf the function has a limit L at a given point, state the value of the limit using the notationlim f(x) = L. If the function does not have a limit at a given point, write a sentence tox→aexplain why.(b) For each of the values of a from part (a) where f has a limit, determine the value of f(a)at each such point. In addition, for each such a value, does f(a) have the same value aslimx→a f(x)?(c) For each of the values a = −3, −2, −1, 0, 1, 2, 3, determine whether or not f ′ (a) exists. Inparticular, based on the given graph, ask yourself if it is reasonable to say that f has atangent line at (a, f(a)) for each of the given a-values. If so, visually estimate the slope ofthe tangent line to find the value of f ′ (a).⊲⊳
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 and 22: 1.5. INTERPRETING THE DERIVATIVE AN
Page 23 and 24: 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: 1.6. THE SECOND DERIVATIVE 25ffxxf
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
Page 73 and 74: 3.1. USING DERIVATIVES TO IDENTIFY
Page 75 and 76: 3.2. USING DERIVATIVES TO DESCRIBE
Page 77 and 78: 3.2. USING DERIVATIVES TO DESCRIBE
Page 79 and 80: 3.3. GLOBAL OPTIMIZATION 75Activity
Page 81 and 82:
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
Page 95 and 96:
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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