Active Calculus - Gvsu - Grand Valley State University

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12 1.3. THE DERIVATIVE OF A FUNCTION AT A POINTActivity 1.9.A rapidly growing city in Arizona has its population P at time t, where t is the number ofdecades after the year 2010, modeled by the formula P (t) = 25000e t/5 . Use this function torespond to the following questions.(a) Sketch an accurate graph of P for t = 0 to t = 5 on the axes provided in Figure 1.6.Label the scale on the axes carefully.ytFigure 1.6: Axes for plotting y = P (t) in Activity 1.9.(b) Compute the average rate of change of P between 2030 and 2050. Include units on youranswer and write one sentence to explain the meaning (in everyday language) of thevalue you found.(c) Use the limit definition to write an expression for the instantaneous rate of change of Pwith respect to time, t, at the instant a = 2. Explain why this limit is difficult to evaluateexactly.(d) Estimate the limit in (c) for the instantaneous rate of change of P at the instant a = 2by using several small h values. Once you have determined an accurate estimate ofP ′ (2), include units on your answer, and write one sentence (using everyday language)to explain the meaning of the value you found.(e) On your graph above, sketch two lines: one whose slope represents the average rate ofchange of P on [2, 4], the other whose slope represents the instantaneous rate of changeof P at the instant a = 2.(f) In a carefully-worded sentence, describe the behavior of P ′ (a) as a increases in value.What does this reflect about the behavior of the given function P ?⊳
1.4. THE DERIVATIVE FUNCTION 131.4 The derivative functionPreview Activity 1.4. Consider the function f(x) = 4x − x 2 .(a) Use the limit definition to compute the following derivative values: f ′ (0), f ′ (1), f ′ (2), andf ′ (3).(b) Observe that the work to find f ′ (a) is the same, regardless of the value of a. Based on yourwork in (a), what do you conjecture is the value of f ′ (4)? How about f ′ (5)? (Note: youshould not use the limit definition of the derivative to find either value.)(c) Conjecture a formula for f ′ (a) that depends only on the value a. That is, in the same waythat we have a formula for f(x) (recall f(x) = 4x − x 2 ), see if you can use your work aboveto guess a formula for f ′ (a) in terms of 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
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Page 19 and 20: 1.4. THE DERIVATIVE FUNCTION 15rsxx
Page 21 and 22: 1.5. INTERPRETING THE DERIVATIVE AN
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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
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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
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Page 61 and 62: 2.7. DERIVATIVES OF FUNCTIONS GIVEN
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Page 65 and 66: 2.8. USING DERIVATIVES TO EVALUATE
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2.8. USING DERIVATIVES TO EVALUATE
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Chapter 3Using Derivatives3.1 Using
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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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