Active Calculus - Gvsu - Grand Valley State University

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16 1.4. THE DERIVATIVE FUNCTIONActivity 1.11.For each of the listed functions, determine a formula for the derivative function. For the firsttwo, determine the formula for the derivative by thinking about the nature of the given functionand its slope at various points; do not use the limit definition. For the latter four, use thelimit definition. Pay careful attention to the function names and independent variables. Itis important to be comfortable with using letters other than f and x. For example, given afunction p(z), we call its derivative p ′ (z).(a) f(x) = 1(b) g(t) = t(c) p(z) = z 2(d) q(s) = s 3(e) F (t) = 1 t(f) G(y) = √ y⊳
1.5. INTERPRETING THE DERIVATIVE AND ITS UNITS 171.5 Interpreting the derivative and its unitsPreview Activity 1.5. One of the longest stretches of straight (and flat) road in North Americacan be found on the Great Plains in the state of North Dakota on state highway 46, which lies justsouth of the interstate highway I-94 and runs through the town of Gackle. A car leaves town (attime t = 0) and heads east on highway 46; its position in miles from Gackle at time t in minutes isgiven by the graph of the function in Figure 1.7. Three important points are labeled on the graph;where the curve looks linear, assume that it is indeed a straight line.100s(104, 106.8)806040(57, 63.8)(68, 63.8)2020 40 60 80 100tFigure 1.7: The graph of y = s(t), the position of the car along highway 46, which tells its distancein miles from Gackle, ND, at time t in minutes.(a) In everyday language, describe the behavior of the car over the provided time interval. Inparticular, discuss what is happening on the time intervals [57, 68] and [68, 104].(b) Find the slope of the line between the points (57, 63.8) and (104, 106.8). What are the unitson this slope? What does the slope represent?(c) Find the average rate of change of the car’s position on the interval [68, 104]. Include unitson your answer.(d) Estimate the instantaneous rate of change of the car’s position at the moment t = 80. Writea sentence to explain your reasoning and the meaning of this value.⊲⊳
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: 1.4. THE DERIVATIVE FUNCTION 15rsxx
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 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
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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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