Active Calculus - Gvsu - Grand Valley State University

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28 1.7. LIMITS, CONTINUITY, AND DIFFERENTIABILITYActivity 1.18.Consider a function that is piecewise-defined according to the formula⎪⎨f(x) =⎧3(x + 2) + 2 for −3 < x < −22323(x + 2) + 1 for −2 ≤ x < −1(x + 2) + 1 for −1 < x < 12 for x = 1⎪⎩4 − x for x > 1Use the given formula to answer the following questions.321-2 -1 1 2-1Figure 1.13: Axes for plotting the function y = f(x) in Activity 1.18.(a) For each of the values a = −2, −1, 0, 1, 2, compute f(a).(b) For each of the values a = −2, −1, 0, 1, 2, determine lim f(x) and lim f(x).x→a− x→a +(c) For each of the values a = −2, −1, 0, 1, 2, determine limx→af(x). If the limit fails to exist,explain why by discussing the left- and right-hand limits at the relevant a-value.(d) For which values of a is the following statement true?lim f(x) ≠ f(a)x→a(e) On the axes provided in Figure 1.13, sketch an accurate, labeled graph of y = f(x). Besure to carefully use open circles (◦) and filled circles (•) to represent key points on thegraph, as dictated by the piecewise formula.⊳
1.7. LIMITS, CONTINUITY, AND DIFFERENTIABILITY 29Activity 1.19.This activity builds on your work in Preview Activity 1.7, using the same function f as givenby the graph that is repeated in Figure 1.14f321-3 -2 -1-11 2 3-2-3Figure 1.14: The graph of y = f(x) for Activity 1.19.(a) At which values of a does lim x→a f(x) not exist?(b) At which values of a is f(a) not defined?(c) At which values of a does f have a limit, but lim x→a f(x) ≠ f(a)?(d) <strong>State</strong> all values of a for which f is not continuous at x = a.(e) Which condition is stronger, and hence implies the other: f has a limit at x = a orf is continuous at x = a? Explain, and hence complete the following sentence: “If fat x = a, then fat x = a,” where youcomplete the blanks with has a limit and is continuous, using each phrase once.⊳
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 and 30: 1.6. THE SECOND DERIVATIVE 25ffxxf
Page 31: 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
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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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