Active Calculus - Gvsu - Grand Valley State University

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102 4.3. THE DEFINITE INTEGRALActivity 4.8.Suppose that the following information is known about the functions f, g, x 2 , and x 3 :• ∫ 20 f(x) dx = −3; ∫ 52f(x) dx = 2• ∫ 20 g(x) dx = 4; ∫ 52g(x) dx = −1• ∫ 20 x2 dx = 8 3 ; ∫ 51172f(x) dx =3• ∫ 20 x3 dx = 4; ∫ 52 x3 dx = 6094Use the provided information and the rules discussed in the preceding section to evaluate eachof the following definite integrals.(a) ∫ 25f(x) dx(b) ∫ 50g(x) dx(c) ∫ 50(f(x) + g(x)) dx(d) ∫ 52 (3x2 − 4x 3 ) dx(e) ∫ 05 (2x3 − 7g(x)) dx⊳
4.3. THE DEFINITE INTEGRAL 103Activity 4.9.Suppose that v(t) = √ 4 − (t − 2) 2 tells us the instantaneous velocity of a moving object on theinterval 0 ≤ t ≤ 4, where t is measured in minutes and v is measured in meters per minute.(a) Sketch an accurate graph of y = v(t). What kind of curve is y = √ 4 − (t − 2) 2 ?(b) Evaluate ∫ 40v(t) dt exactly.(c) In terms of the physical problem of the moving object with velocity v(t), what is themeaning of ∫ 40v(t) dt? Include units on your answer.(d) Determine the exact average value of v(t) on [0, 4]. Include units on your answer.(e) Sketch a rectangle whose base is the line segment from t = 0 to t = 4 on the t-axis suchthat the rectangle’s area is equal to the value of ∫ 40v(t) dt. What is the rectangle’s exactheight?(f) How can you use the average value you found in (d) to compute the total distancetraveled by the moving object over [0, 4]?⊳
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Active CalculusMatt Boelkins, Lead
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Chapter 1Understanding the Derivati
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1.1. HOW DO WE MEASURE VELOCITY? 3A
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1.2. THE NOTION OF LIMIT 51.2 The n
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1.2. THE NOTION OF LIMIT 7Activity
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1.3. THE DERIVATIVE OF A FUNCTION A
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1.3. THE DERIVATIVE OF A FUNCTION A
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1.4. THE DERIVATIVE FUNCTION 131.4
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1.4. THE DERIVATIVE FUNCTION 15rsxx
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1.5. INTERPRETING THE DERIVATIVE AN
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1.5. INTERPRETING THE DERIVATIVE AN
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1.6. THE SECOND DERIVATIVE 21(a) In
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1.6. THE SECOND DERIVATIVE 23• on
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1.6. THE SECOND DERIVATIVE 25ffxxf
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1.7. LIMITS, CONTINUITY, AND DIFFER
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1.7. LIMITS, CONTINUITY, AND DIFFER
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1.8. THE TANGENT LINE APPROXIMATION
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Chapter 2Computing Derivatives2.1 E
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2.1. ELEMENTARY DERIVATIVE RULES 35
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2.2. THE SINE AND COSINE FUNCTIONS
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2.2. THE SINE AND COSINE FUNCTIONS
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2.3. THE PRODUCT AND QUOTIENT RULES
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2.3. THE PRODUCT AND QUOTIENT RULES
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2.4. DERIVATIVES OF OTHER TRIGONOME
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2.4. DERIVATIVES OF OTHER TRIGONOME
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2.5. THE CHAIN RULE 492.5 The chain
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Active Calculus - Gvsu - Grand Valley State University

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