Active Calculus - Gvsu - Grand Valley State University

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106 4.4. THE FUNDAMENTAL THEOREM OF CALCULUSgiven function, f(x)antiderivative, F (x)k, (k ≠ 0)x n , n ≠ −11x , x > 0sin(x)cos(x)sec(x) tan(x)csc(x) cot(x)sec 2 (x)csc 2 (x)e xa x (a > 1)11+x 2√ 11−x 2Activity 4.11.Table 4.1: Familiar basic functions and their antiderivatives.Use your knowledge of derivatives of basic functions to complete the above table of antiderivatives.For each entry, your task is to find a function F whose derivative is the given functionf. When finished, use the FTC and the results in the table to evaluate the three given definiteintegrals.(a)(b)(c)∫ 10∫ π/30∫ 10(x 3 − x − e x + 2 ) dx(2 sin(t) − 4 cos(t) + sec 2 (t) − π) dt( √ x − x 2 ) dx⊳
4.4. THE FUNDAMENTAL THEOREM OF CALCULUS 107Activity 4.12.During a 30-minute workout, a person riding an exercise machine burns calories at a rate ofc calories per minute, where the function y = c(t) is given in Figure 4.9. On the interval 0 ≤t ≤ 10, the formula for c is c(t) = −0.05t 2 + t + 10, while on 20 ≤ t ≤ 30, its formula isc(t) = −0.05t 2 + 2t − 5.15cal/miny = c(t)10510 20 30minFigure 4.9: The rate c(t) at which a person exercising burns calories, measured in calories perminute.(a) What is the exact total number of calories the person burns during the first 10 minutesof her workout?(b) Let C(t) be an antiderivative of c(t). What is the meaning of C(30) − C(0) in the contextof the person exercising? Include units on your answer.(c) Determine the exact average rate at which the person burned calories during the 30-minute workout.(d) At what time(s), if any, is the instantaneous rate at which the person is burning caloriesequal to the average rate at which she burns calories, on the time interval 0 ≤ t ≤ 30?⊳
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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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2.5. THE CHAIN RULE 51Activity 2.14
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2.6. DERIVATIVES OF INVERSE FUNCTIO
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Active Calculus - Gvsu - Grand Valley State University

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