Active Calculus - Gvsu - Grand Valley State University

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100 4.3. THE DEFINITE INTEGRALFigure 4.7: A left Riemann sum with 5 subintervals for the function f(x) = 2x + 1 on the interval[1, 4]. The value of the sum is L 5 = 16.2.1” for the function formula). Use the applet to compute L n , M n , R n for n = 10, n = 100,and n = 1000. What do you conjecture is the exact area bounded by f(x) = x 2 + 1 and thex-axis on [1, 4]?(e) Why can we not compute the exact value of the area bounded by f(x) = x 2 + 1 and thex-axis on [1, 4] using a formula like we did in (b)?⊲⊳
4.3. THE DEFINITE INTEGRAL 101Activity 4.7.Use known geometric formulas and the net signed area interpretation of the definite integralto evaluate each of the definite integrals below.(a)(b)(c)(d)∫ 10∫ 4−1∫ 1−1∫ 4−33x dx(2 − 2x) dx√1 − x 2 dxg(x) dx, where g is the function pictured in Figure 4.8. Assume that each portion ofg is either part of a line or part of a circle.1y = g(x)-3 -2 -1 1 2 3 4-1Figure 4.8: A function g that is piecewise defined; each piece of the function is part of a circle orpart of a line.⊳
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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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Active Calculus - Gvsu - Grand Valley State University

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