Practice Problems

Site: Saylor Academy
Course: MA005: Calculus I
Book: Practice Problems
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Date: Tuesday, July 23, 2024, 5:03 AM

Description

Work through the odd-numbered problems 1-51. Once you have completed the problem set, check your answers for the odd-numbered questions against here.

Table of contents

Practice Problems

Problems 1 – 19 refer to the graph of f in Fig. 13. Use the graph to determine the values of the definite integrals. (The bold numbers represent the area of each region.)

1. \int_{0}^{3} \mathrm{f}(x) \mathrm{dx}

3. \int_{2}^{2} \mathrm{f}(x) \mathrm{dx}

5. \int_{0}^{5} \mathrm{f}(x) \mathrm{dx}

7. \int_{3}^{6} \mathrm{f}(t) \mathrm{dt}

9. \int_{3}^{0} \mathrm{f}(x) \mathrm{dx}

11. \int_{6}^{0} \mathrm{f}(x) \mathrm{dx}

13. \int_{4}^{4} \mathrm{f}^{2}(s) \mathrm{ds}

15. \int_{0}^{3} x+\mathrm{f}(x) \mathrm{dx}

17. \int_{0}^{5} 2+f(x) d x

19. \int_{0}^{5} \operatorname{lf}(x) \mid \mathrm{dx}


Problems 21 – 29 refer to the graph of g in Fig. 14. Use the graph to evaluate the integrals.


21. \int_{0}^{2} g(x) d x

23. \int_{0}^{5} \mathrm{~g}(x) \mathrm{d} \mathrm{x}

25. \int_{0}^{8} g(s) d s

27. \int_{0}^{3} 2 \mathrm{~g}(t) \mathrm{dt}

29. \int_{6}^{3} g(u) d u


For problems 31 – 33, use the constant functions f(x)=4 and g(x)=3 on the interval [0,2]. Calculate each integral and verify that the value obtained in part (a) is not equal to the value in part (b).

31. (a)  \int_{0}^{2} f(x) \mathrm{dx} \cdot \int_{0}^{2} \mathrm{~g}(x) \mathrm{d} \mathrm{x}         (b) \int_{0}^{2} \mathrm{f}(x) \cdot \mathrm{g}(x) \mathrm{dx}\)

33. (a)  \int_{0}^{2} \mathrm{f}^{2}(x) \mathrm{dx}                             (b) \left(\int_{0}^{2} f(x) d x\right)^{2}


For problems 35 – 41, sketch the graph of the integrand function and use it to help evaluate the integral.

35. \int_{0}^{4}|x| d x

37. \int_{-1}^{2}|x| d x

39. \int_{1}^{3} \mathrm{INT}(u) \mathrm{du}

41. \int_{1}^{3} 2+\mathrm{INT}(t) \mathrm{dt}


For problems 43 – 45, (a) Sketch the graph of \mathrm{y}=\mathrm{A}(x)=\int_{0}^{x} \mathrm{f}(t) \mathrm{d} t and (b) sketch the graph of \mathrm{y}=\mathrm{A}^{\prime}(x).

43. \mathrm{f}(x)=x

45. f in Fig. 15


For problems 47 – 49, state whether the function is (a) continuous on [1,4], (b) differentiable on [1,4] , and (c) integrable on [1,4].

47. f in Fig. 15

49. f in Fig. 17


51. Write the total distance traveled by the car in Fig. 19 between 1 pm and 4 pm as a definite integral and estimate the value of the integral.


Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-5.4-Properties-of-the-Definite-Integral.pdf
Creative Commons License This work is licensed under a Creative Commons Attribution 3.0 License.

Answers

1. 5

3. 0

5. 3

7. 0

9. -5

11. -5

13. 0

15. 4.5+5 = 9.5

17. 10+3 = 13

19. 5+2 = 7


21. 1

23. -1

25. 2.5

27. 1

29. 1


31. (a) 8.6 = 48          (b) 24

33. (a) 32                     (b) 8^{2}=64


35. 8

37. 2.5

39. 3

41. 7


43.

(a) graph \mathrm{y}=\mathrm{A}(\mathrm{x})=\mathrm{x}^{2} / 2 

(b) graph y=A^{\prime}(x)=x

45.

(a) graph of \mathrm{y}=\mathrm{A}(\mathrm{x}) in Figure 45a

(b) graph of y=A^{\prime}(x) Figure 45b


47.

(a) f is continuous on [1,4]

(b) f is not differentiable on [1,4] (not diff. at x \approx 2.3 and 3.3

(c) f is integrable on [1,4]


49.

(a) f is not continuous on [1,4] (not cont. at x=2 and 3

(b) f is not differentiable on [1,4] (not diff. at x=2 and 3

(c) f is integrable on [1,4]


51.\int_{1}^{4} \text { velocity } \mathrm{dt}=\int_{1}^{2} \text { velocity } \mathrm{dt}+\int_{2}^{4} \text { velocity } \mathrm{dt}=35+50=85 \text{ miles}