# Practice Problems

 Site: Saylor Academy Course: MA005: Calculus I Book: Practice Problems
 Printed by: Guest user 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.

## 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.

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