Module 5 Review Problems

True or False. Justify your answer with a proof or a counterexample. Assume all functions $f$ and $g$ are continuous over their domains (1-4).

1. If $f(x)>0,{f}^{\prime }(x)>0$ for all $x,$ then the right-hand rule underestimates the integral ${\displaystyle\int }_{a}^{b}f(x).$ Use a graph to justify your answer.

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2. ${\displaystyle\int }_{a}^{b}f{(x)}^{2}dx={\displaystyle\int }_{a}^{b}f(x)dx{\displaystyle\int }_{a}^{b}f(x)dx$

3. If $f(x)\le g(x)$ for all $x\in \left[a,b\right],$ then ${\displaystyle\int }_{a}^{b}f(x)\le {\displaystyle\int }_{a}^{b}g(x).$

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4. All continuous functions have an antiderivative.

Evaluate the Riemann sums ${L}_{4}\text{ and }{R}_{4}$ for the following functions over the specified interval. Compare your answer with the exact answer, when possible, or use a calculator to determine the answer.

5. $y=3{x}^{2}-2x+1$ over $\left[-1,1\right]$

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${L}_{4}=5.25,{R}_{4}=3.25,$ exact answer: 4

6. $y=\text{ln}({x}^{2}+1)$ over $\left[0,e\right]$

7. $y={x}^{2} \sin x$ over $\left[0,\pi \right]$

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${L}_{4}=5.364,{R}_{4}=5.364,$ exact answer: 5.870

8. $y=\sqrt{x}+\frac{1}{x}$ over $\left[1,4\right]$

Evaluate the following integrals.

9. ${\displaystyle\int }_{-1}^{1}({x}^{3}-2{x}^{2}+4x)dx$

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$-\frac{4}{3}$

10. ${\displaystyle\int }_{0}^{4}\frac{3t}{\sqrt{1+6{t}^{2}}}dt$

11. ${\displaystyle\int }_{\pi \text{/}3}^{\pi \text{/}2}2 \sec (2\theta ) \tan (2\theta )d\theta$

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12. ${\displaystyle\int }_{0}^{\pi \text{/}4}{e}^{{ \cos }^{2}x} \sin x \cos{x} dx$

Find the antiderivative.

13. $\displaystyle\int \frac{dx}{{(x+4)}^{3}}$

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$-\frac{1}{2{(x+4)}^{2}}+C$

14. $\displaystyle\int x\text{ln}({x}^{2})dx$

15. $\displaystyle\int \frac{4{x}^{2}}{\sqrt{1-{x}^{6}}}dx$

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$\frac{4}{3}\phantom{\rule{0.05em}{0ex}}{ \sin }^{-1}({x}^{3})+C$

16. $\displaystyle\int \frac{{e}^{2x}}{1+{e}^{4x}}dx$

Find the derivative.

17. $\frac{d}{dt}{\displaystyle\int }_{0}^{t}\frac{ \sin x}{\sqrt{1+{x}^{2}}}dx$

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$\frac{ \sin t}{\sqrt{1+{t}^{2}}}$

18. $\frac{d}{dx}{\displaystyle\int }_{1}^{{x}^{3}}\sqrt{4-{t}^{2}}dt$

19. $\frac{d}{dx}{\displaystyle\int }_{1}^{\text{ln}(x)}(4t+{e}^{t})dt$

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$4\frac{\text{ln}x}{x}+1$

20. $\frac{d}{dx}{\displaystyle\int }_{0}^{ \cos x}{e}^{{t}^{2}}dt$

The following problems consider the historic average cost per gigabyte of RAM on a computer.

Year	5-Year Change ($)
1980	0
1985	−5,468,750
1990	−755,495
1995	−73,005
2000	−29,768
2005	−918
2010	−177

21. If the average cost per gigabyte of RAM in 2010 is $12, find the average cost per gigabyte of RAM in 1980.

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22. The average cost per gigabyte of RAM can be approximated by the function $C(t)=8,500,000{(0.65)}^{t},$ where $t$ is measured in years since 1980, and $C$ is cost in US$. Find the average cost per gigabyte of RAM for 1980 to 2010.

23. Find the average cost of 1GB RAM for 2005 to 2010.

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24. The velocity of a bullet from a rifle can be approximated by $v(t)=6400{t}^{2}-6505t+2686,$ where $t$ is seconds after the shot and $v$ is the velocity measured in feet per second. This equation only models the velocity for the first half-second after the shot: $0\le t\le 0.5.$ What is the total distance the bullet travels in 0.5 sec?

25. What is the average velocity of the bullet for the first half-second?

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$\frac{19117}{12}\text{ft/sec},\text{or}1593\text{ft/sec}$

Integration: Supplemental Content

Module 5 Review Problems

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