True or False. Justify your answer with a proof or a counterexample. Assume all functions [latex]f[/latex] and [latex]g[/latex] are continuous over their domains (1-4).
1. If [latex]f(x)>0,{f}^{\prime }(x)>0[/latex] for all [latex]x,[/latex] then the right-hand rule underestimates the integral [latex]{\displaystyle\int }_{a}^{b}f(x).[/latex] Use a graph to justify your answer.
2. [latex]{\displaystyle\int }_{a}^{b}f{(x)}^{2}dx={\displaystyle\int }_{a}^{b}f(x)dx{\displaystyle\int }_{a}^{b}f(x)dx[/latex]
3. If [latex]f(x)\le g(x)[/latex] for all [latex]x\in \left[a,b\right],[/latex] then [latex]{\displaystyle\int }_{a}^{b}f(x)\le {\displaystyle\int }_{a}^{b}g(x).[/latex]
4. All continuous functions have an antiderivative.
Evaluate the Riemann sums [latex]{L}_{4}\text{ and }{R}_{4}[/latex] 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. [latex]y=3{x}^{2}-2x+1[/latex] over [latex]\left[-1,1\right][/latex]
6. [latex]y=\text{ln}({x}^{2}+1)[/latex] over [latex]\left[0,e\right][/latex]
7. [latex]y={x}^{2} \sin x[/latex] over [latex]\left[0,\pi \right][/latex]
8. [latex]y=\sqrt{x}+\frac{1}{x}[/latex] over [latex]\left[1,4\right][/latex]
Evaluate the following integrals.
9. [latex]{\displaystyle\int }_{-1}^{1}({x}^{3}-2{x}^{2}+4x)dx[/latex]
10. [latex]{\displaystyle\int }_{0}^{4}\frac{3t}{\sqrt{1+6{t}^{2}}}dt[/latex]
11. [latex]{\displaystyle\int }_{\pi \text{/}3}^{\pi \text{/}2}2 \sec (2\theta ) \tan (2\theta )d\theta [/latex]
12. [latex]{\displaystyle\int }_{0}^{\pi \text{/}4}{e}^{{ \cos }^{2}x} \sin x \cos{x} dx[/latex]
Find the antiderivative.
13. [latex]\displaystyle\int \frac{dx}{{(x+4)}^{3}}[/latex]
14. [latex]\displaystyle\int x\text{ln}({x}^{2})dx[/latex]
15. [latex]\displaystyle\int \frac{4{x}^{2}}{\sqrt{1-{x}^{6}}}dx[/latex]
16. [latex]\displaystyle\int \frac{{e}^{2x}}{1+{e}^{4x}}dx[/latex]
Find the derivative.
17. [latex]\frac{d}{dt}{\displaystyle\int }_{0}^{t}\frac{ \sin x}{\sqrt{1+{x}^{2}}}dx[/latex]
18. [latex]\frac{d}{dx}{\displaystyle\int }_{1}^{{x}^{3}}\sqrt{4-{t}^{2}}dt[/latex]
19. [latex]\frac{d}{dx}{\displaystyle\int }_{1}^{\text{ln}(x)}(4t+{e}^{t})dt[/latex]
20. [latex]\frac{d}{dx}{\displaystyle\int }_{0}^{ \cos x}{e}^{{t}^{2}}dt[/latex]
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 $12, find the average cost per gigabyte of RAM in 1980.
22. The average cost per gigabyte of RAM can be approximated by the function [latex]C(t)=8,500,000{(0.65)}^{t},[/latex] where [latex]t[/latex] is measured in years since 1980, and [latex]C[/latex] is cost in US$. Find the average cost per gigabyte of RAM for 1980 to 2010.
23. Find the average cost of 1GB RAM for 2005 to 2010.
24. The velocity of a bullet from a rifle can be approximated by [latex]v(t)=6400{t}^{2}-6505t+2686,[/latex] where [latex]t[/latex] is seconds after the shot and [latex]v[/latex] is the velocity measured in feet per second. This equation only models the velocity for the first half-second after the shot: [latex]0\le t\le 0.5.[/latex] 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?
