### Review Exercises

*True or False.* Justify your answer with a proof or a counterexample. Assume all functions $f$ and $g$ are continuous over their domains.

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

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

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

All continuous functions have an antiderivative.

Evaluate the Riemann sums ${L}_{4}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{R}_{4}$ 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.

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

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

Evaluate the following integrals.

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

${\int}_{\pi \text{/}3}^{\pi \text{/}2}2\phantom{\rule{0.1em}{0ex}}\text{sec}\left(2\theta \right)}\text{tan}\left(2\theta \right)d\theta $

${\int}_{0}^{\pi \text{/}4}{e}^{{\text{cos}}^{2}x}}\text{sin}\phantom{\rule{0.1em}{0ex}}x\phantom{\rule{0.1em}{0ex}}\text{cos}x\phantom{\rule{0.1em}{0ex}}dx$

Find the antiderivative.

$\int x\phantom{\rule{0.1em}{0ex}}\text{ln}\left({x}^{2}\right)dx$

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

Find the derivative.

$\frac{d}{dt}{\displaystyle {\int}_{0}^{t}\frac{\text{sin}\phantom{\rule{0.1em}{0ex}}x}{\sqrt{1+{x}^{2}}}dx}$

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

$\frac{d}{dx}{\displaystyle {\int}_{0}^{\text{cos}\phantom{\rule{0.1em}{0ex}}x}{e}^{{t}^{2}}}dt$

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 |

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

The average cost per gigabyte of RAM can be approximated by the function $C\left(t\right)=8,500,000{\left(0.65\right)}^{t},$ 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.

The velocity of a bullet from a rifle can be approximated by $v\left(t\right)=6400{t}^{2}-6505t+2686,$ 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?