Gilbert Strang; Edwin “Jed” Herman; Gilbert Strang; Edwin “Jed” Herman

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.

439.

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

440.

$\int_{a}^{b} f {(x)}^{2} d x = \int_{a}^{b} f (x) d x \int_{a}^{b} f (x) d x$

441.

If $f (x) \leq g (x)$ for all $x \in [a, b],$ then $\int_{a}^{b} f (x) \leq \int_{a}^{b} g (x) .$

442.

All continuous functions have an antiderivative.

Evaluate the Riemann sums $L_{4} and R_{4}$ 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.

443.

$y = 3 x^{2} - 2 x + 1$ over $[−1, 1]$

444.

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

445.

$y = x^{2} sin x$ over $[0, π]$

446.

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

Evaluate the following integrals.

447.

$\int_{−1}^{1} (x^{3} - 2 x^{2} + 4 x) d x$

448.

$\int_{0}^{4} \frac{3 t}{\sqrt{1 + 6 t^{2}}} d t$

449.

$\int_{π / 3}^{π / 2} 2 sec (2 θ) tan (2 θ) d θ$

450.

$\int_{0}^{π / 4} e^{{cos}^{2} x} sin x cos x d x$

Find the antiderivative.

451.

$\int \frac{d x}{{(x + 4)}^{3}}$

452.

$\int x ln (x^{2}) d x$

453.

$\int \frac{4 x^{2}}{\sqrt{1 - x^{6}}} d x$

454.

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

Find the derivative.

455.

$\frac{d}{d t} \int_{0}^{t} \frac{sin x}{\sqrt{1 + x^{2}}} d x$

456.

$\frac{d}{d x} \int_{1}^{x^{3}} \sqrt{4 - t^{2}} d t$

457.

$\frac{d}{d x} \int_{1}^{ln (x)} (4 t + e^{t}) d t$

458.

$\frac{d}{d x} \int_{0}^{cos x} e^{t^{2}} d t$

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

459.

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

460.

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.

461.

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

462.

The velocity of a bullet from a rifle can be approximated by $v (t) = 6400 t^{2} - 6505 t + 2686,$ 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: $0 \leq t \leq 0.5 .$ What is the total distance the bullet travels in 0.5 sec?

463.

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