Calculus Volume 2

# Review Exercises

Calculus Volume 2Review Exercises

### Review Exercises

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

439.

If $f(x)>0,f′(x)>0f(x)>0,f′(x)>0$ for all $x,x,$ then the right-hand rule underestimates the integral $∫abf(x).∫abf(x).$ Use a graph to justify your answer.

440.

$∫ a b f ( x ) 2 d x = ∫ a b f ( x ) d x ∫ a b f ( x ) d x ∫ a b f ( x ) 2 d x = ∫ a b f ( x ) d x ∫ a b f ( x ) d x$

441.

If $f(x)≤g(x)f(x)≤g(x)$ for all $x∈[a,b],x∈[a,b],$ then $∫abf(x)≤∫abg(x).∫abf(x)≤∫abg(x).$

442.

All continuous functions have an antiderivative.

Evaluate the Riemann sums $L4andR4L4andR4$ 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.

$y=3x2−2x+1y=3x2−2x+1$ over $[−1,1][−1,1]$

444.

$y=ln(x2+1)y=ln(x2+1)$ over $[0,e][0,e]$

445.

$y=x2sinxy=x2sinx$ over $[0,π][0,π]$

446.

$y=x+1xy=x+1x$ over $[1,4][1,4]$

Evaluate the following integrals.

447.

$∫ −1 1 ( x 3 − 2 x 2 + 4 x ) d x ∫ −1 1 ( x 3 − 2 x 2 + 4 x ) d x$

448.

$∫ 0 4 3 t 1 + 6 t 2 d t ∫ 0 4 3 t 1 + 6 t 2 d t$

449.

$∫ π / 3 π / 2 2 sec ( 2 θ ) tan ( 2 θ ) d θ ∫ π / 3 π / 2 2 sec ( 2 θ ) tan ( 2 θ ) d θ$

450.

$∫ 0 π / 4 e cos 2 x sin x cos x d x ∫ 0 π / 4 e cos 2 x sin x cos x d x$

Find the antiderivative.

451.

$∫ d x ( x + 4 ) 3 ∫ d x ( x + 4 ) 3$

452.

$∫ x ln ( x 2 ) d x ∫ x ln ( x 2 ) d x$

453.

$∫ 4 x 2 1 − x 6 d x ∫ 4 x 2 1 − x 6 d x$

454.

$∫ e 2 x 1 + e 4 x d x ∫ e 2 x 1 + e 4 x d x$

Find the derivative.

455.

$d d t ∫ 0 t sin x 1 + x 2 d x d d t ∫ 0 t sin x 1 + x 2 d x$

456.

$d d x ∫ 1 x 3 4 − t 2 d t d d x ∫ 1 x 3 4 − t 2 d t$

457.

$d d x ∫ 1 ln ( x ) ( 4 t + e t ) d t d d x ∫ 1 ln ( x ) ( 4 t + e t ) d t$

458.

$d d x ∫ 0 cos x e t 2 d t d d x ∫ 0 cos x e t 2 d t$

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 $C(t)=8,500,000(0.65)t,C(t)=8,500,000(0.65)t,$ where $tt$ is measured in years since 1980, and $CC$ 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 $v(t)=6400t2−6505t+2686,v(t)=6400t2−6505t+2686,$ where $tt$ is seconds after the shot and $vv$ is the velocity measured in feet per second. This equation only models the velocity for the first half-second after the shot: $0≤t≤0.5.0≤t≤0.5.$ 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?

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