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Calculus Volume 1

Review Exercises

Calculus Volume 1Review 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.

$∫abf(x)2dx=∫abf(x)dx∫abf(x)dx∫abf(x)2dx=∫abf(x)dx∫abf(x)dx$

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.

$∫−11(x3−2x2+4x)dx∫−11(x3−2x2+4x)dx$

448.

$∫043t1+6t2dt∫043t1+6t2dt$

449.

$∫π/3π/22sec(2θ)tan(2θ)dθ∫π/3π/22sec(2θ)tan(2θ)dθ$

450.

$∫0π/4ecos2xsinxcosxdx∫0π/4ecos2xsinxcosxdx$

Find the antiderivative.

451.

$∫dx(x+4)3∫dx(x+4)3$

452.

$∫xln(x2)dx∫xln(x2)dx$

453.

$∫4x21−x6dx∫4x21−x6dx$

454.

$∫e2x1+e4xdx∫e2x1+e4xdx$

Find the derivative.

455.

$ddt∫0tsinx1+x2dxddt∫0tsinx1+x2dx$

456.

$ddx∫1x34−t2dtddx∫1x34−t2dt$

457.

$ddx∫1ln(x)(4t+et)dtddx∫1ln(x)(4t+et)dt$

458.

$ddx∫0cosxet2dtddx∫0cosxet2dt$

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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