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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=3x22x+1y=3x22x+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)=6400t26505t+2686,v(t)=6400t26505t+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: 0t0.5.0t0.5. 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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