5.R: Chapter 5 Review Exercises

In exercises 1 - 4, answer True or False. Justify your answer with a proof or a counterexample. Assume all functions $f$ and $g$ are continuous over their domains.

1) If $f(x)>0,\;f′(x)>0$ for all $x$, then the right-hand rule underestimates the integral $\displaystyle ∫^b_af(x)\,dx.$ Use a graph to justify your answer.

Answer

False

2) $\displaystyle ∫^b_af(x)^2\,dx=∫^b_af(x)\,dx$

3) If $f(x)≤g(x)$ for all $x∈[a,b]$, then $\displaystyle ∫^b_af(x)\,dx≤∫^b_ag(x)\,dx.$

Answer

True

4) All continuous functions have an antiderivative.

In exercises 5 - 8, evaluate the Riemann sums $L_4$ and $R_4$ for the given functions over the specified interval. Compare your answer with the exact answer, when possible, or use a calculator to determine the answer.

5) $y=3x^2−2x+1)$ over $[−1,1]$

Answer

$L_4=5.25, \;R_4=3.25,$ exact answer: 4

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

7) $y=x^2\sin x$ over $[0,π]$

Answer

$L_4=5.364,\;R_4=5.364,$ exact answer: $5.870$

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

In exercises 9 - 12, evaluate the integrals.

9) $\displaystyle ∫^1_{−1}(x^3−2x^2+4x)\,dx$

Answer

$−\frac{4}{3}$

10) $\displaystyle ∫^4_0\frac{3t}{\sqrt{1+6t^2}}\,dt$

11) $\displaystyle ∫^{π/2}_{π/3}2\sec(2θ)\tan(2θ)\,dθ$

Answer

$1$

12) $\displaystyle ∫^{π/4}_0e^{\cos^2x}\sin x\cos x\,dx$

In exercises 13 - 16, find the antiderivative.

13) $\displaystyle ∫\frac{dx}{(x+4)^3}$

Answer

$−\dfrac{1}{2(x+4)^2}+C$

14) $\displaystyle ∫x\ln(x^2)\,dx$

15) $\displaystyle ∫\frac{4x^2}{\sqrt{1−x^6}}\,dx$

Answer

$\displaystyle \frac{4}{3}\sin^{−1}(x^3)+C$

16) $\displaystyle ∫\frac{e^{2x}}{1+e^{4x}}\,dx$

In exercises 17 - 20, find the derivative.

17) $\displaystyle \frac{d}{dt}∫^t_0\frac{\sin x}{\sqrt{1+x^2}}\,dx$

Answer

$\dfrac{\sin t}{\sqrt{1+t^2}}$

18) $\displaystyle \frac{d}{dx}∫^{x^3}_1\sqrt{4−t^2}\,dt$

19) $\displaystyle \frac{d}{dx}∫^{\ln(x)}_1(4t+e^t)\,dt$

Answer

$4\dfrac{\ln x}{x}+1$

20) $\displaystyle \frac{d}{dx}∫^{\cos x}_0e^{t^2}\,dt$

In exercises 21 - 23, 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	$−177$

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

Answer

$$6,328,113$

Solution: $6,328,113

22) 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 dollars. Find the average cost per gigabyte of RAM for the period from 1980 to 2010.

23) Find the average cost of $1$ GB RAM from 2005 to 2010.

Answer

$$73.36$

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

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

Answer

$\frac{19117}{12}$ ft/sec, or about $1593$ ft/sec