$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# Chapter 5 Review Exercises

• • Contributed by OpenStax
• Mathematics at OpenStax CNX
$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

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.

False

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

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

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 $$[−1,1]$$

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

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

7) $$y=x^2\sin x$$ over $$[0,π]$$

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

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

In exercises 9 - 12, evaluate the integrals.

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

$$−\frac{4}{3}$$

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

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

$$1$$

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

In exercises 13 - 16, find the antiderivative.

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

$$−\dfrac{1}{2(x+4)^2}+C$$

14) $$\displaystyle ∫x\ln(x^2)\,dx$$

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

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

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

In exercises 17 - 20, find the derivative.

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

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

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

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

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

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

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.

$$73.36$$

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?

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