5.R: Chapter 5 Review Exercises

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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

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