Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

Chapter 5 Review Exercises

  • Page ID
    10767
  •  
    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    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.

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

     

    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

    Contributors

    • Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.