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Mathematics LibreTexts

3E Chapter Exercises

  • Page ID
    10940
  • [ "stage:draft", "article:topic" ]

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    Chapter Review Exercises

     

    Exercise \(\PageIndex{1}\)

    True or False? Justify your answer with a proof or a counterexample. Assume that \(f(x)\) is continuous and differentiable unless stated otherwise.

    1) If \(f(−1)=−6\) and \(f(1)=2\), then there exists at least one point \(x∈[−1,1]\) such that \(f′(x)=4.\)

    Answer

    True, by Mean Value Theorem

    2) If \(f′(c)=0,\) there is a maximum or minimum at \(x=c.\)

    3) There is a function such that \(f(x)<0,f′(x)>0,\) and \(f''(x)<0.\) (A graphical “proof” is acceptable for this answer.)

    Answer

    True

    4) There is a function such that there is both an inflection point and a critical point for some value \(x=a.\)

     

    Exercise \(\PageIndex{2}\)

    Given the graph of \(f′\), determine where \(f\) is increasing or decreasing.

    Answer

    Increasing: \((−2,0)∪(4,∞)\), decreasing: \((−∞,−2)∪(0,4)\)

     

    Exercise \(\PageIndex{3}\)

    The graph of \(f\) is given below. Draw \(f′\).

    Answer

    Under Construction

     

    Exercise \(\PageIndex{4}\)

    Find the linear approximation \(L(x)\) to \(y=x^2+tan(πx)\) near \(x=\frac{1}{4}.\)

    Answer

    \(L(x)=\frac{17}{16}+\frac{1}{2}(1+4π)(x−\frac{1}{4})\)

     

    Exercise \(\PageIndex{5}\)

    Find the differential of \(y=x^2−5x−6\) and evaluate for \(x=2\) with \(dx=0.1.\)

    Answer

    Under Construction

     

    Exercise \(\PageIndex{6}\)

    Find the critical points and the local and absolute extrema of the following functions on the given interval.

    1) \(f(x)=x+sin^2(x)\) over \([0,π]\)

    Answer

    Critical point: \(x=\frac{3π}{4},\) absolute minimum: \(x=0,\) absolute maximum: \(x=π\)

    2) \(f(x)=3x^4−4x^3−12x^2+6\) over \([−3,3]\)

     

    Exercise \(\PageIndex{7}\)

    Determine over which intervals the following functions are increasing, decreasing, concave up, and concave down.

    1) \(x(t)=3t^4−8t^3−18t^2\)

    Answer

    Increasing: \((−1,0)∪(3,∞)\)

    Decreasing: \((−∞,−1)∪(0,3)\)

    Concave up: \((−∞,\frac{1}{3}(2−\sqrt{13}))∪(\frac{1}{3}(2+\sqrt{13}),∞)\) 

    Concave down: \((\frac{1}{3}(2−\sqrt{13}),\frac{1}{3}(2+\sqrt{13}))\)

    2) \(y=x+sin(πx)\)

    3) \(g(x)=x−\sqrt{x}\)

    Answer

    Increasing: \((\frac{1}{4}∞),\)

    Decreasing: \((0,\frac{1}{4})\)

    Concave up: \((0,∞)\)

    Concave down: nowhere

    4) \(f(θ)=sin(3θ)\

     

    Exercise \(\PageIndex{8}\)

    Evaluate the following limits.

    1) \(lim_{x→∞}\frac{3x\sqrt{x^2+1}}{\sqrt{x4−1}}\)

    Answer

    \(3\)

    2) \(lim_{x→∞}cos(\frac{1}{x})\)

    3) \(lim_{x→1}\frac{x−1}{sin(πx)}\)

    Answer

    \(−\frac{1}{π}\)

    4) \(lim_{x→∞}(3x)^{1/x}\)

     

    Exercise \(\PageIndex{9}\)

    Use Newton’s method to find the first two iterations, given the starting point.

    1) \(y=x^3+1,x_0=0.5\)

    Answer

    \(x_1=−1,x_2=−1\)

    2) \(\frac{1}{x+1}=\frac{1}{2},x_0=0\)

     

    Exercise \(\PageIndex{10}\)

    Find the antiderivatives \(F(x)\) of the following functions.

    1) \(g(x)=\sqrt{x}−\frac{1}{x^2}\)

    Answer

    \(F(x)=\frac{2x^{3/2}}{3}+\frac{1}{x}+C\)

    2) \(f(x)=2x+6cosx,F(π)=π^2+2\

     

    Exercise \(\PageIndex{11}\)

    Graph the following functions by hand. Make sure to label the inflection points, critical points, zeros, and asymptotes.

    1) \(y=\frac{1}{x(x+1)^2}\)

    Answer:

    Inflection points: none

    Critical points: \(x=−\frac{1}{3}\)

    Zeros: none; vertical asymptotes: \(x=−1, x=0\) 

    Horizontal asymptote: \(y=0\)

    2) \(y=x−\sqrt{4−x^2}\)

     

    Exercise \(\PageIndex{12}\)

    A car is being compacted into a rectangular solid. The volume is decreasing at a rate of \(2 m^3/sec\). The length and width of the compactor are square, but the height is not the same length as the length and width. If the length and width walls move toward each other at a rate of \(0.25\) m/sec, find the rate at which the height is changing when the length and width are \(2\) m and the height is \(1.5\) m.

    Answer

    The height is decreasing at a rate of \(0.125\) m/sec

     

    Exercise \(\PageIndex{13}\)

    A rocket is launched into space; its kinetic energy is given by \(K(t)=(\frac{1}{2})m(t)v(t)^2\), where \(K\) is the kinetic energy in joules, \(m\) is the mass of the rocket in kilograms, and \(v\) is the velocity of the rocket in meters/second. Assume the velocity is increasing at a rate of \(15 m/sec^2\) and the mass is decreasing at a rate of \(10\) kg/sec because the fuel is being burned. At what rate is the rocket’s kinetic energy changing when the mass is \(2000\) kg and the velocity is \(5000\) m/sec? Give your answer in mega-Joules (MJ), which is equivalent to \(10^6\) J.

    Answer

    Under Construction

     

    Exercise \(\PageIndex{14}\)

    The famous Regiomontanus’ problem for angle maximization was proposed during the \(15\) th century. A painting hangs on a wall with the bottom of the painting a distance \(a\) feet above eye level, and the top \(b\) feet above eye level. What distance x (in feet) from the wall should the viewer stand to maximize the angle subtended by the painting, \(θ\)?

    Answer

    \(x=\sqrt{ab}\) feet

     

    Exercise \(\PageIndex{15}\)

    An airline sells tickets from Tokyo to Detroit for \($1200.\) There are \(500\) seats available and a typical flight books \(350\) seats. For every \($10\) decrease in price, the airline observes an additional five seats sold. What should the fare be to maximize profit? How many passengers would be onboard?

    Answer

    Under Construction

     
    Exercise \(\PageIndex{16}\)

    For the following functions:

    a. Find the interval(s) on which \(f(x)\) is increasing

    b. Find the interval(s) on which \(f(x)\) is decreasing

    c. Find the local (relative) maximum and local (relative) minimum (if any).

    1) \(f(x)=x^2 e^{-x}\)

    2) \(f(x)=3(x^2-4)^{2/3}\)

    3) \(f(x)=x+\frac{1}{x}\)

    Answer

    Under Construction

     

    Exercise \(\PageIndex{17}\)

    For the following functions:

    a. Find the interval(s) on which \(g(x)\) is concave up

    b. Find the interval(s) on which \(g(x)\) is concave down

    c. Find the inflection points (if any)

     

    1) \(g(x)=10x^3+ 3x^5\)

    2) \(g(x)=e^{-x^2/2}\)

    3) \(g(x)=5x^3+ 2x^5\)

    4) \(g(x)=\ln(1+x^2)\)

    5) \(g(x)=(3-x^2)^2\)

    Answer

    Under Construction

     
    Exercise \(\PageIndex{18}\)

    Find the horizontal asymptote

    \(y=f(x)= x- \sqrt{x^2+2x-6}\)

    Answer

    Under Construction

     

    Exercise \(\PageIndex{19}\)

    \(f(t)=\frac{1}{1-t^2}\)

    a. Find vertical and horizontal asymptotes (if any)

    b. Find the intervals when \(f(t)\) is concave up and concave down

    Answer

    Under Construction

     

    Exercise \(\PageIndex{20}\)

    \(f(t)=e^{-t^2}\)

    a. Find the local maxima and minima for the function \(f(t)\)

    b. Find the intervals when \(f(t)\) is increasing and decreasing

    c. Find the intervals when \(f(t)\) is concave up and concave down 

    d. Find the inflection points for \(f(t)\)

    Answer

    Under Construction

     

    Exercise \(\PageIndex{21}\)

    Find all vertical and horizontal asymptotes for the function and justify your work using limits.

    \(f(x)=\frac{x^2-5x+6}{x^2-4x+3}\)

    Answer

    Under Construction

     

    Exercise \(\PageIndex{22}\)

    Using the following information: 

    \(y=\frac{x}{x^2-1},\qquad y^\prime=-\frac{x^2+1}{\left(x^2-1\right)^2},\mbox{ and }y^{\prime\prime}=\frac{2x\left(x^2+3\right)}{\left(x^2-1\right)^3}\)

    a) Identify any asymptotes for the function \(y\) using limits.

    b) Find where the function \(y\) is increasing/decreasing and any relative extrema.

    c) Find where the function \(y\) is concave up/down and any points of inflection.

    d) Graph the function \(y\)

    Answer

    Under Construction

     

    Exercise \(\PageIndex{23}\)

    Let \(f(x) = x^{5/4}-x^{1/4}\) 

    Determine the intervals on which \(f\) is increasing and decreasing and find any local extrema.

    Answer

    Under Construction

     

    Exercise \(\PageIndex{24}\)

    Let \(g(x) = \ln (x^2+1)\)

    Determine the intervals of concavity of \(g(x)\) and locate any inflection points.

    Answer

    Under Construction

     

    Exercise \(\PageIndex{25}\)

    Determine the intervals of concavity of \(g(x)\) and locate any inflection points:

    1) \(g(x) = (\ln x)^2\)

    2) \(g(x) = x-2 \tan^{-1}(x)\)

    Answer

    Under Construction

     

    Exercise \(\PageIndex{26}\)

    Find interval(s) on which the function is concave up and the interval(s) on which the function is concave down, and inflection point(s):

    \(y=f(x)= x-2tan^{-1}(x)\)

    Answer

    Under Construction

     

    Exercise \(\PageIndex{27}\)

    Let \(f(x) = x^2e^{-16x^2}\) 

    Determine the intervals on which \(f\) is increasing and decreasing and find any local extrema.

    Answer

    Under Construction

     

    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.