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Mathematics LibreTexts

2.5E Exercises

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  • Page ID
    10652
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    Exercise \({1}\)

    For the following exercises, given \(y=f(u)\) and \(u=g(x)\), find dy/dx by using Leibniz’s notation for the chain rule: \(\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}.\)

    1) \(y=3u−6,u=2x^2\)

    2) \(y=6u^3,u=7x−4\)

    3) \(y=sinu,u=5x−1\)

    4) \(y=cosu,u=\frac{−x}{8}\)

    5) \(y=tanu,u=9x+2\)

    6) \(y=\sqrt{4u+3},u=x^2−6x\)

    Answers to even numbered questions

    2. \(18u^2⋅7=18(7x−4)^2⋅7\)

    4. \(−sinu⋅\frac{−1}{8=}−sin(\frac{−x}{8})⋅\frac{−1}{8}\)

    6. \(\frac{8x−24}{2\sqrt{4u+3}}=\frac{4x−12}{\sqrt{4x^2−24x+3}}\)

     

    Exercise \({2}\)

    For each of the following exercises,

    a. decompose each function in the form \(y=f(u)\) and \(u=g(x),\) and

    b. find \(\frac{dy}{dx}\) as a function of \(x\).

    1) \(y=(3x−2)^6\)

    2) \(y=(3x^2+1)^3\)

    3) \(y=sin^5(x)\)

    4) \(y=(\frac{x}{7}+\frac{7}{x})^7\)

    5) \(y=tan(secx)\)

    6) \(y=csc(πx+1)\)

    7) \(y=cot^2x\)

    8) \(y=−6sin^{−3}x\)

    Answers to even numbered questions

    2a. \(u=3x^2+1\)

    b. \(18x(3x^2+1)^2\)

    4a. \(f(u)=u^7,u=\frac{x}{7}+\frac{7}{x}\)

    b. \(7(\frac{x}{7}+\frac{7}{x})^6⋅(\frac{1}{7}−\frac{7}{x^2})\)

    6a.  \(f(u)=cscu,u=πx+1\)

    b. \(−πcsc(πx+1)⋅cot(πx+1)\)

    8a. \(f(u)=−6u^{−3},u=sinx\)

    b. \(18sin^{−4}x⋅cosx\)

     

    Exercise \({3}\)

    For the following exercises, find \(\frac{dy}{dx}\) for each function.

    1) \(y=(3x^2+3x−1)^4\)

    2) \(y=(5−2x)^{−2}\)

    3) \(y=cos^3(πx)\)

    4) \(y=(2x^3−x^2+6x+1)^3\)

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

    6) \(y=(tanx+sinx)^{−3}\)

    7) \(y=x^2cos^4x\)

    8) \(y=sin(cos7x)\)

    9) \(y=\sqrt{6+secπx^2}\)

    10) \(y=cot^3(4x+1)\)

    Answers to even numbered questions

    2. \(\frac{4}{(5−2x)^3}\)

    4. \(y=(2x^3−x^2+6x+1)^3\)

    6. \(−3(tanx+sinx)^{−4}⋅(sec^2x+cosx)\)

    8. \(−7cos(cos7x)⋅sin7x\)

    10. \(−12cot^2(4x+1)⋅csc^2(4x+1)\)

     

    Exercise \({4}\)

    Let \(y=[f(x)]^3\) and suppose that \(f′(1)=4\) and \(\frac{dy}{dx}=10\) for \(x=1\). Find \(f(1)\).

    Answer

    Under Construction

     

    Exercise \({5}\)

    Let \(y=(f(x)+5x^2)^4\) and suppose that \(f(−1)=−4\) and \(\frac{dy}{dx}=3\) when \(x=−1\). Find \(f′(−1)\)

    Answer

    \(10\frac{3}{4}\)

     

    Exercise \({6}\)

    Let \(y=(f(u)+3x)^2\) and \(u=x^3−2x\). If \(f(4)=6\) and \(\frac{dy}{dx}=18\) when \(x=2\), find \(f′(4)\).

    Answer

    Under Construction

     

    Exercise \({7}\)

    Find the equation of the tangent line to \(y=−sin(\frac{x}{2})\) at the origin. Use a calculator to graph the function and the tangent line together.

    Answer

    \(y=\frac{−1}{2}x\)

     

    Exercise \({8}\)

    Find the equation of the tangent line to \(y=(3x+\frac{1}{x})^2\) at the point \((1,16)\). Use a calculator to graph the function and the tangent line together.

    Answer

    Under Construction

     

    Exercise \({9}\)

    Find the \(x\) -coordinates at which the tangent line to \(y=(x−\frac{6}{x})^8\) is horizontal.

    Answer

    \(x=±\sqrt{6}\)

     

    Exercise \({10}\)

    Find an equation of the line that is normal to \(g(θ)=sin2^(πθ)\) at the point \((\frac{1}{4},\frac{1}{2})\). Use a calculator to graph the function and the normal line together

    Answer

    Under Construction

     

    Exercise \(\PageIndex{11}\)

    For the following exercises, use the information in the following table to find \(h′(a)\) at the given value for \(a\).

    \(x\) \(f(x)\) \(f'(x)\) \(g(x)\) \(g'(x)\)
    0 2 5 0 2
    1 1 −2 3 0
    2 4 4 1 −1
    3 3 −3 2 3

    1) \(h(x)=f(g(x));a=0\)

    2) \(h(x)=g(f(x));a=0\)

    3) \(h(x)=(x^4+g(x))^{−2};a=1\)

    4) \(h(x)=(\frac{f(x)}{g(x)})^2;a=3\)

    5) \(h(x)=f(x+f(x));a=1\)

    6) \(h(x)=(1+g(x))^3;a=2\)

    7) \(h(x)=g(2+f(x^2));a=1\)

    8) \(h(x)=f(g(sinx));a=0\)

    Answer to odd numbered questions

    1. \(10\)

    3. \(−\frac{1}{8}\)

    5. \(−4\)

    7. \(−12\)

     

    Exercise \({12}\)

    The position function of a freight train is given by \(s(t)=100(t+1)^{−2}\), with \(s\) in meters and \(t\) in seconds.

    At time \(t=6\)s, find the train’s

    a. velocity and

    b. acceleration.

    c. Using a. and b. is the train speeding up or slowing down?

    Answer

    a. \(−\frac{200}{343}\) m/s

    b. \(\frac{600}{2401}\) m/s^2

    c. The train is slowing down since velocity and acceleration have opposite signs

     

    Exercise \({13}\)

    A mass hanging from a vertical spring is in simple harmonic motion as given by the following position function, where t is measured in seconds and \(s\) is in inches:

    \(s(t)=−3cos(πt+\frac{π}{4}).\)

    a. Determine the position of the spring at \(t=1.5\) s.

    b. Find the velocity of the spring at \(t=1.5\) s.

    Answer

    Under Construction

     

    Exercise \({14}\)

    The total cost to produce \(x\) boxes of Thin Mint Girl Scout cookies is \(C\) dollars, where \(C=0.0001x^3−0.02x^2+3x+300.\) In \(t\) weeks production is estimated to be \(x=1600+100t\) boxes.

    a. Find the marginal cost \(C′(x).\)

    b. Use Leibniz’s notation for the chain rule, \(\frac{dC}{dt}=\frac{dC}{dx}⋅\frac{dx}{dt}\), to find the rate with respect to time \(t\) that the cost is changing.

    c. Use b. to determine how fast costs are increasing when \(t=2\) weeks. Include units with the answer.

    Answer

    a. \(C′(x)=0.0003x^2−0.04x+3\)

    b. \(dCdt=100⋅(0.0003x^2−0.04x+3)\)

    c. Approximately $90,300 per week

     

    Exercise \({15}\)

    The formula for the area of a circle is \(A=πr^2\), where \(r\) is the radius of the circle. Suppose a circle is expanding, meaning that both the area \(A\) and the radius \(r\) (in inches) are expanding.

    a. Suppose \(r=2−\frac{100}{(t+7)^2}\) where \(t\) is time in seconds. Use the chain rule \(\frac{dA}{dt}=\frac{dA}{dr}⋅\frac{dr}{dt}\) to find the rate at which the area is expanding.

    b. Use a. to find the rate at which the area is expanding at \(t=4\) s.

    Answer

    Under Construction

     

    Exercise \({16}\)

    The formula for the volume of a sphere is \(S=\frac{4}{3}πr^3\), where \(r\) (in feet) is the radius of the sphere. Suppose a spherical snowball is melting in the sun.

    a. Suppose \(r=\frac{1}{(t+1)^2}−\frac{1}{12}\) where t is time in minutes. Use the chain rule \(\frac{dS}{dt}=\frac{dS}{dr}⋅\frac{dr}{dt}\) to find the rate at which the snowball is melting.

    b. Use a. to find the rate at which the volume is changing at \(t=1\) min.

    Answer

    a. \(\frac{dS}{dt}=−\frac{8πr^2}{(t+1)^3}\)

    b. The volume is decreasing at a rate of \(−\frac{π}{36}\) \(ft^3\)/min

     

    Exercise \({17}\)

    The daily temperature in degrees Fahrenheit of Phoenix in the summer can be modeled by the function \(T(x)=94−10cos[\frac{π}{12}(x−2)]\), where \(x\) is hours after midnight. Find the rate at which the temperature is changing at 4 p.m.

    Answer

    Under Construction

     

    Exercise \({18}\)

    The depth (in feet) of water at a dock changes with the rise and fall of tides. The depth is modeled by the function \(D(t)=5sin(\frac{π}{6}t−\frac{7π}{6})+8\), where \(t\) is the number of hours after midnight. Find the rate at which the depth is changing at 6 a.m.

    Answer

    \(~2.3\) ft/hr

     

    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.