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Practice Problems for Exam 2

Directions. The following are review problems. We recommend you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it. You can also follow the link for the full video page to find related videos. 
  1. Find the derivative. 
    1. ​\(f(x)=e^{x^2}\)
    2. \(f(x)=x\sin^7(\cos(6x))\)

      1. \(f'(x)=e^{x^2}\cdot 2x\)
      2. \(f'(x)=\sin^7(\cos(6x))+x\cdot7\sin^6(\cos(6x))\cdot\cos(\cos(6x))\cdot(-\sin(6x))\cdot6\)

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      MLC WIR 20B M151 week6 #1ab


    3. ​\(f(x)=\dfrac{(x-1)^2}{e^{x^2+2x}}\)
    4. \(f(x)=x\sec^4(5x)\)​

      1. \(f'(x)=\dfrac{2(x-1)\cdot e^{x^2+2x}-(x-1)^2\cdot e^{x^2+2x}\cdot(2x+2)}{e^{2x^2+4x}}\)
      2. \(f'(x)=\sec^4(5x)+x\cdot 4\sec^3(5x)\cdot\sec(5x)\cdot\tan(5x)\cdot5\)

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      MLC WIR 20B M151 week6 #1cd


    5. \(f(x)=\cos(x+e^{3x})\)
    6. \(f(x)=\dfrac{(4-x)^2}{\tan x}\)​

      1. \(f'(x)=-\sin\left(x+e^{3x}\right)\cdot\left(1+3e^{3x}\right)\)
      2. \(f'(x)=\dfrac{2(4-x)\cdot(-1)\cdot\tan x-(4-x)^2\cdot \sec^2 x}{\tan^2x}\)

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      MLC WIR 20B M151 week6 #1ef


    7. \(f(x)=\ln \left(\sin^2x\right)\)
    8. \(g(x)=\ln \left(xe^{-2x}\right)\)​

      1. \(f'(x)=\dfrac{2\sin x\cdot \cos x}{\sin^2 x} = \dfrac{2\cos x}{\sin x}\)
      2. \(g'(x)=\dfrac{e^{-2x}+x\cdot e^{-2x}(-2)}{x\cdot e^{-2x}}\)

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      MLC WIR 20B M151 week6 #1gh


    9. \(f(x)=\log_5(1+\cos x)\)
    10. \(f(x)=\arcsin\left(\dfrac{1}{x}\right)\)​

      1. \(f'(x)=\dfrac{-\sin x}{(1+\cos x)\ln 5}\)
      2. \(f'(x)=\dfrac{1}{\sqrt{1-\left(\dfrac{1}{x}\right)^2}}\cdot \dfrac{-1}{x^2}\)

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      MLC WIR 20B M151 week6 #1ij


    11. \(f(x)=\sqrt{1-x^2}\arcsin x\)
    12. \(f(x)=\arctan \left(x^2-x\right)\)​

      1. \(f'(x)=\dfrac{1}{2}\left(1-x^2\right)^{-1/2} \cdot (-2x)\cdot \arcsin x + 1\)
      2. \(f'(x)=\dfrac{1}{1+\left(x^2-x\right)^2} \cdot (2x-1)\)

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      MLC WIR 20B M151 week6 #1kl


  2. Find \(\dfrac{dy}{dx}\).
    1. \(x^2y^3-5x^3=\sec(4y)+10^{y^2}\)​

      \(\dfrac{dy}{dx}=\dfrac{2xy^3-15x^2}{4\sec(4y)\tan(4y)+2y\cdot 10^{y^2}\ln 10-3x^2y^2}\)

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      MLC WIR 20B M151 week6 #2a


    2. \(\tan\left(xy^2\right) +\sin y = 6x^2+8y+2\)​

      \(\dfrac{dy}{dx}=\dfrac{12x-y^2\cdot\sec^2\left(xy^2\right)}{2xy\cdot \sec^2\left(xy^2\right)+\cos y-8}\)

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      MLC WIR 20B M151 week6 #2b


  3. If \(f(x)=\cos \left( \ln x^2\right)\), find \(f'(1)\).​

    \(f'(1)=0\)

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    MLC WIR 20B M151 week6 #3


  4. Use logarithmic differentiation to find the derivative of the function.
    1. \(y=x^{\cos x}\).

      \(\dfrac{dy}{dx}=x^{\cos x} \left(-\sin x \ln x + \dfrac{\cos x}{x}\right)\)

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      MLC WIR 20B M151 week6 #4a


    2. \(y=(\ln x)^{\cos \left(x^2+3\right)}\)

      \(y'=(\ln x)^{\cos \left(x^2+3\right)} \left(-2x\sin \left(x^2+3\right) \cdot \ln (\ln x) + \dfrac{\cos \left(x^2+3\right)}{x\ln x}\right) \)

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      MLC WIR 20B M151 week6 #4b


  5. Given that \(h(5)=3\), \(h'(5)=-2\), \(g(5)=-3\) and \(g'(5)=6\), find \(f'(5)\) for each of the following functions.
    1. \(f(x)=g(x)h(x)\).

      \(f'(5)=24\)

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      MLC WIR 20B M151 week6 #5a


    2. \(f(x)=\dfrac{g(x)}{h(x)}\)

      \(f'(5)=\dfrac{4}{3}\)

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      MLC WIR 20B M151 week6 #5b


    3. \(f(x)=g(h(x))\)

      Not enough information

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      MLC WIR 20B M151 week6 #5c


  6. Find \(h''(1)\) if \(h(x)=e^{-x^2}\).

    \(\dfrac{2}{e}\)

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    MLC WIR 20B M151 week6 #6


  7. The vector function \(\mathbf{r}(t)=(t+e^t)\mathbf{i}-2\sin(t)\mathbf{j}\) represents the position of a particle at time \(t\). Find the speed of the object at the point \((1,0)\).

    The speed is \(2\sqrt{2}\)

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    MLC WIR 20B M151 week6 #7


  8. At what point on the graph of \(f(x)=\sqrt{x}\) is the tangent line parallel to the line \(2x-3y=4\)?

    \(\left(\dfrac{9}{16},\dfrac{3}{4}\right)\)

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    MLC WIR 20B M151 week6 #8


  9. For what values of \(x\), with \(0\leq x\leq 2\pi\), does the curve \(y=x+\dfrac{1}{3}\cos(3x)\) have a horizontal tangent?

    \(x=\dfrac{\pi}{6},\dfrac{5\pi}{6}, \dfrac{3\pi}{2}\)

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    MLC WIR 20B M151 week6 #9


  10. Find the \(77\)th derivative of \(g(x)=\sin(2x)\).

    \(g^{(77)}(x)=2^{77}\cos(2x)\)

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    MLC WIR 20B M151 week6 #10


  11. Use a linear approximation at \(x=1\) to approximate the value of \(\sqrt{1.1}\).

    \(\dfrac{21}{20}\)

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    MLC WIR 20B M151 week6 #11


  12. A particle moves according to the equation \(s(t)=t^2-t\), where \(t\) is measured in seconds and \(s\) in feet. What is the total distance the particle travels during the first \(2\) seconds?

    The total distance is \(2.5\) ft. 

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    MLC WIR 20B M151 week6 #12


  13. Find the point(s) on the curve \(x=t^2+4t\), \(y=t^2+2t\) where the tangent line is vertical or horizontal.

    Horizontal at \((-3,1)\)
    Vertical at \((-4,0)\)

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    MLC WIR 20B M151 week6 #13


  14. Find the equation of the tangent line to the curve parametrized by \(x=5t-t^3\), \(y=t^2-2t\) at the point corresponding to \(t=0\).

    \(y=-\dfrac{2}{5}x\)

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    MLC WIR 20B M151 week6 #14


  15. Find the equation of the tangent line to the curve \(2x^2y-3y^2=-11\) at the point \((2,-1)\).

    \(y=\dfrac{4}{7}x-\dfrac{15}{7}\)

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    MLC WIR 20B M151 week6 #15


  16. For the following function, find the value of \(a\) and \(b\) that make \(f(x)\) differentiable everywhere. \[f(x)=\left\{\begin{array}{ll}ax^2+x+1 & \textrm{if } x\leq -1\\ bx-1 & \textrm{if } x> -1\end{array} \right.\]

    \(a=2, \quad b=-3\)

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    MLC WIR 20B M151 week6 #16


  17. Suppose the linear approximation for a function \(f(x)\) at \(a=2\) is given by the tangent line \(y=-3x+11\). If \(g(x)=(f(x))^2\), find the linear approximation for \(g(x)\) at \(a=2\).

    \(L(x)=25-30(x-2)\)

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    MLC WIR 20B M151 week6 #17