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Practice Problems for Module 5

Sections 3.1 and 3.2

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. Differentiate the functions.​
    1. ​\(f(x)=\dfrac{7}{4}x^4-3x^2+12\)
    2. \(H(u)=(3u-1)(u+5)\)

      1. \(f'(x)=7x^3-6x\)
      2. \(H'(u)=3u^2+14u-5\)

      To see the full video page and find related videos, click the following link.
      MLC WIR 20B M151 week4 #1ab


    3. \(F(r)=\dfrac{8}{r^3}\)​
    4. \(y=\dfrac{\sqrt[3]{x}+x}{x^2}\)​

      1. \(F'(r)=-24r^{-4}\)
      2. \(y'=-\dfrac{5}{3}x^{-8/3}-x^{-2}\)

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      MLC WIR 20B M151 week4 #1cd


    5. \(k(x)=e^x+x^e\)​​

      \(k'(x)=e^x+ex^{e-1}\)

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      MLC WIR 20B M151 week4 #1e


    6. \(f(x)=(3x^2-x)e^x\)​​

      \(f'(x)=(6x-1)e^x+(3x^2-x)e^x\)

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      MLC WIR 20B M151 week4 #1f


    7. \(y=\dfrac{e^x}{4-e^x}\)​

      \(y'=\dfrac{e^x(4-e^x)-e^x(-e^x)}{\left(4-e^x\right)^2}\)

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      MLC WIR 20B M151 week4 #1g


    8. \(G(x)=\dfrac{x^2-3}{5x+1}\)​​

      \(G'(x)=\dfrac{(2x)(5x+1)-(x^2-3)(5)}{(5x+1)^2}\)

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      MLC WIR 20B M151 week4 #1h


    9. \(F(y)=\left( \dfrac{1}{y^2}-\dfrac{3}{y^4}\right)\left(y+2y^3\right)\)​​

      \(F'(y)=\left(-2y^{-3}+12y^{-5}\right)\left(y+2y^3\right)+\left(y^{-2}-3y^{-4}\right)\left(1+6y^2\right)\)

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      MLC WIR 20B M151 week4 #1i


    10. \(V(t)=\dfrac{7+t}{te^t}\)​​

      \(V'(t)=\dfrac{(1)\left(te^t\right)-(7+t)\left(e^t+te^t\right)}{\left(te^t\right)^2}\)

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      MLC WIR 20B M151 week4 #1j


  2. Find \(f'(x)\) and \(f''(x)\) for \(f(x)=\left(x^3+1\right)e^x\).

    \(f'(x)=\left(x^3+3x^2+1\right)e^x\)
    \(f''(x)=\left(3x^2+6x\right)e^x +\left(x^3+3x^2+1\right)e^x\)

    To see the full video page and find related videos, click the following link.
    MLC WIR 20B M151 week4 #2a