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Practice Problems for Exam 3

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. The graph below is that of the derivative, \(f'(x)\), of a continous function \(f(x)\). Find the interval(s) where \(f(x)\) is increasing, decreasing, concave upward, concave downward, and find the \(x\)-values of all local extrema and inflection points.
    151_Exam3_Review_p1.png

    Increasing: \((−2, 0)\), \((0, 2)\)
    Decreasing: \((−\infty, −2)\), \((2, \infty)\)
    Local Min at \(x = −2\),
    Local Max at \(x = 2\),
    Concave Up: \((−\infty, −1.4)\), \((0, 1.4)\),
    Concave Down: \((−1.4, 0)\), \((1.4, \infty)\),
    Inflection  Points at \(x = −1.4, 0, 1.4\)

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    MLC WIR 20B M151 week9 #1


  2. Find the absolute maximum and minimum values for \(f(x)=x^2+\dfrac{2}{x}\) over the interal \(\left[\frac{1}{2},2\right]\).

    Absolute Maximum is 5
    Absolute Minimum is 3

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    MLC WIR 20B M151 week9 #2


  3. If \(f'(x) = (x − 2)^3(4 − x)^5(x + 1)^2\), find the interval(s) where \(f(x)\) is increasing and decreasing.

    Increasing: \((2,4)\)
    Decreasing: \((-\infty, -1)\), \((-1,2)\), \((4,\infty)\)

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    MLC WIR 20B M151 week9 #3


  4. If \(f(x)=\dfrac{x^2}{x-1}\), find the interval(s) where \(f(x)\) is increasing and decreasing, and find the
    \(x\)-value(s) of any local maximum and minimum.

    Increasing: \((−\infty,0)\), \((2,\infty)\)
    Decreasing: \((0,1)\), \((1,2)\) 
    Local Maximum at \(x = 0\)
    Local Minimum at \(x = 2\)

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    MLC WIR 20B M151 week9 #4


  5. If \(f(x) = x^4 − 6x^2 + 4\), find the interval(s) where \(f(x)\) is increasing, decreasing, concave upward, and concave downward. And find the local extrema and inflection point(s).

    Increasing: \((−\sqrt{3},0)\), \((\sqrt{3},\infty)\)
    Decreasing: \((-\infty,-\sqrt{3})\), \((0,\sqrt{3})\)
    Local Maximum: \((0,4)\)
    Local Minimum: \((-\sqrt{3},-5)\), \((\sqrt{3},-5)\)
    Concave Up: \((-\infty, -1)\), \((1,\infty)\)
    Concave Down: \((-1,1)\)
    Inflection Point: \((-1,2)\), \((1,2)\)

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    MLC WIR 20B M151 week9 #5


  6. If \(f(x) = 12-x-\dfrac{9}{x}\), find the interval(s) where \(f(x)\) is increasing, decreasing, concave upward, and concave downward. And find the local extrema and inflection point(s).

    Increasing: \((−3,0)\), \((0,3)\)
    Decreasing: \((-\infty,-3)\), \((3,\infty)\)
    Local Maximum: \((3,6)\)
    Local Minimum: \((-3,18)\)
    Concave Up: \((-\infty, 0)\)
    Concave Down: \((0,\infty)\)
    Inflection Point: none

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    MLC WIR 20B M151 week9 #6


  7. Consider the function \(f(x) = \dfrac{(x+2)^3}{(x-1)^2}\) for which \(f'(x)=\dfrac{(x+2)^2(x-7)}{(x-1)^3}\) and \(f''(x)=\dfrac{54(x+2)}{(x-1)^4}\). Find the interval(s) where \(f(x)\) is increasing, decreasing, concave upward, and concave downward. And find the local extrema and inflection point(s).

    Increasing: \((−\infty,-2)\), \((-2,1)\), \((7,\infty)\)
    Decreasing: \((1,7)\)
    Local Maximum: none
    Local Minimum: \(\left( 7, \frac{81}{4}\right)\)
    Concave Up: \((-2, 1)\), \((1,\infty)\)
    Concave Down: \((-\infty,-2)\)
    Inflection Point: \((-2,0)\)

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    MLC WIR 20B M151 week9 #7


  8. Evaluate
    1. \(\displaystyle \lim_{x\rightarrow \infty} \left[ \ln (2+3x)-\ln (4+5x)\right]\)

      \(\ln \left(\dfrac{3}{5}\right)\)

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      MLC WIR 20B M151 week9 #8a


    2. \(\displaystyle \lim_{x\rightarrow \infty} \left( \dfrac{2x^2}{2x+1}-\dfrac{x^2}{x+3}\right)\)

      \(\dfrac{5}{2}\)

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      MLC WIR 20B M151 week9 #8b


    3. \(\displaystyle \lim_{x\rightarrow 1} \dfrac{e^{2x-2}+x^2-2}{\ln x + 2x-2}\)

      \(\dfrac{4}{3}\)

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      MLC WIR 20B M151 week9 #8c


    4. Evaluate \(\displaystyle \lim_{x\rightarrow \infty} \left(1+x+x^2\right)^{\frac{1}{\ln x}}\)

      \(e^2\)

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      MLC WIR 20B M151 week9 #8d


    5. \(\displaystyle \lim_{x\rightarrow \infty} x\sin\left(\dfrac{\pi}{x}\right)\)

      \(\pi\)

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      MLC WIR 20B M151 week9 #8e


    6. \(\displaystyle \lim_{x\rightarrow 0} (1-5x)^{\frac{1}{x}}\)

      \(\dfrac{1}{e^5}\)

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      MLC WIR 20B M151 week9 #8f


  9. Find the number \(c\) that satisfies the conclusion of Mean Value Theorem for \(f(x)=(x-1)^{10}\) on the interval \([0,2]\).

    \(c=1\)

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    MLC WIR 20B M151 week9 #9


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

      \(f(x)=\frac{1}{6}x^6+\frac{1}{3}x^3+C\)

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      MLC WIR 20B M151 week9 #10a


    2. \(f'(x)=\dfrac{3x^4+1}{x^5}\)

      \(f(x)=3\ln|x|-\dfrac{1}{4x^4}+C\)

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      MLC WIR 20B M151 week9 #10b


    3. \(f'(x)=\sec x( \sec x + \tan x)\)

      \(f(x)=\tan x + \sec x + C\)

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      MLC WIR 20B M151 week9 #10c


    4. \(f'(x)=3x^2+4\sin x + e^x, \quad f(0)=10\)

      \(f(x)=x^3-4\cos x + e^x + 13\)

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      MLC WIR 20B M151 week9 #10d


    5. \(f''(x)=8x^3+5\), \(\quad f(1)=0\), \(\quad f'(1)=8\)

      \(f(x)=\frac{2}{5}x^5+\frac{5}{2}x^2+x-\frac{39}{10}\)

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      MLC WIR 20B M151 week9 #10e


    6. \(f''(x)=24x^2+6x+4\), \(\quad f(0)=3\), \(\quad f(1)=10\)

      \(f(x)=2x^4+x^3+2x^2+2x+3\)

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      MLC WIR 20B M151 week9 #10f


  11. Approximate the area under the graph of \(f(x) = 20−x^2\) from \(x = −2\) to \(x = 4\) using \(6\) equal width subintervals and using right endpoints.

    \(89\)

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    MLC WIR 20B M151 week9 #11


  12. Approximate the area under the graph \(f(x)=2\sqrt{x}\) from \(x=1\) to \(x=4\) using \(3\) equal width subintervals and using left endpoints.

    \(2+2\sqrt{2}+2\sqrt{3}\)

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    MLC WIR 20B M151 week9 #12


  13. Use the definition of area to find an expression for the area under the graph of \(f(x)=\dfrac{x^2}{x-3}\) in \([1,3]\) as a limit.

    \(\displaystyle A=\lim_{n\rightarrow \infty} \sum_{i=1}^n \dfrac{\left(1+\frac{2}{n}i\right)^2}{\left(1+\frac{2}{n}i\right)-3}\cdot \frac{2}{n}\)

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    MLC WIR 20B M151 week9 #13


  14. Use geometry to evalute the following integrals.
    1. \(\displaystyle \int_{-1}^2 \! \left( 1-x\right) \, dx\)

      \(\dfrac{3}{2}\)

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      MLC WIR 20B M151 week9 #14a


    2. \(\displaystyle \int_{0}^9 \! \left(\frac{1}{3}x-2\right) \, dx\)

      \(-\dfrac{9}{2}\)

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      MLC WIR 20B M151 week9 #14b