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Practice Problems for the Final Exam

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 vector and scalar projection of \(\langle 4,8 \rangle\) onto \(\langle 2,1 \rangle\).

    \( \left\langle \dfrac{32}{5},\dfrac{16}{5}\right\rangle\)

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    MLC WIR 20B M151 week10 #1

  2. Let \(x=t+1\), \(y=t^2-4\). Find the Cartesian equation.

    \(y=(x-1)^2-4\)

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    MLC WIR 20B M151 week10 #2

  3. Let \(x=2\sin \theta \), \(y=3\cos\theta\). Find the Cartesian equation.

    \( \left(\dfrac{x}{2}\right)^2+\left(\dfrac{y}{3}\right)^2=1\)

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    MLC WIR 20B M151 week10 #3

  4. Find the vector equation of the line passing through the points \((1,2)\) and \((-1,4)\).

    \( \mathbf{r}(t)=\langle 1-2t,2+2t\rangle\)

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    MLC WIR 20B M151 week10 #4

  5. Compute \(\tan \left( \arcsin \dfrac{2}{3} \right)\).

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

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    MLC WIR 20B M151 week10 #5

  6. Find the limit.
    1. \(\displaystyle \lim_{x\rightarrow -2^-} \dfrac{x-1}{x^2(x+2)}\)

      \( \dfrac{1}{16}\)
      Video Errata: In the video, the problem shows the limit as x approaches \(+2\), but the presenter will also solve the limit with \(-2\).

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      MLC WIR 20B M151 week10 #6a

    2. \(\displaystyle \lim_{x\rightarrow -3} \frac{x^2-x-12}{x+3}\)

      \(-7\)

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      MLC WIR 20B M151 week10 #6b

    3. \(\displaystyle \lim_{h\rightarrow 0} \frac{(4+h)^{-1}-4^{-1}}{h}\)

      \(-\dfrac{1}{16}\)

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      MLC WIR 20B M151 week10 #6c

  7. Let \(g(x)=\left\{ \begin{array}{ll}x^2-c^2 & \textrm{if } x<4\\ cx+20 & \textrm{if } x\geq 4\end{array}\right.\). What value(s) of \(c\) make(s) \(g(x)\) continuous everywhere?

    \(c=-2\)

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    MLC WIR 20B M151 week10 #7

  8. Find the limit.
    1. \(\displaystyle \lim_{x\rightarrow \infty}\dfrac{7x^3+4x}{2x^3-x^2+3}\)

      \(\dfrac{7}{2}\)

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      MLC WIR 20B M151 week10 #8a

    2. \(\displaystyle \lim_{x\rightarrow -\infty} \dfrac{\sqrt{x^2+4x}}{4x+1}\)

      \(-\dfrac{1}{4}\)

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      MLC WIR 20B M151 week10 #8b

    3. \(\displaystyle \lim_{x\rightarrow \infty} \dfrac{5e^{3x}-e^{-3x}}{e^{3x}+3e^{-3x}}\)

      \(5\)

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      MLC WIR 20B M151 week10 #8c

    4. \(\displaystyle \lim_{x\rightarrow -\infty} \dfrac{5e^{3x}-e^{-3x}}{e^{3x}+3e^{-3x}}\)

      \(-\dfrac{1}{3}\)

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      MLC WIR 20B M151 week10 #8d

    5. \(\displaystyle \lim_{x\rightarrow \infty} \left( \ln \left(3x^2\right)-\ln \left(6x^4-3x+1\right)\right)\)

      \(-\infty\)

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      MLC WIR 20B M151 week10 #8e

  9. Find the derivative.
    1. \(f(x)=x^3\left(4x^5+5\right)^8\)

      \(f'(x)=3x^2\left(4x^5+5\right)^8+160x^7\left(4x^5+5\right)^7\)

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      MLC WIR 20B M151 week10 #9a

    2. \(h(x)=\sqrt{\cos\left( \sin^2 x\right)}\)

      \(h'(x)=\frac{1}{2}\left(\cos \left(\left(\sin x\right)^2\right)\right)^{-\frac{1}{2}} \left(-\sin \left(\left(\sin x\right)^2\right)\right) \cdot 2(\sin x)\cos x\)

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      MLC WIR 20B M151 week10 #9b

    3. \(f(x)=\ln \left(xe^{-x}\right)\)

      \(f'(x)=\dfrac{1}{x}-1\)

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      MLC WIR 20B M151 week10 #9c

    4. \(g(x)=\dfrac{\ln x}{1-x}\)

      \(g'(x)=\dfrac{\frac{1}{x}(1-x)+\ln x}{(1-x)^2}\)

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      MLC WIR 20B M151 week10 #9d

  10. Find \(\dfrac{dy}{dx}\) for the following equations.
    1. \(e^{y}\sin x = x+xy\)

      \(\dfrac{dy}{dx}=\dfrac{1+y-e^y\cos x}{e^y\sin x- x}\)

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      MLC WIR 20B M151 week10 #10a

    2. \(x\sin y+\cos(2y)=\cos y\).

      \(\dfrac{dy}{dx}=\dfrac{-\sin y}{x\cos y-2\sin(2y)+\sin y}\)

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      MLC WIR 20B M151 week10 #10b

  11. Differentiate \(y=\dfrac{e^x\left(x^2+2\right)^3}{\left(x+1\right)^4\left(x^2+3\right)^2}\).

    \(\dfrac{dy}{dx}=\dfrac{e^x\left(x^2+2\right)^3}{\left(x+1\right)^4\left(x^2+3\right)^2} \left( 1+ \dfrac{6x}{x^2+2}-\dfrac{4}{x+1}-\dfrac{4x}{x^2+3}\right)\)

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    MLC WIR 20B M151 week10 #11

  12. If \(x = 1−t^3\) and \(y = t^2 −3t+1\), find an equation of the tangent line corresponding to \(t = 2\).

    \(y=-\dfrac{1}{12}x-\dfrac{19}{12}\)

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    MLC WIR 20B M151 week10 #12

  13. If \(x = t^3 −3t^2\) and \(y = t^3 −3t\), find all points on the curve where the tangent line is vertical or horizontal.

    Vertical: \((0,0)\), \((-4,2)\)
    Horizontal: \((-2,-2)\), \((-4,2)\)

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    MLC WIR 20B M151 week10 #13

  14. A bacteria culture starts with 1000 bacteria, and after half an hour the population is tripled.
    1. Find an expression for the number of bacteria after t hours.
    2. Find the number of bacteria after 20 minutes.​​

      1. \(y(t)=1000e^{(2\ln 3) t}\)
      2. \(y\left(\frac{1}{3}\right)=1000e^{\frac{2}{3}\ln 3}\)

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      MLC WIR 20B M151 week10 #14

  15. Two sides of triangle have length \(5\) m and \(4\) m. The angle between them is increasing at a rate of \(\pi\) rad/s. Find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length is \(\dfrac{\pi}{6}\).​​

    \(\dfrac{\sqrt{3}}{3}\pi\)

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    MLC WIR 20B M151 week10 #15

  16. Use a linear approximation to find an approximate value of \((1.99)^6\).

    \(62.08\)

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    MLC WIR 20B M151 week10 #16

  17. Given \(f(x) = 4 − x^2\), show \(f(x)\) satisfies the mean value theorem on the interval \([1,2]\) and find all \(c\) that satisfy the conclusion of the Mean Value Theorem.

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

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    MLC WIR 20B M151 week10 #17

  18. The graph of the derivative \(f'\) of a function \(f\) is shown below. 
    1. On what intervals is \(f\) increasing or decreasing?
    2. At what values of \(x\) does \(f\) have a local maximum or minimum?
    3. On what intervals is \(f\) concave upward or downward?
    4. State the \(x\)-coordinates of the points of inflection.
      151_week10_p18.png

      Increasing: \((1,6)\),\((8,\infty)\)
      Decreasing: \((−\infty,1)\),\((6,8)\)
      Local Minimum at \(x = 1, 8\)
      Local Maximum at \(x = 6\)
      Concave Upward: \((−\infty, 2)\), \((3, 5)\), \((7, \infty)\) 
      Concave Downward: \((2, 3)\), \((5, 7)\)
      Inflection Points at \(x = 2,3,5,7\)


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      MLC WIR 20B M151 week10 #18

  19. Find the limit.
    1. \(\displaystyle \lim_{x\rightarrow 0} \dfrac{\sin x - x}{x^3}\)​

      \(-\dfrac{1}{6}\)


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      MLC WIR 20B M151 week10 #19a

    2. \(\displaystyle \lim_{x\rightarrow 0^+} x\ln x\)​

      \(0\)


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      MLC WIR 20B M151 week10 #19b

    3. \(\displaystyle \lim_{x\rightarrow 1} \left( \frac{1}{\ln x} - \frac{1}{x-1}\right) \)​

      \(\dfrac{1}{2}\)


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      MLC WIR 20B M151 week10 #19c

    4. \(\displaystyle \lim_{x\rightarrow \infty} x^{3/x}\)

      \(1\)


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      MLC WIR 20B M151 week10 #19d

  20. Approximate the area under the curve \(f (x) = x^2 + 1\) on the interval \([1, 7]\) using \(3\) equal width-rectangles and using left endpoints.

    \(76\)


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    MLC WIR 20B M151 week10 #20

  21. If \(g(x)=\displaystyle \int_0^x \! f(t)\, dt\), where the graph of \(f(t)\) is given below, where \(0\leq x\leq 10\), evaluate \(g(0)\), \(g(6)\) and \(g(10)\). Where is \(g(x)\) increasing? What is the maximum value of \(g(x)\)?
    151_week10_p21.png

    \(g(0)=0\)
    \(g(3)=9\)
    \(g(6)=6\)
    \(g(10)=12\)
    Increasing: \((0,3)\), \((6,10)\)
    Maximum Value is 12 at \(t=10\). 


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    MLC WIR 20B M151 week10 #21

  22. Find the derivative of \(g(x)=\displaystyle \int_3^{\sin x} \frac{\cos t}{t} \, dt\).

    \(g'(x)=\dfrac{\cos(\sin(x))}{\sin x} \cos x\)


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    MLC WIR 20B M151 week10 #22

  23. Find the general indefinite integral.
    1. \(\displaystyle \int \! \dfrac{1+\sqrt{x}+x}{x} \, dx\)

      \( \ln |x|+2x^{\frac{1}{2}} + x + C\)


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      MLC WIR 20B M151 week10 #23a

    2. \(\displaystyle \int \! \left(2x^3-6x+\dfrac{3}{x^2+1}\right) \, dx\)

      \( \frac{1}{2}x^4-3x^2+3\arctan x + C\)


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      MLC WIR 20B M151 week10 #23b

  24. Evaluate
    1. \(\displaystyle \int_{-1}^0 \left(5x^2-4x+3\right)\, dx\)

      \(\dfrac{20}{3}\)


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      MLC WIR 20B M151 week10 #24a

    2. \(\displaystyle \int_{1}^2 \! \dfrac{x^6-x^2}{x^7}\, dx\)

      \(\ln(2)-\dfrac{15}{64}\)


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      MLC WIR 20B M151 week10 #24b

    3. \(\displaystyle \int_{0}^\pi \! \left( 5e^x+3\sin x\right)\, dx\)

      \(5e^\pi +1\)


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      MLC WIR 20B M151 week10 #24c

    4. \(\displaystyle \int \! x^3\cos \left(x^4+2\right) \, dx\)

      \( \dfrac{1}{4}\sin\left(x^4+2\right)+C\)


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      MLC WIR 20B M151 week10 #24d

    5. \(\displaystyle \int_1^e \! \dfrac{\ln x}{x}\, dx\)

      \( \dfrac{1}{2}\)


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      MLC WIR 20B M151 week10 #24e