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Practice Problems for Module 2

Sections J.3, 1.5, and 2.2

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. Eliminate the parameter to find the Cartesian equation of the curve. Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases.
    1. ​\(x=2t−1,y=2−t,\quad −2≤t≤2.\)
    2. \(x = 2t − 1, y = t^2 − 1\)
    3. \(x=3\cosθ, y=4\sinθ, \quad 0≤θ≤2\pi\)

      a) \( y=-\dfrac{1}{2}x+\dfrac{3}{2}\)

      b) \(y=\dfrac{1}{4} (x+1)^2-1\)

      c) \(\left(\dfrac{x}{3}\right)^2 + \left( \dfrac{y}{4}\right)^2=1\)

      To see the full video page and find related videos, click the following link.
      MLC WIR 20B M151 week2 #01


  2. Find a vector equation for the line passing through \((1, 3)\) and \((−2, 7)\). 

    \(\langle 1-3t, 3+4t\rangle\)

    To see the full video page and find related videos, click the following link.
    MLC WIR 20B M151 week2 #02


  3. Find the exact value of the expression.
    1. \(\sin \left( \arccos \dfrac{4}{5}\right)\)
    2. \(\sin \left( 2 \sin^{-1} \dfrac{3}{5}\right)\) 

      a) \(\dfrac{3}{5}\)
      b) \(\dfrac{24}{25}\)

      To see the full video page and find related videos, click the following link.
      MLC WIR 20B M151 week2 #03


  4. Simplify each expression
    1. \(\tan \left( \sin^{-1} x\right)\)
    2. \(\sin (\arctan x)\)​

      a) \(\dfrac{x}{\sqrt{1-x^2}}\)
      b) \(\dfrac{x}{\sqrt{x^2+1}}\)

      To see the full video page and find related videos, click the following link.
      MLC WIR 20B M151 week2 #04


  5. State the value of the given quantity, if it exists, from the given graph
    PracticeProblems2_p5.png
    1. ​\(\displaystyle \lim_{x\rightarrow 0^-} g(x)\)
    2. ​\(\displaystyle \lim_{x\rightarrow 0^+} g(x)\)
    3. ​\(\displaystyle \lim_{x\rightarrow 0} g(x)\)
    4. ​\(\displaystyle \lim_{x\rightarrow 2^-} g(x)\)
    5. ​\(\displaystyle \lim_{x\rightarrow 2^+} g(x)\)
    6. ​\(\displaystyle \lim_{x\rightarrow 2} g(x)\)
    7. ​\(g(2)\)
    8. ​\(\displaystyle \lim_{x\rightarrow -1^-} g(x)\)
    9. ​\(\displaystyle \lim_{x\rightarrow -1^+} g(x)\)
    10. ​\(\displaystyle \lim_{x\rightarrow -1} g(x)\)

      1. \(\displaystyle \lim_{x\rightarrow 0^-} g(x)=+\infty\)
      2. ​\(\displaystyle \lim_{x\rightarrow 0^+} g(x)=-\infty\)
      3. ​\(\displaystyle \lim_{x\rightarrow 0} g(x) \quad DNE\)
      4. ​\(\displaystyle \lim_{x\rightarrow 2^-} g(x)=1\)
      5. ​\(\displaystyle \lim_{x\rightarrow 2^+} g(x)=1\)
      6. ​\(\displaystyle \lim_{x\rightarrow 2} g(x)=1\)
      7. ​\(g(2)=1.5\)
      8. ​\(\displaystyle \lim_{x\rightarrow -1^-} g(x)=1\)
      9. ​\(\displaystyle \lim_{x\rightarrow -1^+} g(x)=0\)
      10. ​\(\displaystyle \lim_{x\rightarrow -1} g(x) \quad DNE\)
      Video coming soon
       


  6. Find the limit.
    1. \(\displaystyle \lim_{x\rightarrow 3} \frac{1}{(x-3)^8}\)
    2. \(\displaystyle \lim_{x\rightarrow 0} \dfrac{x-1}{x^2(x+2)}\)
    3. \(\displaystyle \lim_{x\rightarrow -2^+} \dfrac{x-1}{x^2(x+2)}\)
    4. \(\displaystyle \lim_{x\rightarrow -2^-} \dfrac{x-1}{x^2(x+2)}\)
    5. \(\displaystyle \lim_{x\rightarrow -2} \dfrac{x-1}{x^2(x+2)}\)​

      1. \(+\infty\)
      2. \(-\infty\)
      3. \(-\infty\)
      4. \(+\infty\)
      5. DNE

      To see the full video page and find related videos, click the following link.
      MLC WIR 20B M151 week2 #06