Scroll to Top

Virtual Math Learning Center

Virtual Math Learning Center Texas A&M University Virtual Math Learning Center

Practice Problems for Module 2


Exercises

Directions: You should try to solve each problem first, and then click "Reveal Answer" to check your answer. You can click "Watch Video" if you need help with a problem.

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

If you would like to see more videos on this topic, click the following link and check the related videos.

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

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

If you would like to see more videos on this topic, click the following link and check the related videos.

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) \(\dfrac{24}{25}\)

If you would like to see more videos on this topic, click the following link and check the related videos.

4. Simplify each expression

  1. \(\tan \left( \sin^{-1} x\right)\)
  2. \(\sin (\arctan x)\)

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

If you would like to see more videos on this topic, click the following link and check the related videos.

5. State the value of the given quantity, if it exists, from the given graph

  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. ​\(\displaystyle \lim_{x\rightarrow -1} g(x) \quad DNE\)

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

If you would like to see more videos on this topic, click the following link and check the related videos.