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

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

$$\langle 1-3t, 3+4t\rangle$$

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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) $$\dfrac{24}{25}$$

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4. Simplify each expression

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

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

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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

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