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

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