 # Practice Problems for Module 10

Sections 4.2, 4.3, and 4.4

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 number $$c$$ that satisfies the conclusion of the Mean Value Theorem for the function on the interval.
1. $$f(x)=2x^2-3x+1$$, $$\quad [0,2]$$.

$$c=1$$

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2. $$f(x)=\ln x$$, $$\quad [1,4]$$

$$c=\dfrac{3}{\ln 4}$$

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2. Sketch a curve satisfying the following conditions.
1. The domain of $$f(x)$$ is all real numbers.
2. $$f(2)=-2$$, $$f(0)=0$$, $$f(4)=0$$, $$f'(2)=0$$
3. $$f'(x)<0$$ if $$0<x<2$$, $$f'(x)>0$$ if $$x>2$$.
4. $$f''(x)<0$$ if $$0\leq x <1$$ of if $$x>4$$
5. $$f''(x)>0$$ if $$1<x<4$$
6. $$\displaystyle \lim_{x\rightarrow \infty} f(x)=2$$
7. The graph of $$f(x)$$ is symmetric about the $$y$$-axis

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3. Find the limit.
1. $$\displaystyle \lim_{x\rightarrow -2} \frac{x^3+8}{x+2}$$​

$$12$$

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2. $$\displaystyle \lim_{x\rightarrow \pi/2} \frac{1-\sin x}{1+\cos 2x}$$​

$$\dfrac{1}{4}$$

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3. $$\displaystyle \lim_{x\rightarrow \infty} \frac{\ln \sqrt{x}}{x^2}$$​

$$0$$

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4. $$\displaystyle \lim_{x\rightarrow 0} \frac{e^x-1-x}{x^2}$$​

$$\dfrac{1}{2}$$

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5. $$\displaystyle \lim_{x\rightarrow \infty} x^3e^{-x^2}$$​

$$0$$

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6. $$\displaystyle \lim_{x\rightarrow0^+} \left( \dfrac{1}{x} - \dfrac{1}{e^x-1}\right)$$​

$$\dfrac{1}{2}$$

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7. $$\displaystyle \lim_{x\rightarrow 1^+} x^{1/(1-x)}$$​

$$\dfrac{1}{e}$$

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8. $$\displaystyle \lim_{x\rightarrow 0^+} (4x+1)^{\cot x}$$​

$$e^4$$

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4. Sketch the graph of $$f(x)=\dfrac{x}{x^2-4}$$ by locating intervals of increase/decrease, local extrema, concavity, and inflection points.

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