# Practice Problems for Module 10

Sections 4.2, 4.3, and 4.4

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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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