# Practice Problems for Exam 3

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1. The graph below is that of the derivative, $$f'(x)$$, of a continous function $$f(x)$$. Find the interval(s) where $$f(x)$$ is increasing, decreasing, concave upward, concave downward, and find the $$x$$-values of all local extrema and inflection points.

Increasing: $$(−2, 0)$$, $$(0, 2)$$
Decreasing: $$(−\infty, −2)$$, $$(2, \infty)$$
Local Min at $$x = −2$$,
Local Max at $$x = 2$$,
Concave Up: $$(−\infty, −1.4)$$, $$(0, 1.4)$$,
Concave Down: $$(−1.4, 0)$$, $$(1.4, \infty)$$,
Inflection  Points at $$x = −1.4, 0, 1.4$$

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2. Find the absolute maximum and minimum values for $$f(x)=x^2+\dfrac{2}{x}$$ over the interal $$\left[\frac{1}{2},2\right]$$.

Absolute Maximum is 5
Absolute Minimum is 3

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3. If $$f'(x) = (x − 2)^3(4 − x)^5(x + 1)^2$$, find the interval(s) where $$f(x)$$ is increasing and decreasing.

Increasing: $$(2,4)$$
Decreasing: $$(-\infty, -1)$$, $$(-1,2)$$, $$(4,\infty)$$

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4. If $$f(x)=\dfrac{x^2}{x-1}$$, find the interval(s) where $$f(x)$$ is increasing and decreasing, and find the
$$x$$-value(s) of any local maximum and minimum.

Increasing: $$(−\infty,0)$$, $$(2,\infty)$$
Decreasing: $$(0,1)$$, $$(1,2)$$
Local Maximum at $$x = 0$$
Local Minimum at $$x = 2$$

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5. If $$f(x) = x^4 − 6x^2 + 4$$, find the interval(s) where $$f(x)$$ is increasing, decreasing, concave upward, and concave downward. And find the local extrema and inflection point(s).

Increasing: $$(−\sqrt{3},0)$$, $$(\sqrt{3},\infty)$$
Decreasing: $$(-\infty,-\sqrt{3})$$, $$(0,\sqrt{3})$$
Local Maximum: $$(0,4)$$
Local Minimum: $$(-\sqrt{3},-5)$$, $$(\sqrt{3},-5)$$
Concave Up: $$(-\infty, -1)$$, $$(1,\infty)$$
Concave Down: $$(-1,1)$$
Inflection Point: $$(-1,2)$$, $$(1,2)$$

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6. If $$f(x) = 12-x-\dfrac{9}{x}$$, find the interval(s) where $$f(x)$$ is increasing, decreasing, concave upward, and concave downward. And find the local extrema and inflection point(s).

Increasing: $$(−3,0)$$, $$(0,3)$$
Decreasing: $$(-\infty,-3)$$, $$(3,\infty)$$
Local Maximum: $$(3,6)$$
Local Minimum: $$(-3,18)$$
Concave Up: $$(-\infty, 0)$$
Concave Down: $$(0,\infty)$$
Inflection Point: none

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7. Consider the function $$f(x) = \dfrac{(x+2)^3}{(x-1)^2}$$ for which $$f'(x)=\dfrac{(x+2)^2(x-7)}{(x-1)^3}$$ and $$f''(x)=\dfrac{54(x+2)}{(x-1)^4}$$. Find the interval(s) where $$f(x)$$ is increasing, decreasing, concave upward, and concave downward. And find the local extrema and inflection point(s).

Increasing: $$(−\infty,-2)$$, $$(-2,1)$$, $$(7,\infty)$$
Decreasing: $$(1,7)$$
Local Maximum: none
Local Minimum: $$\left( 7, \frac{81}{4}\right)$$
Concave Up: $$(-2, 1)$$, $$(1,\infty)$$
Concave Down: $$(-\infty,-2)$$
Inflection Point: $$(-2,0)$$

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8. Evaluate
1. $$\displaystyle \lim_{x\rightarrow \infty} \left[ \ln (2+3x)-\ln (4+5x)\right]$$

$$\ln \left(\dfrac{3}{5}\right)$$

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2. $$\displaystyle \lim_{x\rightarrow \infty} \left( \dfrac{2x^2}{2x+1}-\dfrac{x^2}{x+3}\right)$$

$$\dfrac{5}{2}$$

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3. $$\displaystyle \lim_{x\rightarrow 1} \dfrac{e^{2x-2}+x^2-2}{\ln x + 2x-2}$$

$$\dfrac{4}{3}$$

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4. Evaluate $$\displaystyle \lim_{x\rightarrow \infty} \left(1+x+x^2\right)^{\frac{1}{\ln x}}$$

$$e^2$$

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5. $$\displaystyle \lim_{x\rightarrow \infty} x\sin\left(\dfrac{\pi}{x}\right)$$

$$\pi$$

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6. $$\displaystyle \lim_{x\rightarrow 0} (1-5x)^{\frac{1}{x}}$$

$$\dfrac{1}{e^5}$$

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9. Find the number $$c$$ that satisfies the conclusion of Mean Value Theorem for $$f(x)=(x-1)^{10}$$ on the interval $$[0,2]$$.

$$c=1$$

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10. Find $$f(x)$$.
1. $$f'(x)=x^2\left(x^3+1\right)$$

$$f(x)=\frac{1}{6}x^6+\frac{1}{3}x^3+C$$

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2. $$f'(x)=\dfrac{3x^4+1}{x^5}$$

$$f(x)=3\ln|x|-\dfrac{1}{4x^4}+C$$

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3. $$f'(x)=\sec x( \sec x + \tan x)$$

$$f(x)=\tan x + \sec x + C$$

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4. $$f'(x)=3x^2+4\sin x + e^x, \quad f(0)=10$$

$$f(x)=x^3-4\cos x + e^x + 13$$

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5. $$f''(x)=8x^3+5$$, $$\quad f(1)=0$$, $$\quad f'(1)=8$$

$$f(x)=\frac{2}{5}x^5+\frac{5}{2}x^2+x-\frac{39}{10}$$

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6. $$f''(x)=24x^2+6x+4$$, $$\quad f(0)=3$$, $$\quad f(1)=10$$

$$f(x)=2x^4+x^3+2x^2+2x+3$$

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11. Approximate the area under the graph of $$f(x) = 20−x^2$$ from $$x = −2$$ to $$x = 4$$ using $$6$$ equal width subintervals and using right endpoints.

$$89$$

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12. Approximate the area under the graph $$f(x)=2\sqrt{x}$$ from $$x=1$$ to $$x=4$$ using $$3$$ equal width subintervals and using left endpoints.

$$2+2\sqrt{2}+2\sqrt{3}$$

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13. Use the definition of area to find an expression for the area under the graph of $$f(x)=\dfrac{x^2}{x-3}$$ in $$[1,3]$$ as a limit.

$$\displaystyle A=\lim_{n\rightarrow \infty} \sum_{i=1}^n \dfrac{\left(1+\frac{2}{n}i\right)^2}{\left(1+\frac{2}{n}i\right)-3}\cdot \frac{2}{n}$$

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14. Use geometry to evalute the following integrals.
1. $$\displaystyle \int_{-1}^2 \! \left( 1-x\right) \, dx$$

$$\dfrac{3}{2}$$

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2. $$\displaystyle \int_{0}^9 \! \left(\frac{1}{3}x-2\right) \, dx$$

$$-\dfrac{9}{2}$$

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