## Educators

### Problem 1

If $f(x)=5 x^{\frac{4}{3}},$ then $f^{\prime}(8)=$
(A) 10
(B) $\frac{40}{3}$
(C) 80
(D) $\frac{160}{3}$

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### Problem 2

$$\lim _{x \rightarrow \infty} \frac{5 x^{2}-3 x+1}{4 x^{2}+2 x+5}$$
(A) 0
(B) $\frac{4}{5}$
(C) $\frac{5}{4}$
(D) $\infty$

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### Problem 3

If $f(x)=\frac{3 x^{2}+x}{3 x^{2}-x},$ then $f^{\prime}(x)$ is
(A) 1
(B) $\frac{6 x^{2}+1}{6 x^{2}-1}$
(C) $\frac{-6}{(3 x-1)^{2}}$
(D) $\frac{-2 x^{2}}{\left(x^{2}-x\right)^{2}}$

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### Problem 4

$$\lim _{x \rightarrow 0} \frac{\sin x^{2}}{x}=$$
(A) 1
(B) 0
(C) $\frac{\pi}{2}$
(D) Does Not Exist

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### Problem 5

If $x^{2}-2 x y+3 y^{2}=8,$ then $\frac{d y}{d x}=$
(A) $\frac{8+2 y-2 x}{6 y-2 x}$
(B) $\frac{3 y-x}{y-x}$
(C) $\frac{1}{3}$
(D) $\frac{y-x}{3 y-x}$

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### Problem 6

Which of the following integrals correctly corresponds to the area of the shaded region in the figure
above?
(A) $\int_{1}^{2}\left(x^{2}-4\right) d x$
(B) $\int_{1}^{2}\left(4-x^{2}\right) d x$
(C) $\int_{1}^{5}\left(x^{2}-4\right) d x$
(D) $\int_{1}^{5}\left(4-x^{2}\right) d x$

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### Problem 7

If $f(x)=\sec x+\csc x,$ then $f^{\prime}(x)=$
(A) 0
(B) $\csc x-\sec x$
(C) $\sec x \tan x+\csc x \cot x$
(D) $\sec x \tan x-\csc x \cot x$

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### Problem 8

An equation of the line normal to the graph of $y=\sqrt{\left(3 x^{2}+2 x\right)}$ at $(2,4)$ is
(A) $4 x+7 y=20$
(B) $-7 x+4 y=2$
(C) $7 x+4 y=30$
(D) $4 x+7 y=36$

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### Problem 9

$$\int_{-1}^{1} \frac{4}{1+x^{2}} d x=$$
$\begin{array}{ll}{(\mathrm{A})} & {0} \\ {\text { (B) }} & {\pi}\end{array}$
$\begin{array}{ll}{\text { (C) }} & {2 \pi} \\ {\text { (D) }} & {2}\end{array}$

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### Problem 10

If $f(x)=\cos ^{2} x,$ then $f^{\prime \prime}(\pi)=$
$\begin{array}{ll}{\text { (A) }} & {-2} \\ {\text { (B) }} & {0} \\ {\text { (C) }} & {1} \\ {\text { (D) }} & {2}\end{array}$

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### Problem 11

If $f(x)=\frac{5}{x^{2}+1}$ and $g(x)=3 x,$ then $g(f(2))=$
$\begin{array}{ll}{\text { (A) }} & {\frac{5}{37}} \\ {\text { (B) }} & {3} \\ {\text { (C) }} & {5} \\ {\text { (D) }} & {\frac{37}{5}}\end{array}$

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### Problem 12

$$\int x \sqrt{5 x^{2}-4} d x=$$
(A) $\frac{1}{10}\left(5 x^{2}-4\right)^{\frac{3}{2}}+C$
(B) $\frac{1}{15}\left(5 x^{2}-4\right)^{\frac{3}{2}}+C$
(C) $\frac{20}{3}\left(5 x^{2}-4\right)^{\frac{3}{2}}+C$
(D) $\frac{3}{20}\left(5 x^{2}-4\right)^{\frac{3}{2}}+C$

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### Problem 13

The slope of the line tangent to the graph of $3 x^{2}+5 \ln y=12$ at $(2,1)$ is
(A) $-\frac{12}{5}$
(B) $\frac{12}{5}$
(C) $\frac{5}{12}$
(D) $-7$

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### Problem 14

The equation $y=2-3 \sin \frac{\pi}{4}(x-1)$ has a fundamental period of
(A) $\frac{1}{8}$
(B) $\frac{4}{\pi}$
(C) 8
(D) 2$\pi$

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### Problem 15

If $f(x)=\left\{\begin{array}{l}{x^{2}+5 \text { if } x<2} \\ {7 x-5 \text { if } x \geq 2}\end{array}, \text { for all real numbers } x, \text { which of the following must be true? }\right.$
$\begin{array}{ll}{\text { I. }} & {f(x) \text { is continuous everywhere }} \\ {\text { II. }} & {f(x) \text { is differentiable everywhere. }} \\ {\text { III. }} & {f(x) \text { has a local minimum at } x=2}\end{array}$
(A) I only
(B) I and II only
(C) II and III only
(D) I, II, and III

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### Problem 16

For what value of $x$ does the function $f(x)=x^{3}-9 x^{2}-120 x+6$ have a local minimum?
(A) 10
(B) 4
(C) $-4$
(D) $-10$

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### Problem 17

The acceleration of a particle moving along the $x$ -axis at time $t$ is given by $a(t)=4 t-12$ . If the
velocity is 10 when $t=0$ and the position is 4 when $t=0,$ then the particle is changing direction at
$\begin{array}{ll}{\text { (A) }} & {t=1} \\ {\text { (B) } t} & {=3} \\ {\text { (C) } t} & {=5} \\ {\text { (D) } t} & {=1 \text { and } t=5}\end{array}$

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### Problem 18

The average value of the function $f(x)=(x-1)^{2}$ on the interval from $x=1$ to $x=5$ is
(A) $\frac{16}{3}$
(B) $\frac{64}{3}$
(C) $\frac{66}{3}$
(D) $\frac{256}{3}$

Maruti S.

### Problem 19

$$\int\left(e^{3 \ln x}+e^{3 x}\right) d x=$$
(A) $3+\frac{e^{3 x}}{3}+C$
(B) $\frac{e^{x^{4}}}{4}+3 e^{3 x}+C$
(C) $\frac{e^{x^{4}}}{4}+\frac{e^{3 x}}{3}+C$
(D) $\frac{x^{4}}{4}+\frac{e^{3 x}}{3}+C$

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### Problem 20

If $f(x)=\left(x^{2}+x+11\right) \sqrt{\left(x^{3}+5 x+121\right)},$ then $f(0)=$
(A) $\frac{5}{2}$
(B) $\frac{27}{2}$
(C) 22
(D) $\frac{247}{2}$

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### Problem 21

If $f(x)=5^{3 x},$ then $f^{\prime}(x)=$
(A) $5^{3 x}(\ln 125)$
(B) $\frac{5^{3 x}}{3 \ln 5}$
(C) 3$\left(5^{2 x}\right)$
(D) 3$\left(5^{3 x}\right)$

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### Problem 22

A solid is generated when the region in the first quadrant enclosed by the graph of $y=\left(x^{2}+1\right)^{3}$ , the line $x=1,$ the $x$ -axis, and the $y$ -axis is revolved about the $x$ -axis. Its volume is found by evaluating which of the following integrals?
(A) $\pi \int_{1}^{8}\left(x^{2}+1\right)^{3} d x$
(B) $\pi \int_{1}^{8}\left(x^{2}+1\right)^{6} d x$
(C) $\pi \int_{0}^{1}\left(x^{2}+1\right)^{3} d x$
(D) $\pi \int_{0}^{1}\left(x^{2}+1\right)^{6} d x$

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### Problem 23

$$\lim _{x \rightarrow 0} 4 \frac{\sin x \cos x-\sin x}{x^{2}}=$$
(A) 2
(B) $\frac{40}{3}$
(C) 0
(D) undefined

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### Problem 24

If $\frac{d y}{d x}=\frac{\left(3 x^{2}+2\right)}{y}$ and $y=4$ when $x=2,$ then when $x=3, y=$
(A) 18
(B) 58
(C) $\pm \sqrt{74}$
(D) $\pm \sqrt{58}$

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### Problem 25

$$\int \frac{d x}{9+x^{2}}=$$
(A) $3 \tan ^{-1}\left(\frac{x}{3}\right)+C$
(B) $\frac{1}{3} \tan ^{-1}\left(\frac{x}{3}\right)+C$
(C) $\frac{1}{3} \tan ^{-1}(x)+C$
(D) $\frac{1}{9} \tan ^{-1}(x)+C$

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### Problem 26

If $f(x)=\cos ^{3}(x+1),$ then $f^{\prime}(\pi)=$
(A) $-3 \cos ^{2}(\pi+1) \sin (\pi+1)$
(B) 3 $\cos ^{2}(\pi+1)$
(C) 3 $\cos ^{2}(\pi+1) \sin (\pi+1)$
(D) 0

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### Problem 27

$$\int x \sqrt{x+3} d x=$$
(A) $\frac{2(x+3)^{\frac{3}{2}}}{3}+C$
(B) $\frac{2}{5}(x+3)^{\frac{5}{2}}-2(x+3)^{\frac{3}{2}}+C$
(C) $\frac{3(x+3)^{\frac{3}{2}}}{2}+C$
(D) $\quad \frac{4 x^{2}(x+3)^{\frac{3}{2}}}{3}+C$

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### Problem 28

If $f(x)=\ln (\ln (1-x)),$ then $f^{\prime}(x)=$
(A) $-\frac{1}{\ln (1-x)}$
(B) $\frac{1}{(1-x) \ln (1-x)}$
(C) $\frac{1}{(1-x)^{2}}$
(D) $-\frac{1}{(1-x) \ln (1-x)}$

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### Problem 29

$$\lim _{h \rightarrow 0} \frac{\tan \left(\frac{\pi}{6}+h\right)-\tan \left(\frac{\pi}{6}\right)}{h}=$$
$\begin{array}{ll}{\text { (A) }} & {\frac{4}{3}} \\ {\text { (B) }} & {\sqrt{3}} \\ {\text { (C) }} & {0} \\ {\text { (D) }} & {\frac{3}{4}}\end{array}$

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### Problem 30

$$\int \operatorname{tab}^{6} x \sec ^{2} x d x=$$
(A) $\frac{\tan ^{7} x}{7}+C$
(B) $\frac{\tan ^{7} x}{7}+\frac{\sec ^{3} x}{3}+C$
(C) $\frac{\tan ^{7} x \sec ^{3} x}{21}+C$
(D) $7 \tan ^{7} x+C$

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