## Educators

### Problem 1

$\int_{\frac{\pi}{4}}^{x} \cos (2 t) d t=$
(A) $\cos (2 x)$
(B) $\frac{\sin (2 x)-1}{2}$
(C) $\cos (2 x)-1$
(D) $\frac{\sin 2(x)}{2}$

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### Problem 2

What are the coordinates of the point of inflection on the graph of $y=x^{3}-15 x^{2}+33 x+100 ?$
(A) $\quad(9,0)$
(B) $(5,-48)$
(C) $(9,-89)$
(D) $\quad(5,15)$

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### Problem 3

If $3 x^{2}-2 x y+3 y=1,$ then when $x=2, \frac{d y}{d x}=$
(A) $\quad-12$
(B) $\quad-10$
(C) $-\frac{10}{7}$
(D) 12

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### Problem 4

$\int_{1}^{3} \frac{8}{x^{3}} d x=$
(A) $\frac{32}{9}$
(B) $\frac{40}{9}$
(C) 0
(D) $-\frac{32}{9}$

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### Problem 5

The graph of a piecewise linear function $f,$ for $0 \leq x \leq 8,$ is shown above. What is the value of $\int_{0}^{8} f(x) d x ?$
(A) 1
(B) 4
(C) 8
(D) 10

Sarah B.

### Problem 6

$\lim _{x \rightarrow 0} \frac{x-\sin x}{x^{3}}$
(A) 0
(B) 1
(C) 2
(D) Does not exist

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### Problem 7

If $f(x)=x^{2} \sqrt{3 x+1},$ then $f^{\prime}(x)=$
(A) $\frac{9 x^{2}+2 x}{\sqrt{3 x+1}}$
(B) $\frac{-9 x^{2}+4 x}{2 \sqrt{3 x+1}}$
(C) $\frac{15 x^{2}+4 x}{2 \sqrt{3 x+1}}$
(D) $\frac{-9 x^{2}-4 x}{2 \sqrt{3 x+1}}$

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### Problem 8

What is the instantaneous rate of change at $t=-1$ of the function $f,$ if $f(t)=\frac{t^{3}+t}{4 t+1} ?$
(A) $\frac{12}{9}$
(B) $\frac{4}{9}$
(C) $-\frac{4}{9}$
(D) $-\frac{12}{9}$

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### Problem 9

$\int_{2}^{e+1}\left(\frac{4}{x-1}\right) d x=$
(A) 4
(B) 4$e$
(C) 0
(D) $-4$

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### Problem 10

A car's velocity is shown on the graph above. Which of the following gives the total distance traveled from $t=0$ to $t=16$ (in kilometers)?
(A) $\quad 360$
(B) 390
(C) 780
(D) $1,000$

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### Problem 11

$\frac{d}{d x} \tan ^{2}(4 x)=$
(A) 8 $\tan (4 x)$
(B) 4 $\sec ^{4}(4 x)$
(C) 8 $\tan (4 x) \sec ^{2}(4 x)$
(D) 4 $\tan (4 x) \sec ^{2}(4 x)$

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### Problem 12

What is the equation of the line tangent to the graph of $y=\sin ^{2} x$ at $x=\frac{\pi}{4} ?$
(A) $y-\frac{1}{2}=\left(x-\frac{\pi}{4}\right)$
(B) $y-\frac{1}{\sqrt{2}}=\left(x-\frac{\pi}{4}\right)$
(C) $y-\frac{1}{\sqrt{2}}=\frac{1}{2}\left(x-\frac{\pi}{4}\right)$
(D) $\quad y-\frac{1}{2}=\frac{1}{2}\left(x-\frac{\pi}{4}\right)$

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### Problem 13

If the function $f(x)=\left\{\begin{array}{l}{3 a x^{2}+2 b x+1 ; x \leq 1} \\ {a x^{4}-4 b x^{2}-3 x ; x>1}\end{array}\right.$
(A) $-\frac{11}{4}$
(B) $\frac{1}{4}$
(C) 0
(D) $-\frac{1}{4}$

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### Problem 14

The graph of $y=x^{4}+8 x^{3}-72 x^{2}+4$ is concave down for
(A) $-6<x<2$
(B) $x>2$
(C) $x<-6$
(D) $-3-3 \sqrt{5}<x<-3+3 \sqrt{5}$

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### Problem 15

$\lim _{x \rightarrow \infty} \frac{\ln (x+1)}{\log _{2} x}$
$\begin{array}{ll}{\text { (A) }} & {\frac{1}{\ln 2}} \\ {\text { (B) }} & {0} \\ {\text { (C) }} & {1} \\ {\text { (D) }} & {\ln 2}\end{array}$

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### Problem 16

The graph of $f(x)$ is shown in the figure above. Which of the following could be the graph of $f^{\prime}(x) ?$
A) graph IS NOT AVAILABLE TO COPY
B) graph IS NOT AVAILABLE TO COPY
C) graph IS NOT AVAILABLE TO COPY
D) graph IS NOT AVAILABLE TO COPY

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### Problem 17

If $f(x)=\ln (\cos (3 x)),$ then $f^{\prime}(x)=$
$\begin{array}{ll}{\text { (A) }} & {3 \sec (3 x)} \\ {\text { (B) }} & {3 \tan (3 x)} \\ {\text { (C) }} & {-3 \tan (3 x)} \\ {\text { (D) }} & {-3 \cot (3 x)}\end{array}$

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### Problem 18

If $f(x)=\int_{0}^{x+1} \sqrt[3]{t^{2}-1},$ then $f^{\prime}(-4)=$
(A) $-2$
(B) 2
(C) $\quad \sqrt[3]{15}$
(D) 0

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### Problem 19

A particle moves along the $x$ -axis so that its position at time $t,$ in seconds, is given by $x(t)=t^{2}-7 t+$ $6 .$ For what value(s) of $t$ is the velocity of the particle zero?
$\begin{array}{ll}{\text { (A) }} & {1} \\ {\text { (B) }} & {6} \\ {\text { (C) }} & {1 \text { or } 6} \\ {\text { (D) }} & {3.5}\end{array}$

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### Problem 20

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

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### Problem 21

The average value of $\sec ^{2} x$ on the interval $\left[\frac{\pi}{6}, \frac{\pi}{4}\right]$ is
(A) $\frac{12 \sqrt{3}-12}{\pi}$
(B) $\frac{12-4 \sqrt{3}}{\pi}$
(C) $\frac{6 \sqrt{2}-6}{\pi}$
(D) $\frac{6-6 \sqrt{2}}{\pi}$

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### Problem 22

Find the area of the region bounded by the parabolas $y=x^{2}$ and $y=6 x-x^{2}$
(A) 9
(B) 27
(C) $-9$
(D) $-18$

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### Problem 23

The function $f$ is given by $f(x)=x^{4}+4 x^{3} .$ On which of the following intervals is $f$ decreasing?
(A) $\quad(-3,0)$
(B) $(0, \infty)$
(C) $(-3, \infty)$
(D) $(-\infty,-3)$

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### Problem 24

$\lim _{x \rightarrow 0} \frac{\tan (3 x)+3 x}{\sin (5 x)}=$
(A) 0
(B) $\frac{3}{5}$
(C) $\frac{6}{5}$
(D) Nonexistent

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### Problem 25

If the region enclosed by the $y$ -axis, the curve $y=4 \sqrt{x},$ and the line $y=8$ is revolved about the $x-$ axis, the volume of the solid generated is
$\begin{array}{ll}{\text { (A) }} & {\frac{32 \pi}{3}} \\ {\text { (B) }} & {128 \pi} \\ {\text { (C) }} & {\frac{128}{3}} \\ {\text { (D) }} & {\frac{128 \pi}{3}}\end{array}$

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### Problem 26

The maximum velocity attained on the interval $0 \leq t \leq 5,$ by the particle whose displacement is given by $s(t)=2 t^{3}-12 t^{2}+16 t+2$ is
$\begin{array}{ll}{(\mathrm{A})} & {286} \\ {\text { (B) }} & {46} \\ {\text { (C) }} & {16} \\ {\text { (D) }} & {0}\end{array}$

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### Problem 27

The value of $c$ that satisfies the Mean Value Theorem for derivatives on the interval $[0,5]$ for the function $f(x)=x^{3}-6 x$ is
(A) 0
(B) 1
(C) $\frac{5}{3}$
(D) $\frac{5}{\sqrt{3}}$

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### Problem 28

If $f(x)=\sec (4 x),$ then $f\left(\frac{\pi}{16}\right)$ is
(A) 4$\sqrt{2}$
(B) $\sqrt{2}$
(C) $\frac{1}{\sqrt{2}}$
(D) $\frac{4}{\sqrt{2}}$

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### Problem 29

$\frac{d}{d x} \int_{2 x}^{5 x} \cos t d t=$
(A) $5 \cos 5 x-2 \cos 2 x$
(B) $5 \sin 5 x-2 \sin 2 x$
(C) $\cos 5 x-\cos 2 x$
(D) $\sin 5 x-\sin 2 x$

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### Problem 30

$\quad$ Let $g(x)=\int_{0}^{x} f(t) d t$ where $f(t)$ has the graph shown above. Which of the following could be the graph of $g ?$
A) graph IS NOT AVAILABLE TO COPY
B) graph IS NOT AVAILABLE TO COPY
C) graph IS NOT AVAILABLE TO COPY
D) graph IS NOT AVAILABLE TO COPY

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