## Educators

Problem 1

Directions: Solve each of the following problems, using the available space for scratchwork. After examining the form of the choices, decide which is the best of the choices given and fill in the corresponding oval on the answer sheet. No credit will be given for anything written in the test book. Do not spend too much time on any one problem.
In this test: Unless otherwise specified, the domain of a function $f$ is assumed to be the set of all real numbers $x$ for which $f(x)$ is a real number.
If $g(x)=\frac{1}{32} x^{4}-5 x^{2},$ find $g^{\prime}(4)$
$\begin{array}{ll}{\text { (A) }} & {-72} \\ {\text { (B) }} & {-32} \\ {\text { (C) }} & {24} \\ {\text { (D) }} & {32}\end{array}$

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Problem 2

Directions: Solve each of the following problems, using the available space for scratchwork. After examining the form of the choices, decide which is the best of the choices given and fill in the corresponding oval on the answer sheet. No credit will be given for anything written in the test book. Do not spend too much time on any one problem.
In this test: Unless otherwise specified, the domain of a function $f$ is assumed to be the set of all real numbers $x$ for which $f(x)$ is a real number.
$\lim _{x \rightarrow 0} \frac{8 x^{2}}{\cos x-1}=$
(A) $-16$
(B) $-1$
(C) 8
(D) 6

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Problem 3

Directions: Solve each of the following problems, using the available space for scratchwork. After examining the form of the choices, decide which is the best of the choices given and fill in the corresponding oval on the answer sheet. No credit will be given for anything written in the test book. Do not spend too much time on any one problem.
In this test: Unless otherwise specified, the domain of a function $f$ is assumed to be the set of all real numbers $x$ for which $f(x)$ is a real number.
$\lim _{x \rightarrow 5} \frac{x^{2}-25}{x-5}$
$\begin{array}{ll}{\text { (A) }} & {0} \\ {\text { (B) }} & {10}\end{array}$
(C) 5
(D) The limit does not exist.

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Problem 4

Directions: Solve each of the following problems, using the available space for scratchwork. After examining the form of the choices, decide which is the best of the choices given and fill in the corresponding oval on the answer sheet. No credit will be given for anything written in the test book. Do not spend too much time on any one problem.
In this test: Unless otherwise specified, the domain of a function $f$ is assumed to be the set of all real numbers $x$ for which $f(x)$ is a real number.
If $f(x)=\frac{x^{5}-x+2}{x^{3}+7},$ find $f^{\prime}(x)$
(A) $\frac{\left(5 x^{4}-1\right)}{\left(3 x^{2}\right)}$
(B) $\quad \frac{\left(x^{3}+7\right)\left(5 x^{4}-1\right)-\left(x^{5}-x+2\right)\left(3 x^{2}\right)}{\left(x^{3}+7\right)}$
(C) $\quad \frac{\left(x^{5}-x+2\right)\left(3 x^{2}\right)-\left(x^{3}+7\right)\left(5 x^{4}-1\right)}{\left(x^{3}+7\right)^{2}}$
(D) $\quad \frac{\left(x^{3}+7\right)\left(5 x^{4}-1\right)-\left(x^{5}-x+2\right)\left(3 x^{2}\right)}{\left(x^{3}+7\right)^{2}}$

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Problem 5

Directions: Solve each of the following problems, using the available space for scratchwork. After examining the form of the choices, decide which is the best of the choices given and fill in the corresponding oval on the answer sheet. No credit will be given for anything written in the test book. Do not spend too much time on any one problem.
In this test: Unless otherwise specified, the domain of a function $f$ is assumed to be the set of all real numbers $x$ for which $f(x)$ is a real number.
Evaluate $5\left(\frac{1}{2}+h\right)^{4}-5\left(\frac{1}{2}\right)^{4}$
$\lim _{h \rightarrow 0} \frac{5\left(\frac{1}{2}+h\right)^{4}-5\left(\frac{1}{2}\right)^{4}}{h}$
(A) $\frac{5}{2}$
(B) $\frac{5}{16}$
(C) 160
(D) The limit does not exist.

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Problem 6

Directions: Solve each of the following problems, using the available space for scratchwork. After examining the form of the choices, decide which is the best of the choices given and fill in the corresponding oval on the answer sheet. No credit will be given for anything written in the test book. Do not spend too much time on any one problem.
In this test: Unless otherwise specified, the domain of a function $f$ is assumed to be the set of all real numbers $x$ for which $f(x)$ is a real number.
$\int x \sqrt{3 x} d x=$
(A) $\frac{2 \sqrt{3}}{5} x^{\frac{5}{2}}+C$
(B) $\frac{5 \sqrt{3}}{2} x^{\frac{5}{2}}+C$
(C) $\frac{\sqrt{3}}{2} x^{\frac{1}{2}}+C$
(()) $\frac{5 \sqrt{3}}{2} x^{\frac{3}{2}}+C$

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Problem 7

Find $k$ so that $f(x)=\left\{\begin{array}{ll}{\frac{x^{2}-16}{x-4} ;} & {x \neq 4} \\ {k} & { ; x=4}\end{array}\right.$
(A) 0
(B) 16
(C) 8
(D) There is no real value of $k$ that makes $f(x)$ continuous for all $x$ .

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Problem 8

Which of the following integrals correctly gives the area of the region consisting of all points above the $x$ -axis and below the curve $y=8+2 x-x^{2} ?$
(A) $\int_{-2}^{4}\left(x^{2}-2 x-8\right) d x$
(B) $\int_{-4}^{2}\left(8+2 x-x^{2}\right) d x$
(C) $\int_{-2}^{4}\left(8+2 x-x^{2}\right) d x$
(D) $\int_{-4}^{2}\left(x^{2}-2 x-8\right) d x$

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Problem 9

If $f(x)=x^{2} \cos 2 x,$ find $f^{\prime}(x)$
(A) $-2 x \cos 2 x+2 x^{2} \sin 2 x$
(B) $-4 x \sin 2 x$
(C) $2 x \cos 2 x-2 x^{2} \sin 2 x$
(D) $2 x-2 \sin 2 x$

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Problem 10

An equation of the line tangent to $y=4 x^{3}-7 x^{2}$ at $x=3$ is
(A) $y+45=66(x+3)$
(B) $y-45=66(x-3)$
(C) $y=66 x$
(D) $y+45=\frac{-1}{66}(x-3)$

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Problem 11

$\int_{0}^{\frac{1}{2}} \frac{2}{\sqrt{1-x^{2}}} d x=$
$\begin{array}{ll}{(\mathrm{A})} & {\frac{\pi}{3}} \\ {\text { (B) }} & {\frac{\pi}{3}} \\ {\text { (C) }} & {\frac{2 \pi}{3}}\end{array}$
(D) $\quad-\frac{2 \pi}{3}$

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Problem 12

Find a positive value $c,$ for $x$ , that satisfies the conclusion of the Mean Value Theorem for
Derivatives for $f(x)=3 x^{2}-5 x+1$ on the interval $[2,5] .$
$\begin{array}{ll}{(\mathrm{A})} & {1} \\ {\text { (B) }} & {\frac{11}{6}} \\ {\text { (C) }} & {\frac{23}{6}} \\ {\text { (D) }} & {\frac{7}{2}}\end{array}$

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Problem 13

Given $f(x)=2 x^{2}-7 x-10,$ find the absolute maximum of $f(x)$ on $[-1,3]$
(A) $-1$
(B) $\frac{7}{4}$
(C) $-\frac{129}{8}$
(D) 0

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Problem 14

Find $\frac{d y}{d x}$ if $x^{3} y+x y^{3}=-10$
(A) $\left(3 x^{2}+3 x y^{2}\right)$
(B) $\frac{\left(3 x^{2} y+y^{3}\right)}{\left(3 x y^{2}+x^{3}\right)}$
(C) $-\frac{\left(3 x^{2} y+y^{3}\right)}{\left(3 x y^{2}+x^{3}\right)}$
(D) $-\frac{\left(x^{2} y+y^{3}\right)}{\left(x y^{2}+x^{3}\right)}$

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Problem 15

$\lim _{x \rightarrow 0} \frac{x \cdot 2^{x}}{2^{x}-1}=$
(A) $\ln 2$
(B) 1
(C) 2
(D) $\quad \frac{1}{\ln 2}$

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Problem 16

$\int 7 x e^{3 x^{2}} d x=$
(A) $\frac{6}{7} e^{3 x^{2}}+C$
(B) $\frac{7}{6} e^{3 x^{2}+C}$
(C) $7 e^{3} x^{2}+C$
(D) $42 e^{3 x^{2}}+C$

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Problem 17

Find the equation of the tangent line to $9 x^{2}+16 y^{2}=52$ through $(2,-1)$
(A) $-9 x+8 y-26=0$
(B) $9 x-8 y-26=0$
(C) $9 x-8 y-106=0$
(D) $8 x+9 y-17=0$

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Problem 18

A particle's position is given by $s=t^{3}-6 t^{2}+9 t .$ What is its acceleration at time $t=4 ?$
(A) 0
(B) $-9$
(C) $-12$
(D) 12

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Problem 19

If $f(x)=3^{\pi x},$ then $f^{\prime}(x)=$
(A) $\frac{3^{\pi x}}{\ln 3}$
(B) $\frac{3^{\pi x}}{\pi}$
(C) $\pi\left(3^{\pi x-1}\right)$
(D) $\pi \ln 3\left(3^{\pi r}\right)$

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Problem 20

The average value of $f(x)=\frac{1}{x}$ from $x=1$ to $x=e$ is
(A) $\quad \frac{1}{e+1}$
$\begin{array}{ll}{\text { (B) }} & {\frac{1}{1-e}} \\ {\text { (C) }} & {e-1} \\ {\text { (D) }} & {\frac{1}{e-1}}\end{array}$

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Problem 21

If $f(x)=\sin ^{2} x,$ find $f^{\prime \prime \prime}(x)$
(A) $\quad-\sin ^{2} x$
(B) $\cos 2 x$
(C) $-4 \sin 2 x$
(D) $-\sin 2 x$

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Problem 22

Find the slope of the normal line to $y=x+\cos x y$ at $(0,1)$
(A) 1
(B) $-1$
(C) 0
(D) Undefined

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Problem 23

$\int e^{x}\left(e^{3 x}\right) d x=$
(A) $\frac{1}{3} e^{3 x}+C$
(B) $\frac{1}{4} e^{4 x}+C$
(C) $\frac{1}{4} e^{5 x}+C$
(D) $4 e^{4 x}+C$

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Problem 24

$\lim _{x \rightarrow 0} \frac{\tan ^{3}(2 x)}{x^{3}}=$
(A) $-8$
(B) 2
(C) 8
(D) The limit does not exist.

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Problem 25

A solid is generated when the region in the first quadrant bounded by the graph of $y=1+\sin ^{2} x,$ the line $x=\frac{\pi}{2},$ 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) $\pi \int_{0}^{1}\left(1+\sin ^{4} x\right) d x$
(B) $\pi \int_{0}^{1}\left(1+\sin ^{2} x\right)^{2} d x$
(C) $\pi \int_{0}^{\frac{\pi}{2}}\left(1+\sin ^{4} x\right) d x$
(D) $\pi \int_{0}^{\frac{\pi}{2}}\left(1+\sin ^{2} x\right)^{2} d x$

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Problem 26

If $y=\left(\frac{x^{3}-2}{2 x^{5}-1}\right)^{4},$ find $\frac{d y}{d x}$ at $x=1$
$\begin{array}{ll}{\text { (A) }} & {-52} \\ {\text { (B) }} & {-28} \\ {\text { (C) }} & {13} \\ {\text { (D) }} & {52}\end{array}$

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Problem 27

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

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Problem 28

If $\frac{d y}{d x}=\frac{x^{3}+1}{y}=$ and $y=2$ when $x=1,$ then, when $x=2, y=$
(A) $\sqrt{\frac{27}{2}}$
(B) $\sqrt{\frac{27}{8}}$
(C) $\pm \sqrt{\frac{27}{8}}$
(D) $\pm \sqrt{\frac{27}{2}}$

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Problem 29

The graph of $y=5 x^{4}-x^{5}$ has an inflection point (or points) at
(A) $x=3$ only
(B) $x=0,3$
(C) $x=-3$ only
(D) $x=0,-3$

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Problem 30

$\int_{0}^{1} \tan x d x=$
(A) 0
(B) $\ln (\cos (1))$
(C) $\ln (\sec (1))$
(D) $\ln (\sec (1))-1$

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