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Section 1 - Problem 1

If $g(x)=\frac{1}{32} x^{4}-5 x^{2},$ find $g^{\prime}(4)$
(A) $\quad-72$
(B) $\quad-32$
(C) $\quad 24$
(D) $\quad 32$

Amrita B.

Section 1 - Problem 2

$\lim _{x \rightarrow 0} \frac{8 x^{2}}{\cos x-1}=$
(A) $\quad-16$
(B) $\quad-1$
(C) $\quad 8$
(D) $\quad 6$

Amrita B.

Section 1 - Problem 3

$\lim _{x \rightarrow 5} \frac{x^{2}-25}{x-5}$
(A) 0
(B) 10
(C) 5
(D) The limit does not exist.

Amrita B.

Section 1 - Problem 4

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) $\frac{\left(x^{3}+7\right)\left(5 x^{4}-1\right)-\left(x^{5}-x+2\right)\left(3 x^{2}\right)}{\left(x^{3}+7\right)}$
(C) $\frac{\left(x^{5}-x+2\right)\left(3 x^{2}\right)-\left(x^{3}+7\right)\left(5 x^{4}-1\right)}{\left(x^{3}+7\right)^{2}}$
(D) $\frac{\left(x^{3}+7\right)\left(5 x^{4}-1\right)-\left(x^{5}-x+2\right)\left(3 x^{2}\right)}{\left(x^{3}+7\right)^{2}}$

Amrita B.

Section 1 - Problem 5

Evaluate $\lim _{h \rightarrow 0} \frac{5\left(\frac{1}{2}+h\right)^{4}-5\left(\frac{1}{2}\right)^{4}}{h}$
(A) $\frac{5}{2}$
(B) $\frac{5}{16}$
(C) 160
(D) The limit does not exist.

Amrita B.

Section 1 - Problem 6

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

Amrita B.

Section 1 - Problem 7

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

Amrita B.

Section 1 - 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) $\int_{-1}^{2}\left(8+2 x-x^{2}\right) d x$
(C) $\int_{-2}^{4}\left(8+2 x-x^{2}\right) d x$
(D) $\int_{-1}^{2}\left(x^{2}-2 x-8\right) d x$

Amrita B.

Section 1 - 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) $2 x-2 \sin 2 x$

Amrita B.

Section 1 - 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) $y+45=\frac{-1}{66}(x-3)$

Amrita B.

Section 1 - Problem 11

$\int_{0}^{\frac{1}{2}} \frac{2}{\sqrt{1-x^{2}}} d x=$
(A) $\frac{\pi}{3}$
(B) $\frac{\pi}{3}$
(C) $\frac{2 \pi}{3}$
(D) $\quad-\frac{2 \pi}{3}$

Amrita B.

Section 1 - 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 $[2,5]$ .
(A) 1
(B) $\frac{11}{6}$
(C) $\frac{23}{6}$
(D) $\frac{7}{2}$

Amrita B.

Section 1 - 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

Amrita B.

Section 1 - Problem 14

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

Amrita B.

Section 1 - Problem 15

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

Amrita B.

Section 1 - Problem 16

$\int 7 x e^{3 x^{2}} d x=$
(A) $\frac{6}{7} e^{5 x^{2}+C}$
(B) $\frac{7}{6} e^{3 x^{2}+C}$
(C) $7 e^{3} x^{2}+C$
(D) 42$e^{3 x^{2}+C}$

Amrita B.

Section 1 - 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) $8 x+9 y-17=0$

Amrita B.

Section 1 - 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

Amrita B.

Section 1 - 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) $\pi \ln 3\left(3^{\pi x}\right)$

Amrita B.

Section 1 - Problem 20

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

Amrita B.

Section 1 - Problem 21

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

Amrita B.

Section 1 - Problem 22

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

Amrita B.

Section 1 - 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) $4 e^{4 x}+C$

Amrita B.

Section 1 - 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.

Amrita B.
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
(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) $\pi \int_{0}^{\frac{\pi}{2}}\left(1+\sin ^{2} x\right)^{2} d x$