Question 1

$\displaystyle\lim_{x\rightarrow\pi}\frac{3\cos{x}+3}{x-\pi}$

Accepted Answer

$0$

Answer

$\displaystyle -\frac{1}{3}$

Answer

3

Answer

$-3$

Question 2

$\displaystyle \lim_{x\rightarrow-\infty}\frac{\sqrt{{4x^2+3}}}{2x-1}$

Accepted Answer

$-1$

Answer

$1$

Answer

$-2$

Answer

nonexistent

Question 3

$\displaystyle\int_{-1}^1\frac{2}{1+x^2} \ dx=$

Accepted Answer

$\pi$

Answer

$0$

Answer

$-\pi$

Answer

$\displaystyle \frac{\pi}{2}$

Question 4

If $\displaystyle f(x)=8\sin^3{x}$, then $\displaystyle f^{\prime}\left(\frac{\pi}{3}\right)$ is

Accepted Answer

$9$

Answer

$6\sqrt{3}$

Answer

$18$

Answer

$6$

Question 5

Given the function $\displaystyle f(x)=x^3-3x^2+3x-1$, which of the following statements is true?

Accepted Answer

The graph of $f$ has a point of inflection at $x=1$.

Answer

The graph of $f$ has a relative minimum at $x=0$.

Answer

The graph of $f$ has a relative minimum at $x=1$.

Answer

The graph of $f$ has a relative maximum at $x=1$.

Question 6

If $\displaystyle \int_{8}^{-8} f(x) \ dx=-10$ and $\displaystyle \int_{-8}^{-2} f(x) \ dx = 4$, what is the value of $\displaystyle \int_{-2}^8 f(x) \ dx$?

Accepted Answer

$6$

Answer

$8$

Answer

$-6$

Answer

$-14$

Question 7

What is the slope of the tangent line to the curve $\displaystyle 2x^2y-3xy^2-2=0$ at the point $(2, 1)$?

Accepted Answer

$\displaystyle \frac{5}{4}$

Answer

$\displaystyle  -\frac{2}{5}$

Answer

$\displaystyle -\frac{5}{4}$

Answer

$\displaystyle \frac{2}{3}$

Question 8

Air is pumped into a spherical balloon at a rate of 10 cubic centimeters per second. How fast is the diameter, in centimeters per second, increasing when the radius is 5 cm?

Accepted Answer

$\displaystyle \frac{1}{5\pi}$

Answer

$\displaystyle \frac{1}{10\pi}$

Answer

$\displaystyle 5\pi$

Answer

$\displaystyle \frac{1}{\pi}$

Question 9

$\displaystyle \lim_{h\rightarrow 0}\frac{\sin\left(\frac{\pi}{3}+h\right)-\sin\left(\frac{\pi}{3}\right)}{h}$ is

Accepted Answer

$\displaystyle \frac{1}{2}$

Answer

$\displaystyle \frac{\sqrt{3}}{2}$

Answer

$0$

Answer

$1$

Question 10

$\displaystyle \int\frac{4}{e^{2x}} \ dx=$

Accepted Answer

$\displaystyle -\frac{2}{e^{2x}}+C$

Answer

$\displaystyle \frac{2}{e^{2x}}+C$

Answer

$\displaystyle -\frac{4}{e^{2x}}+C$

Answer

$\displaystyle 4e^{2x}+C$

Question 11

If $\displaystyle f(x)=\frac{2x}{\left(2x-1\right)^2}$, then $\displaystyle f^{\prime}(x)=$

Accepted Answer

$\displaystyle \frac{-2(2x+1)}{(2x-1)^3}$

Answer

$\displaystyle \frac{2}{2x-1}$

Answer

$\displaystyle \frac{1}{2(2x+1)}$

Answer

$\displaystyle \frac{2x+1}{(2x-1)^3}$

Question 12

The definite integral $\displaystyle \int_0^3(x^2+1) \ dx$ is equivalent to which of the following limits?

Accepted Answer

$\displaystyle \lim_{n\rightarrow\infty}\sum_{k=1}^n\frac{3}{n}\left(\left(\frac{3k}{n}\right)^2+1\right)$

Answer

$\displaystyle \lim_{n\rightarrow\infty}\sum_{k=1}^n\frac{3}{n}\left(\left(\frac{k}{n}\right)^2+1\right)$

Answer

$\displaystyle \lim_{x\rightarrow\infty}\sum_{k=1}^n\frac{1}{n}\left(\frac{k}{n}\right)^2$

Answer

$\displaystyle \lim_{n\rightarrow\infty}\sum_{k=1}^n\frac{3}{n}\left(\frac{3k}{n}\right)^2$

Question 13

The graph of a continuous function $y=f(x)$ passes through the point $(e, 4)$ and $\displaystyle \frac{dy}{dx}=\frac{3}{x}$. Which of the following is the function $f(x)$?

Accepted Answer

$\displaystyle y=3\ln{x}+1$

Answer

$\displaystyle y=e^x+4$

Answer

$\displaystyle  y=3\ln{x}+e$

Answer

$\displaystyle y=\ln{x}+4$

Question 14

The figure shows the graph of a piecewise function. The graph has a horizontal tangent at $x=-4$. What are all the values of $x$, such that $-6<x<6$, for which $f$ is continuous but not differentiable?

Accepted Answer

$x=0$ and $x=2$

Answer

$x=-4$ and $x=0$

Answer

$x=-2$

Answer

$x=-2$ and $x=0$

Question 15

At which of the four points on the graph is $\displaystyle \frac{dy}{dx}>0$ and $\displaystyle \frac{d^2y}{dx^2}<0$?

Accepted Answer

C

Answer

A

Answer

B

Answer

D

Question 16

The table shows the values of the functions $h$ and $k$ and their derivatives at $x=4$ and $x=8$. What is the value of $\displaystyle \left.\frac{d}{dx}h\left(k(x)\right)\right|_{x=4}$?

Accepted Answer

$3$

Answer

$24$

Answer

$4$

Answer

$12$

Question 17

The figure shows the graph of a continuous function $f$ for $-4\leq x\leq 4$. On what interval(s) is $f^{\prime}(x)>0$ for all values of $x$?

Accepted Answer

$(-2, 0)$ only

Answer

$(-4, -2)$ and $(0, 2)$

Answer

$(-4, 2)$ only

Answer

$(0, 4)$

Question 18

The graph of a function $f$ is shown. The graph of $f$ consists of two line segments, and $f$ is continuous on the interval $[-3, 3]$. If $\displaystyle g(x)=\int_0^x f(t) \ dt$, then $g(-3)$ is

Accepted Answer

$\displaystyle \frac{3}{2}$

Answer

$1$

Answer

$\displaystyle \frac{5}{2}$

Answer

$\displaystyle -\frac{5}{2}$

Question 19

The $f$ and $g$ be differentiable functions such that $\displaystyle f^{-1}(x)=g(x)$ for all $x$. The table shows selected values of $f(x)$ and $f^{\prime}(x)$. What is the value of $g^{\prime}(2)$?

Accepted Answer

$\displaystyle -\frac{1}{3}$

Answer

$-3$

Answer

$-5$

Answer

$2$

Question 20

Given that $\displaystyle f(x)=\begin{cases}x^3+x+a & 	extrm{ for }x\leq 1\2bx-1 & 	extrm{ for }x>1\end{cases}$, if $f$ is differentiable at $x=1$, what is the value of $a+2b$?

Accepted Answer

$5$

Answer

$4$

Answer

$2$

Answer

$3$

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