AB MCQ Set 80 (CB Official 2013)

Question 1 of 18 | MCQ · Level 2

What is $\operatorname*{lim}\limits_{h \rightarrow 0} \dfrac{\cos\left(\dfrac{3 \pi}{2} + h\right) - \cos\left(\dfrac{3 \pi}{2}\right)}{h}$?

$1$

$\dfrac{\sqrt{2}}{2}$

$0$

$-1$

The limit does not exist.

Question 2 of 18 | MCQ · Level 3

The slope of the tangent to the curve $y^3 x + y^2 x^2 = 6$ at $(2, 1)$ is

$-\dfrac{3}{2}$

$-1$

$-\dfrac{5}{14}$

$-\dfrac{3}{14}$

$0$

Question 3 of 18 | MCQ · Level 3

Let $S$ be the region enclosed by the graphs of $y = 2 x$ and $y = 2 x^2$ for $0 \leq x \leq 1$. What is the volume of the solid generated when $S$ is revolved about the line $y = 3$?

$\pi \displaystyle\int_{0}^{1} [(3 - 2 x^2)^2 - (3 - 2 x)^2] d x$

$\pi \displaystyle\int_{0}^{1} [(3 - 2 x)^2 - (3 - 2 x^2)^2] d x$

$\pi \displaystyle\int_{0}^{1} (4 x^2 - 4 x^4) d x$

$\pi \displaystyle\int_{0}^{2} [\left(3 - \dfrac{y}{2}\right)^2 - \left(3 - \sqrt{\dfrac{y}{2}}\right)^2] d y$

$\pi \displaystyle\int_{0}^{2} [\left(3 - \sqrt{\dfrac{y}{2}}\right)^2 - \left(3 - \dfrac{y}{2}\right)^2] d y$

Question 4 of 18 | MCQ · Level 3

Which of the following statements about the function given by $f(x) = x^4 - 2 x^3$ is true?

The function has no relative extremum.

The graph of the function has one point of inflection and the function has two relative extrema.

The graph of the function has two points of inflection and the function has one relative extremum.

The graph of the function has two points of inflection and the function has two relative extrema.

The graph of the function has two points of inflection and the function has three relative extrema.

Question 5 of 18 | MCQ · Level 2

If $f(x) = \sin^2(3 - x)$, then $f'(0) =$

$-2 \cos 3$

$-2 \sin 3 \cos 3$

$6 \cos 3$

$2 \sin 3 \cos 3$

$6 \sin 3 \cos 3$

Question 6 of 18 | MCQ · Level 3

Which of the following is the solution to the differential equation $\dfrac{d y}{d x} = \dfrac{4 x}{y}$, where $y(2) = -2$?

$y = 2 x$ for $x > 0$

$y = 2 x - 6$ for $x \neq 3$

$y = -\sqrt{4 x^2 - 12}$ for $x > \sqrt{3}$

$y = \sqrt{4 x^2 - 12}$ for $x > \sqrt{3}$

$y = -\sqrt{4 x^2 - 6}$ for $x > \sqrt{1.5}$

Question 7 of 18 | MCQ · Level 1

What is the average rate of change of the function $f$ given by $f(x) = x^4 - 5 x$ on the closed interval $[0, 3]$?

$8.5$

$8.7$

$22$

$33$

$66$

Question 8 of 18 | MCQ · Level 3

The position of a particle moving along a line is given by $s(t) = 2 t^3 - 24 t^2 + 90 t + 7$ for $t \geq 0$. For what values of $t$ is the speed of the particle increasing?

$3 < t < 4$ only

$t > 4$ only

$t > 5$ only

$0 < t < 3$ and $t > 5$

$3 < t < 4$ and $t > 5$

Question 9 of 18 | MCQ · Level 2

$\int (x - 1) \sqrt{x} d x =$

$\dfrac{3}{2} \sqrt{x} - \dfrac{1}{\sqrt{x}} + C$

$\dfrac{2}{3} x^{\dfrac{3}{2}} + \dfrac{1}{2} x^{\dfrac{1}{2}} + C$

$\dfrac{1}{2} x^2 - x + C$

$\dfrac{2}{5} x^{\dfrac{5}{2}} - \dfrac{2}{3} x^{\dfrac{3}{2}} + C$

$\dfrac{1}{2} x^2 + 2 x^{\dfrac{3}{2}} - x + C$

Question 10 of 18 | MCQ · Level 2

What is $\operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{x^2 - 4}{2 + x - 4 x^2}$?

$-2$

$-\dfrac{1}{4}$

$\dfrac{1}{2}$

$1$

The limit does not exist.

Question 11 of 18 | MCQ · Level 3

If $y = 5 + \displaystyle\int_{2}^{2 x} e^{-t^2} d t$, which of the following is true?

$\dfrac{d y}{d x} = e^{-x^2}$ and $y(0) = 5$

$\dfrac{d y}{d x} = e^{-x^2}$ and $y(1) = 5$

$\dfrac{d y}{d x} = e^{-4 x^2}$ and $y(1) = 5$

$\dfrac{d y}{d x} = 2 e^{-4 x^2}$ and $y(0) = 5$

$\dfrac{d y}{d x} = 2 e^{-4 x^2}$ and $y(1) = 5$

Question 12 of 18 | MCQ · Level 3

[Calculator] A particle travels along a straight line with a velocity of $v(t) = 3 e^{-\dfrac{t}{2}} \sin(2 t)$ meters per second. What is the total distance, in meters, traveled by the particle during the time interval $0 \leq t \leq 2$ seconds?

$0.835$

$1.850$

$2.055$

$2.261$

$7.025$

Question 13 of 18 | MCQ · Level 4

A city is built around a circular lake that has a radius of $1$ mile. The population density of the city is $f(r)$ people per square mile, where $r$ is the distance from the center of the lake, in miles. Which of the following expressions gives the number of people who live within $1$ mile of the lake?

$2 \pi \displaystyle\int_{0}^{1} r f(r) d r$

$2 \pi \displaystyle\int_{0}^{1} r (1 + f(r)) d r$

$2 \pi \displaystyle\int_{0}^{2} r (1 + f(r)) d r$

$2 \pi \displaystyle\int_{1}^{2} r f(r) d r$

$2 \pi \displaystyle\int_{1}^{2} r (1 + f(r)) d r$

Question 14 of 18 | MCQ · Level 3

Let $f$ be a function such that $f''(x) < 0$ for all $x$ in the closed interval $[1, 2]$. Selected values of $f$ are shown: $f(1.1)=4.18$, $f(1.2)=4.38$, $f(1.3)=4.56$, $f(1.4)=4.73$. Which of the following must be true about $f'(1.2)$?

$f'(1.2) < 0$

$0 < f'(1.2) < 1.6$

$1.6 < f'(1.2) < 1.8$

$1.8 < f'(1.2) < 2.0$

$f'(1.2) > 2.0$

Question 15 of 18 | MCQ · Level 3

[Calculator] Two particles start at the origin and move along the x-axis. For $0 \leq t \leq 10$, their respective position functions are given by $x_1 = \sin t$ and $x_2 = e^{-2 t} - 1$. For how many values of $t$ do the particles have the same velocity?

None

One

Two

Three

Four

Question 16 of 18 | MCQ · Level 2

A differentiable function $f$ has the property that $f(5) = 3$ and $f'(5) = 4$. What is the estimate for $f(4.8)$ using the local linear approximation for $f$ at $x = 5$?

$2.2$

$2.8$

$3.4$

$3.8$

$4.6$

Question 17 of 18 | MCQ · Level 3

[Calculator] Oil is leaking from a tanker at the rate of $R(t) = 2,000 e^{-0.2 t}$ gallons per hour, where $t$ is measured in hours. How much oil leaks out of the tanker from time $t = 0$ to $t = 10$?

$54$ gallons

$271$ gallons

$865$ gallons

\$8,647$ gallons

\$14,778$ gallons

Question 18 of 18 | MCQ · Level 4

[Calculator] If $f'(x) = \sin\left(\dfrac{\pi e^x}{2}\right)$ and $f(0) = 1$, then $f(2) =$

$-1.819$

$-0.843$

$-0.819$

$0.157$

$1.157$

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