AB MCQ Set 100

Question 1 of 45 | Integration and Accumulation of Change · Level 1

$\int (x^3 - 3 x) d x =$

$3 x^2 - 3 + C$

$4 x^4 - 6 x^2 + C$

$\dfrac{x^4}{3} - 3 x^2 + C$

$\dfrac{x^4}{4} - 3 x + C$

$\dfrac{x^4}{4} - \dfrac{3 x^2}{2} + C$

Question 2 of 45 | Differentiation: Composite, Implicit, and Inverse Functions · Level 1

If $f(x) = x^3 + 3 x^2 + 4 x + 5$ and $g(x) = 5$, then $g(f(x)) =$

$5 x^2 + 15 x + 25$

$5 x^3 + 15 x^2 + 20 x + 25$

$1125$

$225$

$5$

Question 3 of 45 | Differentiation: Composite, Implicit, and Inverse Functions · Level 2

The slope of the line tangent to the graph of $y = \ln(x^2)$ at $x = e^2$ is

$\dfrac{1}{e^2}$

$\dfrac{2}{e^2}$

$\dfrac{4}{e^2}$

$\dfrac{1}{e^4}$

$\dfrac{4}{e^4}$

Question 4 of 45 | Differentiation: Definition and Fundamental Properties · Level 1

If $f(x) = x + \sin x$, then $f''(x) =$

$1 + \cos x$

$1 - \cos x$

$\cos x$

$\sin x - x \cos x$

$\sin x + x \cos x$

Question 5 of 45 | Limits and Continuity · Level 2

If $f(x) = e^x$, which of the following lines is an asymptote to the graph of $f$?

$y = 0$

$x = 0$

$y = x$

$y = -x$

$y = 1$

Question 6 of 45 | Differentiation: Definition and Fundamental Properties · Level 2

If $f(x) = \dfrac{x - 1}{x + 1}$ for all $x \neq -1$, then $f'(1) =$

$-1$

$-\dfrac{1}{2}$

$0$

$\dfrac{1}{2}$

$1$

Question 7 of 45 | Limits and Continuity · Level 2

Which of the following equations has a graph that is symmetric with respect to the origin?

$y = \dfrac{x + 1}{x}$

$y = -x^5 + 3 x$

$y = x^4 - 2 x^2 + 6$

$y = (x - 1)^3 + 1$

$y = (x^2 + 1)^2 - 1$

Question 8 of 45 | Applications of Integration · Level 2

A particle moves in a straight line with velocity $v(t) = t^2$. How far does the particle move between times $t = 1$ and $t = 2$?

$\dfrac{1}{3}$

$\dfrac{7}{3}$

$3$

$7$

$8$

Question 9 of 45 | Differentiation: Composite, Implicit, and Inverse Functions · Level 2

If $y = \cos^2 3 x$, then $\dfrac{d y}{d x} =$

$-6 \sin 3 x \cos 3 x$

$-2 \cos 3 x$

$2 \cos 3 x$

$6 \cos 3 x$

$2 \sin 3 x \cos 3 x$

Question 10 of 45 | Analytical Applications of Differentiation · Level 3

The derivative of $f(x) = \dfrac{x^4}{3} - \dfrac{x^5}{5}$ attains its maximum value at $x =$

$-1$

$0$

$1$

$\dfrac{4}{3}$

$\dfrac{5}{3}$

Question 11 of 45 | MCQ · Level 3

If the line $3 x - 4 y = 0$ is tangent in the first quadrant to the curve $y = x^3 + k$, then $k$ is

$\dfrac{1}{2}$

$\dfrac{1}{4}$

$0$

$-\dfrac{1}{8}$

$-\dfrac{1}{2}$

Question 12 of 45 | Differentiation: Definition and Fundamental Properties · Level 3

If $f(x) = 2 x^3 + A x^2 + B x - 5$ and if $f(2) = 3$ and $f(-2) = -37$, what is the value of $A + B$?

$-6$

$-3$

$-1$

$2$

It cannot be determined from the information given.

Question 13 of 45 | MCQ · Level 3

The acceleration $\alpha$ of a body moving in a straight line is given in terms of time $t$ by $\alpha = 8 - 6 t$. If the velocity of the body is $25$ at $t = 1$ and if $s(t)$ is the distance of the body from the origin at time $t$, what is $s(4) - s(2)$?

$20$

$24$

$28$

$32$

$42$

Question 14 of 45 | Differentiation: Definition and Fundamental Properties · Level 3

If $f(x) = x^{\dfrac{1}{3}} (x - 2)^{\dfrac{2}{3}}$ for all $x$, then the domain of $f'$ is

$\{x | x \neq 0\}$

$\{x | x > 0\}$

$\{x | 0 \leq x \leq 2\}$

$\{x | x \neq 0$ and $x \neq 2\}$

$\{x | x$ is a real number$\}$

Question 15 of 45 | Applications of Integration · Level 2

The area of the region bounded by the lines $x = 0$, $x = 2$, and $y = 0$ and the curve $y = e^{\dfrac{x}{2}}$ is

$\dfrac{e - 1}{2}$

$e - 1$

$2(e - 1)$

$2 e - 1$

$2 e$

Question 16 of 45 | Integration and Accumulation of Change · Level 2

The number of bacteria in a culture is growing at a rate of $3000 e^{2 \dfrac{t}{5}}$ per unit of time $t$. At $t = 0$, the number of bacteria present was \$7,500$. Find the number present at $t = 5$.

\$1,200 e^2$

\$3,000 e^2$

\$7,500 e^2$

\$7,500 e^5$

$\dfrac{15,000}{7} e^7$

Question 17 of 45 | Applications of Integration · Level 3

What is the area of the region completely bounded by the curve $y = -x^2 + x + 6$ and the line $y = 4$?

$\dfrac{3}{2}$

$\dfrac{7}{3}$

$\dfrac{9}{2}$

$\dfrac{31}{6}$

$\dfrac{33}{2}$

Question 18 of 45 | Differentiation: Composite, Implicit, and Inverse Functions · Level 2

$\dfrac{d}{d x}(\arcsin 2 x) =$

$\dfrac{-1}{2 \sqrt{1 - 4 x^2}}$

$\dfrac{-2}{\sqrt{4 x^2 - 1}}$

$\dfrac{1}{2 \sqrt{1 - 4 x^2}}$

$\dfrac{2}{\sqrt{1 - 4 x^2}}$

$\dfrac{2}{\sqrt{4 x^2 - 1}}$

Question 19 of 45 | Differentiation: Composite, Implicit, and Inverse Functions · Level 2

Suppose that $f$ is a function that is defined for all real numbers. Which of the following conditions assures that $f$ has an inverse function?

The function $f$ is periodic.

The graph of $f$ is symmetric with respect to the y-axis.

The graph of $f$ is concave up.

The function $f$ is a strictly increasing function.

The function $f$ is continuous.

Question 20 of 45 | Integration and Accumulation of Change · Level 1

If $F$ and $f$ are continuous functions such that $F'(x) = f(x)$ for all $x$, then $\displaystyle\int_{a}^{b} f(x) d x$ is

$F'(a) - F'(b)$

$F'(b) - F'(a)$

$F(a) - F(b)$

$F(b) - F(a)$

none of the above

Question 21 of 45 | Integration and Accumulation of Change · Level 3

$\displaystyle\int_{0}^{1} (x + 1) e^{x^2 + 2 x} d x =$

$\dfrac{e^3}{2}$

$\dfrac{e^3 - 1}{2}$

$\dfrac{e^4 - e}{2}$

$e^3 - 1$

$e^4 - e$

Question 22 of 45 | Analytical Applications of Differentiation · Level 3

Given the function defined by $f(x) = 3 x^5 - 20 x^3$, find all values of $x$ for which the graph of $f$ is concave up.

$x > 0$

$-\sqrt{2} < x < 0$ or $x > \sqrt{2}$

$-2 < x < 0$ or $x > 2$

$x > \sqrt{2}$

$-2 < x < 2$

Question 23 of 45 | Differentiation: Definition and Fundamental Properties · Level 3

$\operatorname*{lim}\limits_{h \rightarrow 0} \dfrac{1}{h} \ln\left(\dfrac{2 + h}{2}\right)$ is

$e^2$

$1$

$\dfrac{1}{2}$

$0$

nonexistent

Question 24 of 45 | Differentiation: Composite, Implicit, and Inverse Functions · Level 3

Let $f(x) = \cos(\arctan x)$. What is the range of $f$?

$\{x | -\dfrac{\pi}{2} < x < \dfrac{\pi}{2}\}$

$\{x | 0 < x \leq 1\}$

$\{x | 0 \leq x \leq 1\}$

$\{x | -1 < x < 1\}$

$\{x | -1 \leq x \leq 1\}$

Question 25 of 45 | Integration and Accumulation of Change · Level 3

$\displaystyle\int_{0}^{\dfrac{\pi}{4}} \tan^2 x d x =$

$\dfrac{\pi}{4} - 1$

$1 - \dfrac{\pi}{4}$

$\dfrac{1}{3}$

$\sqrt{2} - 1$

$\dfrac{\pi}{4} + 1$

Question 26 of 45 | Contextual Applications of Differentiation · Level 3

The radius $r$ of a sphere is increasing at the uniform rate of $0.3$ inches per second. At the instant when the surface area $S$ becomes $100 \pi$ square inches, what is the rate of increase, in cubic inches per second, in the volume $V$? ($S = 4 \pi r^2$ and $V = \dfrac{4}{3} \pi r^3$)

$10 \pi$

$12 \pi$

$22.5 \pi$

$25 \pi$

$30 \pi$

Question 27 of 45 | Integration and Accumulation of Change · Level 3

$\displaystyle\int_{0}^{\dfrac{1}{2}} \dfrac{2 x}{\sqrt{1 - x^2}} d x =$

$1 - \dfrac{\sqrt{3}}{2}$

$\dfrac{1}{2} \ln \dfrac{3}{4}$

$\dfrac{\pi}{6}$

$\dfrac{\pi}{6} - 1$

$2 - \sqrt{3}$

Question 28 of 45 | Applications of Integration · Level 3

A point moves in a straight line so that its distance at time $t$ from a fixed point of the line is $8 t - 3 t^2$. What is the total distance covered by the point between $t = 1$ and $t = 2$?

$1$

$\dfrac{4}{3}$

$\dfrac{5}{3}$

$2$

$5$

Question 29 of 45 | Analytical Applications of Differentiation · Level 3

Let $f(x) = |\sin x - \dfrac{1}{2}|$. The maximum value attained by $f$ is

$\dfrac{1}{2}$

$1$

$\dfrac{3}{2}$

$\dfrac{\pi}{2}$

$\dfrac{3 \pi}{2}$

Question 30 of 45 | Integration and Accumulation of Change · Level 2

$\displaystyle\int_{1}^{2} \dfrac{x - 4}{x^2} d x =$

$-\dfrac{1}{2}$

$\ln 2 - 2$

$\ln 2$

$2$

$\ln 2 + 2$

Question 31 of 45 | Differentiation: Composite, Implicit, and Inverse Functions · Level 3

If $\log_a (2^a) = \dfrac{a}{4}$, then $a =$

$2$

$4$

$8$

$16$

$32$

Question 32 of 45 | Integration and Accumulation of Change · Level 1

$\int \dfrac{5}{1 + x^2} d x =$

$\dfrac{-10 x}{(1 + x^2)^2} + C$

$\dfrac{5}{2 x} \ln(1 + x^2) + C$

$5 x - \dfrac{5}{x} + C$

$5 \arctan x + C$

$5 \ln(1 + x^2) + C$

Question 33 of 45 | MCQ · Level 3

Suppose that $f$ is an odd function; i.e., $f(-x) = -f(x)$ for all $x$. Suppose that $f'(x_0)$ exists. Which of the following must necessarily be equal to $f'(-x_0)$?

$f'(x_0)$

$-f'(x_0)$

$\dfrac{1}{f'(x_0)}$

$\dfrac{-1}{f'(x_0)}$

None of the above

Question 34 of 45 | Applications of Integration · Level 2

The average value of $\sqrt{x}$ over the interval $0 \leq x \leq 2$ is

$\dfrac{1}{3} \sqrt{2}$

$\dfrac{1}{2} \sqrt{2}$

$\dfrac{2}{3} \sqrt{2}$

$1$

$\dfrac{4}{3} \sqrt{2}$

Question 35 of 45 | Applications of Integration · Level 4

The region in the first quadrant bounded by the graph of $y = \sec x$, $x = \dfrac{\pi}{4}$, and the axes is rotated about the x-axis. What is the volume of the solid generated?

$\dfrac{\pi^2}{4}$

$\pi - 1$

$\pi$

$2 \pi$

$\dfrac{8 \pi}{3}$

Question 36 of 45 | Differentiation: Definition and Fundamental Properties · Level 2

If $y = e^{n x}$, then $\dfrac{d^n y}{d x^n} =$

$n^n e^{n x}$

$n! e^{n x}$

$n e^{n x}$

$n^n e^x$

$n! e^x$

Question 37 of 45 | Differential Equations · Level 2

If $\dfrac{d y}{d x} = 4 y$ and if $y = 4$ when $x = 0$, then $y =$

$4 e^{4 x}$

$e^{4 x}$

$3 + e^{4 x}$

$4 + e^{4 x}$

$2 x^2 + 4$

Question 38 of 45 | Integration and Accumulation of Change · Level 2

If $\displaystyle\int_{1}^{2} f(x - c) d x = 5$ where $c$ is a constant, then $\displaystyle\int_{1-c}^{2-c} f(x) d x =$

$5 + c$

$5$

$5 - c$

$c - 5$

$-5$

Question 39 of 45 | Analytical Applications of Differentiation · Level 3

The point on the curve $2 y = x^2$ nearest to $(4, 1)$ is

$(0, 0)$

$(2, 2)$

$(\sqrt{2}, 1)$

$(2 \sqrt{2}, 4)$

$(4, 8)$

Question 40 of 45 | Differentiation: Composite, Implicit, and Inverse Functions · Level 3

If $\tan(x y) = x$, then $\dfrac{d y}{d x} =$

$\dfrac{1 - y \tan(x y) \sec(x y)}{x \tan(x y) \sec(x y)}$

$\dfrac{\sec^2(x y) - y}{x}$

$\cos^2(x y)$

$\dfrac{\cos^2(x y)}{x}$

$\dfrac{\cos^2(x y) - y}{x}$

Question 41 of 45 | Integration and Accumulation of Change · Level 3

Given $\begin{cases} f(x) = x + 1 \text{for} x < 0 \\ f(x) = \cos \pi x \text{for} x \geq 0 \end{cases}$, $\displaystyle\int_{-1}^1 f(x) d x =$

$\dfrac{1}{2} + \dfrac{1}{\pi}$

$-\dfrac{1}{2}$

$\dfrac{1}{2} - \dfrac{1}{\pi}$

$\dfrac{1}{2}$

$-\dfrac{1}{2} + \pi$

Question 42 of 45 | Integration and Accumulation of Change · Level 3

Calculate the approximate area under $y = x^2$ from $x = 1$ to $x = 2$ by the trapezoidal rule, using divisions at $x = \dfrac{4}{3}$ and $x = \dfrac{5}{3}$.

$\dfrac{50}{27}$

$\dfrac{251}{108}$

$\dfrac{7}{3}$

$\dfrac{127}{54}$

$\dfrac{77}{27}$

Question 43 of 45 | Limits and Continuity · Level 2

If the solutions of $f(x) = 0$ are $-1$ and $2$, then the solutions of $f\left(\dfrac{x}{2}\right) = 0$ are

$-1$ and $2$

$-\dfrac{1}{2}$ and $\dfrac{5}{2}$

$-\dfrac{3}{2}$ and $\dfrac{3}{2}$

$-\dfrac{1}{2}$ and $1$

$-2$ and $4$

Question 44 of 45 | Contextual Applications of Differentiation · Level 3

For small values of $h$, the function $\sqrt[4]{16 + h}$ is best approximated by which of the following?

$4 + \dfrac{h}{32}$

$2 + \dfrac{h}{32}$

$\dfrac{h}{32}$

$4 - \dfrac{h}{32}$

$2 - \dfrac{h}{32}$

Question 45 of 45 | Limits and Continuity · Level 2

If $f$ is a continuous function on $[a, b]$, which of the following is necessarily true?

$f'$ exists on $(a, b)$.

If $f(x_0)$ is a maximum of $f$, then $f'(x_0) = 0$.

$\operatorname*{lim}\limits_{x \rightarrow x_0} f(x) = f(\operatorname*{lim}\limits_{x \rightarrow x_0} x)$ for $x_0 \in (a, b)$

$f'(x) = 0$ for some $x \in [a, b]$

The graph of $f'$ is a straight line.

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