AB MCQ Set 190 (iLearnMath Sets 1-20)

Question 1 of 67 | MCQ · Level 1

$\operatorname*{lim}\limits_{x \rightarrow 3} \dfrac{x - 3}{x^2 - 2 x - 3}$ is

$0$

$1$

$\dfrac{1}{4}$

$\infty$

none of these

Question 2 of 67 | MCQ · Level 2

$\operatorname*{lim}\limits_{x \rightarrow 0} \dfrac{|x|}{x}$ is

$0$

nonexistent

$1$

$-1$

none of these

Question 3 of 67 | MCQ · Level 2

$\operatorname*{lim}\limits_{x \rightarrow 7} \dfrac{x - 7}{\sqrt{x} - 7}$ is

$2 \sqrt{7}$

$\sqrt{7}$

$0$

$-2 \sqrt{7}$

nonexistent

Question 4 of 67 | MCQ · Level 2

$\operatorname*{lim}\limits_{x \rightarrow 1} \dfrac{x}{\ln x}$ is

$0$

$\dfrac{1}{e}$

$1$

$e$

nonexistent

Question 5 of 67 | MCQ · Level 3

If $a \neq 0$, then $\operatorname*{lim}\limits_{x \rightarrow a} \dfrac{x^2 - a^2}{x^4 - a^4}$ is

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

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

$\dfrac{1}{6 a^2}$

$0$

nonexistent

Question 6 of 67 | MCQ · Level 1

$\operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{x^3 - 2 x^2 + 3 x - 4}{4 x^3 - 3 x^2 + 2 x - 1} =$

$4$

$1$

$\dfrac{1}{4}$

$0$

$-1$

Question 7 of 67 | MCQ · Level 1

Let $f(x) = 4 - 3 x$. Which of the following is equal to $f'(-1)$?

$-7$

$7$

$-3$

$3$

nonexistent

Question 8 of 67 | MCQ · Level 3

Which of the following is true about the graph of $f(x) = x^{\dfrac{4}{5}}$ at $x = 0$?

It has a corner.

It has a cusp.

It has a vertical tangent.

It has a discontinuity.

$f(0)$ does not exist.

Question 9 of 67 | MCQ · Level 2

Let $f$ be the function given by $f(x) = |x|$. Which of the following statements about $f$ are true? I. $f$ is continuous at $x = 0$. II. $f$ is differentiable at $x = 0$. III. $f$ has an absolute minimum at $x = 0$.

I only

II only

III only

I and III only

II and III only

Question 10 of 67 | MCQ · Level 2

If the line normal to the graph of $f$ at the point $(1, 2)$ passes through the point $(-1, 1)$, then which of the following gives the value of $f'(1)$?

$-2$

$2$

$-\dfrac{1}{2}$

$\dfrac{1}{2}$

$3$

Question 11 of 67 | MCQ · Level 2

Find $\dfrac{d y}{d x}$ if $y = \dfrac{4 x - 3}{2 x + 1}$.

$\dfrac{10}{(4 x - 3)^2}$

$-\dfrac{10}{(4 x - 3)^2}$

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

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

$2$

Question 12 of 67 | MCQ · Level 1

Let $f(x) = 1 - 3 x^2$. Which of the following is equal to $f'(1)$?

$-6$

$-5$

$5$

$6$

Does not exist

Question 13 of 67 | MCQ · Level 3

If the $n$-th derivative of $y$ is denoted as $y^{(n)}$ and $y = -\sin x$, then $y^{(7)}$ is the same as

$y$

$\dfrac{d y}{d x}$

$\dfrac{d^2 y}{d x^2}$

$\dfrac{d^3 y}{d x^3}$

none of the above

Question 14 of 67 | MCQ · Level 1

Find $\dfrac{d y}{d x}$ if $y = \dfrac{4}{x^3}$.

$-4 x^2$

$-\dfrac{12}{x^2}$

$\dfrac{12}{x^2}$

$\dfrac{12}{x^4}$

$-\dfrac{12}{x^4}$

Question 15 of 67 | MCQ · Level 3

Use the table to find $\dfrac{d}{d x}(f \cdot g)$ at $x = 3$. Table values: at $x = 1$, $f = 4$, $g = 2$, $f' = 5$, $g' = \dfrac{1}{2}$; at $x = 3$, $f = 7$, $g = -4$, $f' = \dfrac{3}{2}$, $g' = -1$.

$\dfrac{5}{2}$

$-\dfrac{3}{2}$

$-13$

$12$

$\dfrac{21}{2}$

Question 16 of 67 | MCQ · Level 2

What does the limit statement $\operatorname*{lim}\limits_{x \rightarrow 1} \dfrac{\ln(x + 1) - \ln 2}{x - 1}$ represent?

$0$

$\dfrac{d}{d x}[\ln(x + 1)]$

$f'(1)$, if $f(x) = \ln(x + 1)$

$1$

The limit does not exist

Question 17 of 67 | MCQ · Level 2

Find $\dfrac{d^2 y}{d x^2}$ if $f(x) = (2 x + 3)^4$.

$4 (2 x + 3)^3$

$8 (2 x + 3)^3$

$12 (2 x + 3)^2$

$24 (2 x + 3)^2$

$48 (2 x + 3)^2$

Question 18 of 67 | MCQ · Level 2

Find $\dfrac{d y}{d x}$ for $y = 4 \sin^2(3 x)$.

$8 \sin(3 x)$

$24 \sin(3 x)$

$8 \sin(3 x) \cos(3 x)$

$12 \sin(3 x) \cos(3 x)$

$24 \sin(3 x) \cos(3 x)$

Question 19 of 67 | MCQ · Level 3

If $x^2 + y^2 = 25$, what is the value of $\dfrac{d^2 y}{d x^2}$ at the point $(4, 3)$?

$-\dfrac{25}{27}$

$-\dfrac{7}{27}$

$\dfrac{7}{27}$

$\dfrac{3}{4}$

$\dfrac{25}{27}$

Question 20 of 67 | MCQ · Level 2

What is the instantaneous rate of change at $x = 2$ of the function $f(x) = \dfrac{x^2 - 2}{x - 1}$?

$-2$

$\dfrac{1}{6}$

$\dfrac{1}{2}$

$2$

$6$

Question 21 of 67 | MCQ · Level 3

Find $\dfrac{d y}{d x}$ if $3 x y = 4 x + y^2$.

$\dfrac{4 - 3 y}{2 y - 3 x}$

$\dfrac{3 x - 4}{2 x}$

$\dfrac{3 y - x}{2}$

$\dfrac{3 y - 4}{2 y - 3 x}$

$\dfrac{4 + 3 y}{2 y + 3 x}$

Question 22 of 67 | MCQ · Level 3

The function $f$ is continuous on the closed interval $[0, 2]$ with values $f(0) = 1$, $f(1) = k$, $f(2) = 2$. The equation $f(x) = \dfrac{1}{2}$ must have at least two solutions in $[0, 2]$ if $k =$

$0$

$\dfrac{1}{2}$

$1$

$2$

$3$

Question 23 of 67 | MCQ · Level 3

[Calc] A particle moves along a straight line with velocity given by $v(t) = 7 - (1.01)^{-t^2}$ at time $t \geq 0$. What is the acceleration of the particle at $t = 3$?

$-0.914$

$0.055$

$5.486$

$6.086$

$18.087$

Question 24 of 67 | MCQ · Level 2

[Calc] Which of the following gives $\dfrac{d y}{d x}$ at $x = 1$ if $x^3 + 2 x y = 9$?

$\dfrac{11}{2}$

$\dfrac{5}{2}$

$\dfrac{3}{2}$

$-\dfrac{5}{2}$

$-\dfrac{11}{2}$

Question 25 of 67 | MCQ · Level 2

Which of the following gives $\dfrac{d y}{d x}$ if $y = \cos^3(3 x - 2)$?

$-9 \cos^2(3 x - 2) \sin(3 x - 2)$

$-3 \cos^2(3 x - 2) \sin(3 x - 2)$

$9 \cos^2(3 x - 2) \sin(3 x - 2)$

$-9 \cos^2(3 x - 2)$

$-3 \cos^2(3 x - 2)$

Question 26 of 67 | MCQ · Level 2

If $y = \sin^{-1}(2 x)$, find $\dfrac{d y}{d x}$.

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

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

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

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

$\dfrac{2 x}{1 + 4 x^2}$

Question 27 of 67 | MCQ · Level 3

For what value of $k$ is the function $g(x) = \begin{cases} k x - 2 & \text{if } x \leq -1 \\ k x^2 + 3 & \text{if } x > -1 \end{cases}$ continuous?

$-\dfrac{5}{2}$

$\dfrac{5}{2}$

$-1$

$\dfrac{1}{2}$

$-\dfrac{1}{2}$

Question 28 of 67 | MCQ · Level 2

$f$ is continuous for $a \leq x \leq b$ but not differentiable for some $c$ such that $a < c < b$. Which of the following could be true?

$x = c$ is a vertical asymptote of the graph of $f$.

$\operatorname*{lim}\limits_{x \rightarrow c} f(x) \neq f(c)$

The graph of $f$ has a cusp at $x = c$.

$f(c)$ is undefined.

None of the above

Question 29 of 67 | MCQ · Level 2

What is the instantaneous rate of change at $x = 3$ of the function $f(x) = \dfrac{x^2 - 2}{x + 1}$?

$-\dfrac{17}{16}$

$-\dfrac{1}{8}$

$\dfrac{1}{8}$

$\dfrac{13}{16}$

$\dfrac{17}{16}$

Question 30 of 67 | MCQ · Level 3

If $f(x) = \begin{cases} \ln(3 x) & \text{if } 0 < x \leq 3 \\ x \ln 3 & \text{if } 3 < x \leq 4 \end{cases}$ then $\operatorname*{lim}\limits_{x \rightarrow 3} f(x)$ is

$\ln 9$

$\ln 27$

$3 \ln 3$

$3 + \ln 3$

nonexistent

Question 31 of 67 | MCQ · Level 2

If $f(x) = \tan(3 x)$, then $f'\left(\dfrac{\pi}{9}\right) =$

$\dfrac{4}{3}$

$4$

$6$

$12$

$6 \sqrt{3}$

Question 32 of 67 | MCQ · Level 3

If $\operatorname*{lim}\limits_{h \rightarrow 0} \dfrac{f(x + h) - f(x)}{h} = 3 x^2 + x$, then $\operatorname*{lim}\limits_{h \rightarrow 0} \dfrac{f(x + h) - f(x - h)}{h} =$

$-3 x^2 - x$

$3 x^2 + x$

$-6 x^2 - 2 x$

$6 x^2 + 2 x$

None of the above

Question 33 of 67 | MCQ · Level 2

$\operatorname*{lim}\limits_{x \rightarrow 3} \dfrac{x^3 - 2 x^2 - 3 x}{x^3 - 9 x} =$

$0$

$\dfrac{2}{3}$

$\dfrac{3}{4}$

$1$

$\infty$

Question 34 of 67 | MCQ · Level 2

If $f(x) = \dfrac{x^2}{e^x}$, then $f'(1) =$

$0$

$\dfrac{1}{e}$

$\dfrac{2}{e}$

$2$

$2 e$

Question 35 of 67 | MCQ · Level 3

The tangent line to the curve $y = 3 x^4 - 10 x + 3$ at $x = 1$ intersects the $x$-axis at the point

$(-6, 0)$

$(-4, 0)$

$(0, -6)$

$(3, 0)$

$(4, 0)$

Question 36 of 67 | MCQ · Level 3

If $f(x) = \dfrac{e^{2 x}}{2 x}$, then $f'(x) =$

$1$

$\dfrac{e^{2 x}(1 - 2 x)}{2 x^2}$

$e^{2 x}$

$\dfrac{e^{2 x}(2 x + 1)}{x^2}$

$\dfrac{e^{2 x}(2 x - 1)}{2 x^2}$

Question 37 of 67 | MCQ · Level 3

[Calc] If the derivative of $f$ is given by $f'(x) = e^x - 3 x^2$, at which of the following values of $x$ does $f$ have a relative maximum value?

$-0.46$

$0.20$

$0.91$

$0.95$

$3.73$

Question 38 of 67 | MCQ · Level 3

[Calc] Let $f$ be the function given by $f(x) = 2 e^{4 x^2}$. For what value of $x$ is the slope of the line tangent to the graph of $f$ at $(x, f(x))$ equal to $3$?

$0.168$

$0.276$

$0.318$

$0.342$

$0.551$

Question 39 of 67 | MCQ · Level 2

Let $f$ be the function given by $f(x) = 2 x e^x$. The graph of $f$ is concave down when

$x < -2$

$x > -2$

$x < -1$

$x > -1$

$x < 0$

Question 40 of 67 | MCQ · Level 3

Let $f$ be the function with derivative given by $f'(x) = x^2 - \dfrac{2}{x}$. On which of the following intervals is $f$ decreasing?

$(-\infty, -1]$ only

$(-\infty, 0)$

$[-1, 0)$ only

$(0, \sqrt[3]{2}]$ only

$[\sqrt[3]{2}, \infty)$

Question 41 of 67 | MCQ · Level 1

$\operatorname*{lim}\limits_{x \rightarrow 0} \dfrac{x^2 - 1}{1 - 2 x^2} =$

$-1$

$-\dfrac{1}{2}$

$\dfrac{1}{2}$

$1$

non-existent

Question 42 of 67 | MCQ · Level 3

The sum of two non-negative numbers is $6$. If the square of one of the numbers is multiplied by the second number, then the largest possible product is

$32$

$36$

$38$

$45$

$64$

Question 43 of 67 | MCQ · Level 3

The minimum value of the function $y = \sqrt{x^2 + 2 a x + 10 a^2}$, where $a > 0$, is

$-a$

$a$

$3 a$

$6 a$

$9 a^2$

Question 44 of 67 | MCQ · Level 2

If $y = \sin x + e^{-x}$, then $y + y' =$

$0$

$\sin x + \cos x$

$2 e^{-x}$

$2 \sin x + 2 e^{-x}$

$2 \sin x - 2 e^{-x}$

Question 45 of 67 | MCQ · Level 3

The expression $\dfrac{1}{50}\left(\sqrt{\dfrac{1}{50}} + \sqrt{\dfrac{2}{50}} + \sqrt{\dfrac{3}{50}} + \cdots.c + \sqrt{\dfrac{50}{50}}\right)$ is a Riemann sum approximation for

$\displaystyle\int_{0}^{1} \sqrt{\dfrac{x}{50}} d x$

$\displaystyle\int_{0}^{1} \sqrt{x} d x$

$\dfrac{1}{50} \displaystyle\int_{0}^{1} \sqrt{\dfrac{x}{50}} d x$

$\dfrac{1}{50} \displaystyle\int_{0}^{1} \sqrt{x} d x$

$\dfrac{1}{50} \displaystyle\int_{0}^{50} \sqrt{x} d x$

Question 46 of 67 | MCQ · Level 3

Let $f$ be a function defined for all real numbers $x$. If $f'(x) = \dfrac{|4 - x^2|}{x - 2}$, then $f$ is decreasing on the interval

$(-\infty, 2)$

$(-\infty, \infty)$

$(-2, 4)$

$(-2, \infty)$

$(2, \infty)$

Question 47 of 67 | MCQ · Level 2

If $f(x) = 6 x^2 + \dfrac{16}{x^2}$, then $\operatorname*{lim}\limits_{h \rightarrow 0} \dfrac{f(2 + h) - f(2)}{h} =$

$0$

$20$

$24$

$32$

$\infty$

Question 48 of 67 | MCQ · Level 2

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

$\dfrac{12 x + 13}{(3 x + 2)^2}$

$\dfrac{12 x - 13}{(3 x + 2)^2}$

$\dfrac{5}{(3 x + 2)^2}$

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

$\dfrac{2}{3}$

Question 49 of 67 | MCQ · Level 2

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

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

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

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

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

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

Question 50 of 67 | MCQ · Level 3

[Calc] A rectangle is inscribed in the semicircle $y = \sqrt{4 - x^2}$. Find the largest possible area.

$1.4$

$\sqrt{3}$

$2 \sqrt{3}$

$4$

undefined

Question 51 of 67 | MCQ · Level 3

[Calc] Find the value of $c$ guaranteed by the Mean Value Theorem for $f(x) = \dfrac{2 x}{x^2 + 1}$ on the interval $[0, 1]$.

$0.475$

$0.486$

$0.488$

$0.577$

$1.000$

Question 52 of 67 | MCQ · Level 2

If $F(x) = \displaystyle\int_{0}^{x} \sin^2(2 t) d t$, then $F'(x) =$

$-\cos^2(2 x)$

$\cos^2(2 x)$

$\sin^2(2 x)$

$\dfrac{1}{2} \sin^2(2 x)$

$4 \sin(2 x) \cos(2 x)$

Question 53 of 67 | MCQ · Level 2

Find the average value of $f(x) = \sqrt{x}$ on the interval $[1, 4]$.

$\dfrac{1}{3}$

$\dfrac{7}{9}$

$\dfrac{14}{9}$

$\dfrac{7}{2}$

$\dfrac{14}{3}$

Question 54 of 67 | MCQ · Level 3

[Calc] The rate of natural gas sales for the year 1993 at a certain gas company is given by $P(t) = t^2 - 400 t + 160000$, where $P(t)$ is measured in gallons per day and $t$ is the number of days in 1993 (from day 0 to day 365). To the nearest gallon, what is the total number of gallons of natural gas sales at this company for the 31 days (day 0 to day 31) of January 1993?

\$4,777,730$

\$4,617,930$

\$154,120$

\$148,965$

\$148,561$

Question 55 of 67 | MCQ · Level 1

$\int \cos(7 t + 3) d t =$

$7 \sin(7 t + 3) + C$

$\sin(7 t + 3) + C$

$\dfrac{1}{7} \sin(7 t + 3) + C$

$-7 \sin(7 t + 3) + C$

$-\dfrac{1}{7} \sin(7 t + 3) + C$

Question 56 of 67 | MCQ · Level 3

If $f(x) = \sin x$, $g(x) = \cos(2 x)$, and $h(x) = f(g(x))$, what is $h'\left(\dfrac{\pi}{4}\right)$?

$-2$

$-\sqrt{2}$

$0$

$\sqrt{2}$

$2$

Question 57 of 67 | MCQ · Level 2

The position of a particle moving in a line is $s(t) = t^3 - 5 t^2 + 2 t - 13$. What is the speed of the particle at $t = 2$?

$-21$

$-6$

$6$

$10$

$32$

Question 58 of 67 | MCQ · Level 2

[Calc] If $k > 1$, the area under the curve $y = k x^2$ from $x = 0$ to $x = k$ is

$\dfrac{1}{3} k^4$

$\dfrac{1}{3} k^3$

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

$\dfrac{1}{3} k^3 - k$

$k^3$

Question 59 of 67 | MCQ · Level 3

[Calc] A continuous function $g(t)$ is defined on the closed interval $[0, 6]$ with table values: $g(0) = 4$, $g(1) = 7$, $g(2) = 8$, $g(3) = 12$, $g(4) = 15$, $g(5) = 22$, $g(6) = 26$. Using a midpoint Riemann sum with three subintervals of equal length, the approximate value of $\displaystyle\int_{0}^{6} g(t) d t$ is

$68$

$82$

$89$

$94$

$153$

Question 60 of 67 | MCQ · Level 2

$\displaystyle\int_{1}^{6} \sqrt{x + 3} d x =$

$-\dfrac{5}{36}$

$1$

$\dfrac{58}{5}$

$\dfrac{38}{3}$

$19$

Question 61 of 67 | MCQ · Level 3

[Calc] A young girl, $5$ feet tall, is walking away from a lamppost which is $12$ feet tall. She walks at a constant rate of $2$ feet per second and notices that, as she moves away from the lamppost, the length of her shadow is increasing. How fast is the length of her shadow increasing in feet per second when she is $20$ feet from the post?

$\dfrac{7}{10}$ ft/sec

$\dfrac{10}{7}$ ft/sec

$2$ ft/sec

$\dfrac{34}{7}$ ft/sec

$\dfrac{27}{10}$ ft/sec

Question 62 of 67 | MCQ · Level 3

If $f(x) = 3 x^3 + 5 x$ and $g(x) = f^{-1}(x)$, what is $g'(8)$?

$\dfrac{1}{14}$

$\dfrac{1}{11}$

$\dfrac{1}{8}$

$11$

$14$

Question 63 of 67 | MCQ · Level 3

[Calc] The base of a solid is the region in the first quadrant bounded by the $y$-axis, the graph of $y = \tan^{-1} x$, the horizontal line $y = 3$, and the vertical line $x = 1$. For this solid, each cross section perpendicular to the $x$-axis is a square. What is the volume of the solid?

$2.561$

$6.612$

$8.046$

$8.755$

$20.773$

Question 64 of 67 | MCQ · Level 2

$\dfrac{d}{d x}(\displaystyle\int_{0}^{x^2} \sin(t^3) d t) =$

$-\cos(x^6)$

$\sin(x^3)$

$\sin(x^6)$

$2 x \sin(x^3)$

$2 x \sin(x^6)$

Question 65 of 67 | MCQ · Level 3

Region $Q$ is bounded by $y = \sin(2 x)$, $y = 0$, $x = 0$, $x = \dfrac{\pi}{2}$. Which of the following expressions gives the volume of a solid whose base in the $x y$-plane is region $Q$ and whose cross sections, perpendicular to the $x$-axis, are squares with a side in the $x y$-plane?

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

$\displaystyle\int_{0}^{\dfrac{\pi}{2}} \sin^2(2 x) d x$

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

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

$\pi \displaystyle\int_{0}^{\dfrac{\pi}{2}} \sin(2 x)^2 d x$

Question 66 of 67 | MCQ · Level 2

If $f(x) = x^2 + 2 x$, then $\dfrac{d}{d x}(f(\ln x)) =$

$\dfrac{2 \ln x + 2}{x}$

$2 x \ln x + 2 x$

$2 \ln x + 2$

$2 \ln x + \dfrac{2}{x}$

$\dfrac{2 x + 2}{x}$

Question 67 of 67 | MCQ · Level 3

If the region enclosed by the $y$-axis, the line $y = 2$, and the curve $y = \sqrt{x}$ is revolved about the $y$-axis, the volume of the solid generated is

$\dfrac{32 \pi}{5}$

$\dfrac{16 \pi}{3}$

$\dfrac{16 \pi}{5}$

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

$\pi$

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Area & Circumference

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