BC MCQ Set 40 (Brandeis)

Question 1 of 27 | MCQ · Level 2

Find $\operatorname*{lim}\limits_{x \rightarrow 2} \dfrac{x - 2}{x^2 - x - 2}$.

$0$

$3$

$\dfrac{1}{2}$

$\dfrac{1}{3}$

doesn't exist

Question 2 of 27 | MCQ · Level 2

Find $\operatorname*{lim}\limits_{x \rightarrow +\infty} \dfrac{x^3 - 5}{4 x^3 + x + 1}$.

$\dfrac{3}{4}$

$\dfrac{1}{4}$

$1$

$0$

$+\infty$

Question 3 of 27 | MCQ · Level 1

Find $\operatorname*{lim}\limits_{x \rightarrow -\infty} e^x$.

$0$

$e$

$1$

$+\infty$

$-\infty$

Question 4 of 27 | MCQ · Level 2

$f'(x)$ can be defined as:

$\operatorname*{lim}\limits_{\Delta x \rightarrow 0} \dfrac{f(x + \Delta x) - f(x)}{x}$

$\operatorname*{lim}\limits_{x \rightarrow 0} \dfrac{f(x + \Delta x) - f(x)}{\Delta x}$

$\operatorname*{lim}\limits_{x \rightarrow \Delta x} \dfrac{f(x + \Delta x) - f(x)}{\Delta x}$

$\operatorname*{lim}\limits_{\Delta x \rightarrow 0} \dfrac{f(x + \Delta x) - f(x)}{\Delta x}$

none of the above

Question 5 of 27 | MCQ · Level 2

Let $f(x) = \ln x \cdot \cos x$. Find $f'(x)$.

$\dfrac{1}{\cos x} \cdot (-\sin x)$

$\dfrac{1}{x} \cdot (-\sin x)$

$\dfrac{1}{x} \cos x - \ln x \sin x$

$\dfrac{1}{x \cos x}(\cos x - x \sin x)$

none of the above

Question 6 of 27 | MCQ · Level 2

Let $y = 4 e^{\tan x}$. Find $\dfrac{d y}{d x}$.

$4 e^{\tan x} \cdot \sec^2 x$

$4 e^{\tan x} \cdot \dfrac{1}{1 + x^2}$

$4 e^{\tan x}$

$4 e^{\tan x} \cdot \cot x$

$4 \sec^2 x$

Question 7 of 27 | MCQ · Level 2

Let $f(x) = \sin^{-1} x$. Find $f'(0)$.

$\pi$

$1$

$\dfrac{1}{2}$

$0$

none of the above

Question 8 of 27 | MCQ · Level 2

The equation of the line tangent to the graph of $f(x) = x^2 + 5 x$ at the point with x-coordinate $x = 2$ is:

$y = 9 x - 14$

$y = 9 x$

$y = 9 x - 4$

$y = -\dfrac{1}{9} x - \dfrac{2}{9}$

none of the above

Question 9 of 27 | MCQ · Level 3

Let $f(x) = x^3 - 3 x$. Which of the following statements are true? I. $f(x)$ has local maxima at both $x = -1$ and $x = 1$. II. $f(x)$ has a local minimum at $x = 1$ and an inflection point at $x = 0$. III. $f(x)$ has both a local minimum and an inflection point at $x = 0$.

only I is true

only II is true

only III is true

only I and III are true

none of the statements is true

Question 10 of 27 | MCQ · Level 3

A commercial nursery has $1000$ yards of fencing which the owners plan to use to enclose as large a rectangular garden as possible. The garden will be bounded on one side by a barn, so no fencing is needed on that side. How large will the garden be (in square yards)?

\$125,000$ sq yds

\$250,000$ sq yds

\$111,088.89$ sq yds

\$62,500$ sq yds

none of the above

Question 11 of 27 | MCQ · Level 2

The width of a rectangle is increasing at a rate of $2$ cm/sec, and its length is increasing at a rate of $3$ cm/sec. At what rate is the area of the rectangle increasing when its width is $4$ cm and its length is $5$ cm?

$31$ cm$^2$/sec

$23$ cm$^2$/sec

$5$ cm$^2$/sec

$22$ cm$^2$/sec

none of the above

Question 12 of 27 | MCQ · Level 2

A rock is dropped from a height of $400$ feet and falls toward the earth in a straight line; $t$ seconds after it is dropped, it has fallen a distance of $s(t) = 16 t^2$ feet. At what speed is the rock traveling when it hits the ground?

$20$ ft/sec

$32$ ft/sec

$640$ ft/sec

$320$ ft/sec

$160$ ft/sec

Question 13 of 27 | MCQ · Level 3

Which of the following gives the area between the curves $y = x^2$ and $y = 2 x$ over the interval $[-2, 2]$?

$\displaystyle\int_{-2}^2 (x^2 - 2 x) d x$

$\displaystyle\int_{-2}^2 (2 x - x^2) d x$

$\displaystyle\int_{-2}^0 (x^2 - 2 x) d x + \displaystyle\int_{0}^{2} (2 x - x^2) d x$

$|\displaystyle\int_{-2}^2 (x^2 - 2 x) d x|$

none of the above

Question 14 of 27 | MCQ · Level 4

Suppose that $f(x)$ is a continuous function with the following properties: $f''(x) = \cos x$, $f'(\pi) = 2$ and $f(0) = 4$. What is $f(\pi)$?

$2$

$2 \pi$

$\pi + 2$

$6 + 2 \pi$

$0$

Question 15 of 27 | MCQ · Level 3

Suppose that the function $f(x)$ is defined by $f(x) = \displaystyle\int_{1}^{x} \dfrac{e^t}{t} d t$. Find $f'(x)$.

$e^x \ln x$

$e^x \ln x - \dfrac{e^x}{x}$

$\dfrac{e^x}{x}$

$\dfrac{e^x}{x} - e$

the integral can't be computed, so it's impossible to give the answer

Question 16 of 27 | MCQ · Level 2

Suppose that $f(x) = \dfrac{x}{x^2 + 1}$. Find $\displaystyle\int_{0}^{2} f'(x) d x$.

$\dfrac{2}{5}$

$-\dfrac{28}{25}$

$\dfrac{28}{25}$

$0$

none of the above

Question 17 of 27 | MCQ · Level 3

Which of the following statements about indefinite integrals are true? I. $\int f(x) + g(x) d x = \int f(x) d x + \int g(x) d x$ II. $\int f(x) g(x) d x = \int f(x) d x \cdot \int g(x) d x$ III. $\int f'(g(x)) g'(x) d x = f(g(x)) + C$ IV. $\int [f(x)]^n d x = \dfrac{[f(x)]^{n+1}}{n + 1} + C$

only I and II are true

only I and III are true

only I and IV are true

only I, II and IV are true

only I, III and IV are true

Question 18 of 27 | MCQ · Level 3

Find the volume of the solid obtained by rotating the region bounded by $y = x^2$ and $y = x$ over the interval $[0, 1]$ around the x-axis.

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

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

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

$\displaystyle\int_{0}^{1} \pi (\sqrt{y} - y) d y$

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

Question 19 of 27 | MCQ · Level 2

The integral $\int \dfrac{1}{x \ln x} d x$ can be found by

making the substitution $u = \ln x$

making the substitution $u = \dfrac{1}{x}$

using integration by parts, with $u = \ln x$ and $d v = x$

taking the reciprocal of $\int x \ln x d x$

none of the above

Question 20 of 27 | MCQ · Level 2

The integral $\int x \sin x d x$ can be found by

making the substitution $u = x$

making the substitution $u = \sin x$

using integration by parts, with $u = \sin x$ and $d v = x d x$

using integration by parts, with $u = x$ and $d v = \sin x d x$

none of the above

Question 21 of 27 | MCQ · Level 3

Find $\displaystyle\int_{0}^{\ln \sqrt{3}} \dfrac{e^x}{1 + e^{2 x}} d x$.

$\ln 2$

$1$

$\dfrac{\pi}{12}$

$\dfrac{\pi}{4}$

$0$

Question 22 of 27 | MCQ · Level 3

Find $\operatorname*{lim}\limits_{x \rightarrow 0} \dfrac{x - \sin x}{x^3}$.

$0$

$1$

$\dfrac{1}{6}$

$\dfrac{1}{3}$

does not exist

Question 23 of 27 | MCQ · Level 2

Find $\displaystyle\int_{1}^{\infty} \dfrac{1}{x^2} d x$.

$1$

$2$

$10$

$20$

the integral diverges

Question 24 of 27 | MCQ · Level 3

Which of the following improper integrals converge to a finite value? (I) $\displaystyle\int_{1}^{\infty} e^{-x} d x$ (II) $\displaystyle\int_{-\infty}^{\infty} x^3 d x$ (III) $\displaystyle\int_{-\infty}^{\infty} \dfrac{1}{1 + x^2} d x$

I only

III only

I and II only

I and III only

all of them

Question 25 of 27 | MCQ · Level 2

The second order Taylor polynomial at $x = 0$ for $f(x) = e^{-x}$ is

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

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

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

$1 - x + x^2$

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

Question 26 of 27 | MCQ · Level 3

Which of the following series converge? (I) $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^2}$ (II) $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n}$ (III) $\displaystyle\sum_{n=1}^{\infty} \dfrac{n}{2^n}$

(I) only

(III) only

(I) and (II) only

(I) and (III) only

all of them

Question 27 of 27 | MCQ · Level 2

The radius of convergence of the power series $\displaystyle\sum_{n=0}^{\infty} x^n$ is

$0$

$1$

$2$

$3$

$\infty$

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Area & Circumference

Volume

Triangles

Other Facts

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Area & Circumference

Volume

Triangles

Other Facts

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