Question 1

A tank has the form of an inverted cone, with height 24 feet and radius 5 feet. Water enters the tank at a constant rate of $2\pi$ cubic feet per minute. At what rate is the depth of water in the tank increasing at the instant when the tank holds $288\pi$ cubic feet of water?

Accepted Answer

$\frac{1}{3}$ ft/min

Question 2

Let $f(x) = \sin^2(x)$. What is $f''(x)$?

Accepted Answer

$2\cos(2x)$

Question 3

Suppose $f(x) = \begin{cases} x^2-1, & x \leq 1 \ 2x-1, & x > 1 \end{cases}$. For what value(s) of $a$ is $f(x)$ differentiable at $x=1$?

Accepted Answer

$a=0$

Question 4

A 25-foot ladder rests against a vertical wall. The bottom of the ladder slides away from the wall at a rate of 2 ft/sec. How fast is the top of the ladder sliding down the wall when the bottom of the ladder is 7 feet from the wall?

Accepted Answer

$-\frac{7}{24}$ ft/sec

Question 5

Let $f(x) = \ln(x^2+1)$. At what value(s) of $x$ does the graph of $f$ have a horizontal tangent line?

Accepted Answer

$x=0$

Question 6

Given $f(x) = 	an^{-1}(x)$, what is $\frac{d}{dx}f(x)$?

Accepted Answer

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

Question 7

A particle moves along the x-axis according to the position function $x(t) = 2t^3 - 6t^2$. For $t>0$, at what time is the particle's velocity zero?

Accepted Answer

$t=2$

Question 8

Let $f(x) = \sin(\sqrt{x})$. What is $f'(x)$?

Accepted Answer

$\frac{1}{2\sqrt{x}}\cos(\sqrt{x})$

Question 9

Suppose $y^2 - 3x^2 = 4$ is a rotated ellipse. Using implicit differentiation, what is $\frac{dy}{dx}$?

Accepted Answer

$\frac{3x}{y}$

Question 10

Let $f(x) = \frac{x}{x+1}$. How many tangent lines to the curve pass through the point $(1,2)$?

Accepted Answer

2

Question 11

Given $f(x) = \sin(x)$, use the symmetric difference quotient to estimate $f'(0)$. Which of the following is closest to the value?

Accepted Answer

1

Question 12

A rock is dropped and allowed to fall so that its vertical position is $h(t) = 70 - gt^2$ meters in $t$ seconds, where $g$ is a constant. Suppose the rock falls 70 meters to the bottom of a crevasse in $a$ seconds. What is the value of $g$?

Accepted Answer

$g = \frac{70}{a^2}$

Question 13

Let $f(x) = \frac{x^2-1}{x-1}$. What is $\lim_{x 	o 1} f(x)$?

Accepted Answer

2

Question 14

For which value(s) of $x$ does the graph of $f'(x)$, the derivative of $f$, have a horizontal tangent, given that $f'(x)$ is shown in the graph and $-10 < x < 10$?

Accepted Answer

Where $f''(x) = 0$

Question 15

Let $f(x) = 2x^4 - x^3$. For what value(s) of $x$ is the graph of $f$ concave down?

Accepted Answer

$x < 0$ or $x > \frac{3}{8}$

Question 16

Given $f(x) = \sin(x)$, what is the equation of the tangent line to $f$ at $x = \frac{\pi}{4}$?

Accepted Answer

$y = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}(x - \frac{\pi}{4})$

Question 17

If $f(x)$ is continuous on $[a,b]$ and differentiable on $(a,b)$, which theorem guarantees that there exists $c \in (a,b)$ such that $f'(c) = \frac{f(b)-f(a)}{b-a}$?

Accepted Answer

Mean Value Theorem

Question 18

Let $f(x) = \ln(x)$. For which value(s) of $x$ does $f$ have a vertical tangent?

Accepted Answer

$x=0$

Question 19

Suppose $f(x)$ is a function such that $f'(x) > 0$ for $x < 1$ and $f'(x) < 0$ for $x > 1$. What can you conclude about $f$ at $x=1$?

Accepted Answer

$f$ has a local maximum at $x=1$

BC Calculus Review: Differentiation, Applications, and Graphical Analysis

Study Guide - Practice Questions

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