Question 1

Suppose $f(x)$ is defined as $f(x) = \frac{2x^2 - 3x + 4}{x - 1}$. What is $\lim\limits_{x 	o 1} f(x)$?

Accepted Answer

$3$

Question 2

Given $g(x) = \sqrt{x^2 + 7}$, what is $g'(x)$?

Accepted Answer

$\frac{x}{\sqrt{x^2 + 7}}$

Question 3

Find the equation of the tangent line to $f(x) = 3x^2 - 5x + 8$ at $x = 2$. Express your answer in slope-intercept form.

Accepted Answer

$y = 7x - 4$

Question 4

Evaluate $\lim\limits_{x 	o 2} \frac{x^2 - 4x + 3}{x - 2}$.

Accepted Answer

$-1$

Question 5

A ball is thrown from the roof of a building. Its height after $t$ seconds is $h(t) = -16t^2 + 40t + 100$. What is the velocity of the ball after 3 seconds?

Accepted Answer

$-56$ ft/s

Question 6

Find $\lim\limits_{x 	o 0} \frac{1 - \cos x}{x^2}$.

Accepted Answer

$\frac{1}{2}$

Question 7

Given $f(x) = \frac{2x - 3}{4x + 5}$, use the quotient rule to find $f'(x)$.

Accepted Answer

$\frac{8x + 22}{(4x + 5)^2}$

Question 8

The number of people playing a new game $t$ weeks after it is introduced is given by $g(t) = \frac{8000t}{t^2 + 8}$. What is the instantaneous rate of change in the number of people playing the game exactly 4 weeks after introduction?

Accepted Answer

$120$

Question 9

Find the derivative of $f(x) = \cot x$.

Accepted Answer

$-\csc^2 x$

Question 10

Find $\lim\limits_{x 	o 3} \frac{3x^2 + 5x - 2}{x^2 - 3x - 10}$.

Accepted Answer

$\frac{22}{-1}$

Question 11

If $f(x) = 2x^2 - 3x + 4$, find and simplify the difference quotient $\frac{f(x + h) - f(x)}{h}$.

Accepted Answer

$2h + 2x - 3$

Question 12

Find the average rate of change of $F(u)$ from $u = 1$ to $u = 3$ if $F(u)$ is given by the table: $F(1) = 12$, $F(3) = 16$.

Accepted Answer

$2$

Question 13

Find the derivative of $f(x) = (5x - 8)^{10}$.

Accepted Answer

$50(5x - 8)^9$

Question 14

Find $\lim\limits_{x 	o 2} \frac{2x^2 + x - 10}{x - 2}$.

Accepted Answer

$5$

Question 15

A car's position $s(t)$ in feet after $t$ seconds is given by $s(t) = 6t^2$. What is the average velocity from $t = 0$ to $t = 4$?

Accepted Answer

$24$ ft/s

Question 16

Find the derivative of $f(x) = 	an(2x)$.

Accepted Answer

$2\sec^2(2x)$

Question 17

Find $\lim\limits_{x 	o 0} \frac{\sin(2x)}{x}$.

Accepted Answer

$2$

Question 18

Find the derivative of $h(x) = \sqrt{10 - x^3}$.

Accepted Answer

$-\frac{3x^2}{2\sqrt{10 - x^3}}$

Question 19

Find the derivative of $f(x) = \sec^2 x$.

Accepted Answer

$2\sec^2 x 	an x$

Business Calculus: Limits, Derivatives, and Applications Study Guide

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