Q: Let $h(x) = \dfrac{\tan(x) - \sin(x)}{x^2}$. What is $h'(x)$?

$h'(x) = \dfrac{\sec^2(x) - \cos(x)}{x^2} - \dfrac{2(\tan(x) - \sin(x))}{x^3}$

Q: The pH level of a solution over time (in hours) is given by $H(t) = \ln(0.5t + 1)$. What is the rate of change of the pH level after 8 hours?

$H'(8) = \dfrac{0.5}{0.5 \times 8 + 1}$

Question 1

Let $f(x) = 5 + e^{x^2}$. What is $f'(x)$?

Accepted Answer

$f'(x) = 2x e^{x^2}$

Question 2

Suppose $g(x) = \ln(1 - 5x^2 + x^3)$. What is $g'(x)$?

Accepted Answer

$g'(x) = \dfrac{-10x + 3x^2}{1 - 5x^2 + x^3}$

Question 3

A company’s profit function is $P(x) = 3x + e^{x^2 + 1}$. What is the marginal profit when $x = 0$?

Accepted Answer

$P'(0) = 3 + 2e$

Question 4

Given $y = \log_{10}(2x^4 + 5x)$, what is $\dfrac{dy}{dx}$?

Accepted Answer

$\dfrac{8x^3 + 5}{(2x^4 + 5x) \ln 10}$

Question 5

If $f(x) = (3x + 7) \ln(2x - 1)$, what is $f'(x)$?

Accepted Answer

$f'(x) = 3 \ln(2x - 1) + \dfrac{3x + 7}{2x - 1} \cdot 2$

Question 6

Let $h(x) = \dfrac{	an(x) - \sin(x)}{x^2}$. What is $h'(x)$?

Accepted Answer

$h'(x) = \dfrac{\sec^2(x) - \cos(x)}{x^2} - \dfrac{2(	an(x) - \sin(x))}{x^3}$

Question 7

For $f(x) = x^3 - 3x^2 + 2x^2 - 8$, find $f''(0)$.

Accepted Answer

$f''(0) = 2$

Question 8

The pH level of a solution over time (in hours) is given by $H(t) = \ln(0.5t + 1)$. What is the rate of change of the pH level after 8 hours?

Accepted Answer

$H'(8) = \dfrac{0.5}{0.5 	imes 8 + 1}$

Question 9

The height $H(t)$ (in inches) of a young tree based on its age $t$ (in months) is modeled by $H(t) = 145(1 - e^{-0.03t})$. What happens to the height of the tree as $t 	o \infty$?

Accepted Answer

The height approaches $145$ inches.

Question 10

Given the graph of a function $g(x)$, how do you determine the intervals where $g(x)$ is increasing?

Accepted Answer

Where $g'(x) > 0$

Question 11

Suppose $f(x) = x^4 - 2x^2 + 6x + 4$. What are the critical points?

Accepted Answer

Where $f'(x) = 0$

Question 12

If $g(x) = \dfrac{1}{x^2}$, what is the interval where $g(x)$ is decreasing?

Accepted Answer

$x > 0$

Question 13

A company tracks the number of downloads $D(t)$ in days by $D(t) = 500 - 2t^2$. At what time is the download rate zero?

Question 14

Given $y = \ln(x^4 - 5)$, what is $y''$?

Accepted Answer

$y'' = \dfrac{12x^2(x^4 + 5)}{(x^4 - 5)^2}$

Question 15

If $f(x)$ has $f'(x) > 0$ on $(-\infty, -3) \cup (3, \infty)$ and $f'(x) = 0$ at $x = -3, 3$, what can you say about $f(x)$ at $x = -3$ and $x = 3$?

Accepted Answer

$x = -3$ and $x = 3$ are critical points

Question 16

Which rule is used to differentiate $y = x^2 \sin(x)$?

Accepted Answer

Product Rule

Question 17

If $f(x) = \sin^2(x)$, what is $f'(x)$?

Accepted Answer

$f'(x) = 2\sin(x)\cos(x)$

Question 18

Suppose $f(x)$ is continuous and differentiable everywhere, and $f'(x) = 0$ only at $x = -3, 3$. What does this imply about the function?

Accepted Answer

The function has horizontal tangents only at $x = -3$ and $x = 3$

Question 19

If $f(x) = \sin(2x)$, what is $f''(x)$?

Accepted Answer

$f''(x) = -4\sin(2x)$

Business Calculus: Derivatives, Applications, and Graph Analysis Study Guide

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