Q: The height $H(t)$ (in inches) of a young tree is modeled by $H(t) = 145(1 - e^{-0.03t})$. What is the rate at which the height is changing after 48 months?

$H'(48) = 145 \times 0.03 \times e^{-0.03 \times 48}$

Q: A solution's pH level 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?

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

Question 1

Which of the following is the correct derivative of $f(x) = 5 + e^{x^2}$?

Accepted Answer

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

Question 2

Given $p(x) = \ln(1 - 5x^2 + x^3)$, what is $p'(x)$?

Accepted Answer

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

Question 3

A company tracks the number of downloads $D(t)$ of its new app, where $D(t) = 500 - 2t^2$. What is the instantaneous rate of change of downloads at $t = 5$ days?

Accepted Answer

$D'(5) = -20$

Question 4

If $y = 3 + e^{4x+1}$, what is $\frac{dy}{dx}$?

Accepted Answer

$\frac{dy}{dx} = 4e^{4x+1}$

Question 5

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

Accepted Answer

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

Question 6

For $g(x) = \sin^2(2x)$, what is $g'(x)$?

Accepted Answer

$g'(x) = 4\sin(2x)\cos(2x)$

Question 7

The height $H(t)$ (in inches) of a young tree is modeled by $H(t) = 145(1 - e^{-0.03t})$. What is the rate at which the height is changing after 48 months?

Accepted Answer

$H'(48) = 145 	imes 0.03 	imes e^{-0.03 	imes 48}$

Question 8

Which of the following intervals describes where a function is concave up?

Accepted Answer

Where $f''(x) > 0$

Question 9

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

Accepted Answer

$f''(0) = 4$

Question 10

If $y = \ln(x^4 - 5)$, what is $\frac{d^2y}{dx^2}$?

Accepted Answer

$\frac{d^2y}{dx^2} = \frac{4x^2(x^4 - 9)}{(x^4 - 5)^2}$

Question 11

A solution's pH level 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

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

Question 12

Which rule is used to find the derivative of $f(x) = x^2 \sin(x)$?

Accepted Answer

Product Rule

Question 13

Given $f(x) = \frac{	an(x) - \sin(x)}{x^2}$, which rule(s) must be used to find $f'(x)$?

Accepted Answer

Quotient Rule and derivatives of trigonometric functions

Question 14

If $f'(x) = 0$ at $x = a$, what does this indicate about the function $f(x)$ at $x = a$?

Accepted Answer

There may be a local maximum or minimum at $x = a$

Question 15

Which of the following is the correct second derivative of $f(x) = -9x^4 + 3x^2 - 2$?

Accepted Answer

$f''(x) = -108x^2 + 6$

Question 16

Suppose a function $f(x)$ is continuous and differentiable everywhere, has a relative minimum at $x = -3$, a relative maximum at $x = 0$, and $f'(x) > 0$ on $(-3,0)$ and $(0,8)$. Which of the following statements is true?

Accepted Answer

The function is increasing on $(-3,0)$ and $(0,8)$

Question 17

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

Accepted Answer

$\frac{dy}{dx} = \frac{8x^3 + 5}{2x^4 + 5x}$

Question 18

A company observes that the number of downloads is increasing at a rate of $D'(t) = 3t^2 + 2t$. How many total downloads have occurred when the download rate is 20 downloads per day?

Accepted Answer

Solve $3t^2 + 2t = 20$ for $t$, then compute $D(t)$

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