Q: A biologist is studying the population growth of bacteria in a petri dish. The population $P(t)$ at time $t$ (in hours) is modeled by $P(t) = 500e^{0.4t}$. What is the instantaneous rate of change of the population at $t = 3$ hours?

$P'(3) = 0.4 \times 500e^{0.4 \times 3}$

Q: A medication's concentration in the bloodstream is modeled by $C(t) = \ln(2t + 1)$. What is the rate of change of concentration with respect to time at $t = 2$?

$C'(2) = \frac{1}{2 \times 2 + 1} \times 2 = \frac{2}{5}$

Q: A radioactive substance decays according to $N(t) = 1000e^{-0.2t}$. What is the instantaneous rate of decay at $t = 5$?

$N'(5) = -0.2 \times 1000e^{-0.2 \times 5}$

Question 1

A biologist is studying the population growth of bacteria in a petri dish. The population $P(t)$ at time $t$ (in hours) is modeled by $P(t) = 500e^{0.4t}$. What is the instantaneous rate of change of the population at $t = 3$ hours?

Accepted Answer

$P'(3) = 0.4 	imes 500e^{0.4 	imes 3}$

Question 2

Evaluate the following limit: $\lim_{x 	o 2} \frac{x^2 - 4}{x - 2}$.

Accepted Answer

$4$

Question 3

Which of the following functions is NOT continuous at $x = 0$?

Accepted Answer

$f(x) = \frac{1}{x}$

Question 4

A car's position at time $t$ (in seconds) is given by $s(t) = 3t^2 + 2t + 1$. What is the average rate of change of the car's position from $t = 1$ to $t = 4$?

Accepted Answer

$\frac{s(4) - s(1)}{4 - 1} = \frac{57 - 6}{3} = 17$

Question 5

Find the derivative of $f(x) = x^3 - 5x + 2$ using the power and sum-difference rules.

Accepted Answer

$f'(x) = 3x^2 - 5$

Question 6

Given $f(x) = x^2$ and $g(x) = \sin(x)$, find the derivative of $h(x) = f(x)g(x)$ using the product rule.

Accepted Answer

$h'(x) = 2x\sin(x) + x^2\cos(x)$

Question 7

A medication's concentration in the bloodstream is modeled by $C(t) = \ln(2t + 1)$. What is the rate of change of concentration with respect to time at $t = 2$?

Accepted Answer

$C'(2) = \frac{1}{2 	imes 2 + 1} 	imes 2 = \frac{2}{5}$

Question 8

Which of the following is the correct application of the chain rule for $f(x) = \sin(3x^2)$?

Accepted Answer

$f'(x) = \cos(3x^2) \cdot 6x$

Question 9

Find the derivative of $y = e^{2x} \cdot \ln(x)$ using the product rule.

Accepted Answer

$y' = e^{2x} \cdot \frac{1}{x} + 2e^{2x} \cdot \ln(x)$

Question 10

A chemical reaction's rate is given by $R(t) = 4e^{-0.5t}$. What is the derivative $R'(t)$ with respect to $t$?

Accepted Answer

$R'(t) = -2e^{-0.5t}$

Question 11

Find the second derivative of $f(x) = x^4 - 3x^2 + 2$.

Accepted Answer

$f''(x) = 12x^2 - 6$

Question 12

Which of the following is the correct derivative of $f(x) = \ln(e^{3x})$?

Accepted Answer

$f'(x) = 3$

Question 13

A company’s revenue in thousands of dollars is given by $R(x) = 10x - x^2$, where $x$ is the number of units sold. What is the marginal revenue when $x = 3$?

Accepted Answer

$R'(3) = 10 - 2 	imes 3 = 4$

Question 14

Find the derivative of $f(x) = \frac{x^2 + 1}{x}$ using the quotient rule.

Accepted Answer

$f'(x) = \frac{2x \cdot x - (x^2 + 1) \cdot 1}{x^2} = \frac{2x^2 - x^2 - 1}{x^2} = \frac{x^2 - 1}{x^2}$

Question 15

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

Accepted Answer

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

Question 16

A patient’s temperature $T$ (in degrees Celsius) after taking medication is modeled by $T(t) = 37 + 2e^{-0.3t}$, where $t$ is time in hours. What is the rate at which the temperature is changing at $t = 0$?

Accepted Answer

$T'(0) = -0.6e^{0}$

Question 17

Which of the following is the correct Leibniz notation for the derivative of $y = x^5$?

Accepted Answer

$\frac{dy}{dx} = 5x^4$

Question 18

Find the derivative of $f(x) = \ln(x^2 + 1)$ using the chain rule.

Accepted Answer

$f'(x) = \frac{2x}{x^2 + 1}$

Question 19

A radioactive substance decays according to $N(t) = 1000e^{-0.2t}$. What is the instantaneous rate of decay at $t = 5$?

Accepted Answer

$N'(5) = -0.2 	imes 1000e^{-0.2 	imes 5}$

Calculus Exam 1 & 2 Review: Limits, Differentiation, and Exponential/Logarithmic Functions

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