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 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$?

$$R'(3) = 10 - 2 \times 3 = 4$$

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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