Q: Which of the following statements about the continuity of a function $$f(x) is $$true?

A function is continuous at $$x = a if$$ $$\lim_{x \to a} f(x)$$ exists and equals $$f(a).$$

Q: A restaurant purchased kitchen equipment for $$14,650 on $$January 1, 2017. Its value after $$t$$ years is modeled by $$V(t) = 14,650e$$^{-0.15t}$$. What is the rate of change in value on January 1, 2019?

$$V'(2) = -0.15 \times 14,650e$$^{-0.3}$$

Question 1

A hospital monitors the concentration of a drug in a patient's bloodstream over time, modeled by the function $$C(t) = 12e$$^{-0.5t}$$, where t$ is in hours. What is the instantaneous rate of change of the drug concentration at t = 4$ hours?

Accepted Answer

$$-6e^{-2}$$

Question 2

Given $$f(x) = \dfrac{2x^2$ - 8}{6x - 8}$$, find $$\lim_{x 	o 2} f(x).$$

Accepted Answer

Question 3

A company’s revenue function is $$R(x) = 75x - 9x \ln x$$, where $$x is $$the number of units sold. What is the marginal revenue when $$x = 10$$?

Accepted Answer

$$75 - 9 \ln 10 - 9$$

Question 4

Which of the following statements about the continuity of a function $$f(x) is $$true?

Accepted Answer

A function is continuous at $$x = a if$$ $$\lim_{x 	o a} f(x)$$ exists and equals $$f(a).$$

Question 5

Find the derivative of $$y = \dfrac{8x^3$ - 5}{x^2$}.$$

Accepted Answer

Question 6

A population grows according to $$P(t) = 8000e$$^{0.05t}$$. What does the value of P$$'(20)$$ represent in a real-world context?

Accepted Answer

The rate at which the population is growing at $$t = 20$$ years.

Question 7

Evaluate $$\lim_{x 	o 1} \dfrac{2x^2$ + x - 3}{x^2$ - 1}.$$

Accepted Answer

$$-2$$

Question 8

A car’s position is given by $$s(t) = t^3$ - 6t^2$ + 9t$$, where $$t is in $$hours and $$s in $$miles. What is the velocity at $$t = 2$$ hours?

Accepted Answer

$$v(2) = 3(2)^2$ - 12(2) + 9$$

Question 9

Which of the following best describes uniform motion?

Accepted Answer

The velocity remains constant and there is no acceleration.

Question 10

Find the derivative of $$y = (5x - 11)^5$.

Accepted Answer

$$25(5x - 11)^4$

Question 11

A restaurant purchased kitchen equipment for $$14,650 on $$January 1, 2017. Its value after $$t$$ years is modeled by $$V(t) = 14,650e$$^{-0.15t}$$. What is the rate of change in value on January 1, 2019?

Accepted Answer

$$V'(2) = -0.15 	imes 14,650e$$^{-0.3}$$

Question 12

Find the derivative of $$y = \sqrt{x^2$ + 11}.$$

Accepted Answer

$$\dfrac{2x}{2\sqrt{x^2$ + 11}}$$

Question 13

Solve for $$x$$: $$3 \ln x = 12.$$

Accepted Answer

$$x = e^4$

Question 14

Find $$\dfrac{d}{dx} \left( (x^2$ - x$$)^{10}$$ \right).$$

Accepted Answer

$$10(x^2$$ - $$x)^9 (2x - 1$$)$$

Question 15

Which of the following is the correct domain of the function $$y = \ln(x + 7)$$?

Accepted Answer

$$(-7, \infty)$$

Question 16

Find the derivative of $$y = 4 \ln x.$$

Accepted Answer

$$\dfrac{4}{x}$$

Question 17

Evaluate $$\lim_{x 	o 0} \dfrac{\sin x}{x}.$$

Accepted Answer

$$1$$

Question 18

A chemical reaction’s rate is modeled by $$R(t) = 3t^2$ - 5t + 2. $$What is the instantaneous rate of change at $$t = 4$$?

Accepted Answer

$$R'(4) = 6(4) - 5$$

Question 19

Find the derivative of $$y = (x^2$ + 4$$)^{11}$$.

Accepted Answer

$$22x(x^2$$ + $$4)^{10}$$

Calculus Study Guide: Limits, Continuity, and Differentiation

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