Q: Which of the following statements about continuity is TRUE?

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

Q: A biologist models population growth with $$P(t) = 100e$$^{0.05t}$$. What is the rate of change of the population at t = 10$?

$$5 \times 100e$$^{0.5}$$ = 500e$$^{0.5}$$

Q: Which of the following is NOT a property of limits?

$$\lim_{x \to a} f(x) = f(a)$$ for all functions $$f$$

Q: Which of the following is the correct definition of the derivative of $$f(x) at$$ $$x=a$$?

$$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

Question 1

Which of the following best describes the concept of a limit in calculus?

Accepted Answer

The value a function approaches as the input approaches a certain point.

Question 2

Evaluate the following limit: $$\lim_{x 	o 0} \frac{\sin x}{x}$$

Accepted Answer

$$1$$

Question 3

Which of the following is an indeterminate form when evaluating limits?

Accepted Answer

$$\frac{0}{0}$$

Question 4

A hospital monitors the concentration $$C(t) of a $$drug in a patient’s bloodstream over time $$t. If$$ $$C(t)$$ approaches $$5 mg/L as$$ $$t$$ approaches $$12$$ hours, what is $$\lim_{t 	o 12} C(t)$$?

Accepted Answer

$$5 mg/L$$

Question 5

Which of the following statements about continuity is TRUE?

Accepted Answer

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

Question 6

Find the derivative of $$f(x) = x^3$ - 2x + 5.$$

Accepted Answer

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

Question 7

A car’s position is given by $$s(t) = 4t^2$ + 2t. $$What is the instantaneous velocity at $$t = 3$$ seconds?

Accepted Answer

$$26 m/s$$

Question 8

Which rule is used to differentiate the product of two functions $$f(x)$$ and $$g(x)$$?

Accepted Answer

Product Rule: $$\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$$

Question 9

If $$y = \sin(x^2$)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$2x \cos(x^2$)$$

Question 10

A biologist models population growth with $$P(t) = 100e$$^{0.05t}$$. What is the rate of change of the population at t = 10$?

Accepted Answer

$$5 	imes 100e$$^{0.5}$$ = 500e$$^{0.5}$$

Question 11

Which of the following is the correct application of the Chain Rule for $$f(x) = \cos(3x)$$?

Accepted Answer

$$f'(x) = -3\sin(3x)$$

Question 12

A chemical reaction’s rate is modeled by $$R(t) = t^2$e^{-t}. $$What is $$\frac{dR}{dt}$?

Accepted Answer

$$2te^{-t}$$ - $$t^2e$$^{-t}$

Question 13

Which of the following is NOT a property of limits?

Accepted Answer

$$\lim_{x 	o a} f(x) = f(a)$$ for all functions $$f$$

Question 14

Find $$\lim_{x 	o 2} \frac{x^2$ - 4}{x - 2}.$$

Accepted Answer

$$4$$

Question 15

Which of the following is the correct definition of the derivative of $$f(x) at$$ $$x=a$$?

Accepted Answer

$$f'(a) = \lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$$

Question 16

A company’s profit function is $$P(x) = 200x - x^2$. What is the marginal profit when x = 10$?

Accepted Answer

$$P'(x) = 200 - 2x$$, so $$P'(10) = 180$$

Question 17

Which of the following is the correct derivative of $$f(x) = \ln(x^2$ + 1)$$?

Accepted Answer

$$\frac{2x}{x^2$ + 1}$$

Question 18

A patient’s heart rate $$H(t) is $$modeled by $$H(t) = 70 + 10\sin(\pi t/12). $$What is the rate of change of heart rate at $$t = 6$$ hours?

Accepted Answer

$$H'(t) = 10 \cdot \frac{\pi}{12} \cos(\pi t/12)$$, so $$H'(6) = 10 \cdot \frac{\pi}{12} \cos(\pi/2) = 0$$

Single Variable Calculus: Limits, Derivatives, and Applications

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