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

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

Q: A biologist models the population of bacteria in a petri dish by $$P(t) = 100e$$^{0.05t}$$, where t$ is in hours. What is the instantaneous rate of change of the population at t=10$ hours?

$$P'(10) = 5 \times 100e$$^{0.05 \times 10}$$

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 car's position is given by $$s(t) = 5t^2$ + 2t. $$What is the car's velocity at $$t=4$$ seconds?

$$v(4) = 10 \times 4 + 2 = 42$$

Q: A medication's concentration in the bloodstream is modeled by $$C(t) = 10e$$^{-0.2t}$$. What is the rate at which the concentration is decreasing at t=5$ hours?

$$C'(5) = -0.2 \times 10e$$^{-0.2 \times 5}$$

Q: A company’s profit is modeled by $$P(x) = 200x - x^2$, where x$ is the number of units sold. What is the marginal profit when x=50$?

$$P'(50) = 200 - 2 \times 50 = 100$$

Question 1

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

Accepted Answer

A value that a function approaches as the input approaches some value.

Question 2

Given the function $$f(x) = \frac{2x^2$ - 8}{x - 2}$$, what is $$\lim_{x 	o 2} f(x)$$?

Accepted Answer

$$4$$

Question 3

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

Accepted Answer

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

Question 4

A biologist models the population of bacteria in a petri dish by $$P(t) = 100e$$^{0.05t}$$, where t$ is in hours. What is the instantaneous rate of change of the population at t=10$ hours?

Accepted Answer

$$P'(10) = 5 	imes 100e$$^{0.05 	imes 10}$$

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 $$f(x) = x^3$ - 5x^2$ + 6x - 2.$$

Accepted Answer

$$f'(x) = 3x^2$ - 10x + 6$$

Question 7

A chemical reaction's rate is modeled by $$R(t) = 4t^2$ + 3t. $$What is the average rate of change of $$R$$ from $$t=1 to$$ $$t=3$$?

Accepted Answer

$$R_{avg} = \frac{R(3) - R(1)}{3-1}$$

Question 8

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

Accepted Answer

$$1$$

Question 9

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

Accepted Answer

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

Question 10

A hospital monitors the temperature of a patient using $$T(t) = 98.6 + 0.5\sin(\pi t/12)$$, where $$t is in $$hours. What is the rate of change of temperature at $$t=6$$ hours?

Accepted Answer

$$T'(6) = 0.5 \cos(\pi/2) \cdot \frac{\pi}{12}$$

Question 11

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

Accepted Answer

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

Question 12

A car's position is given by $$s(t) = 5t^2$ + 2t. $$What is the car's velocity at $$t=4$$ seconds?

Accepted Answer

$$v(4) = 10 	imes 4 + 2 = 42$$

Question 13

Which of the following is the correct definition of the definite integral of $$f(x)$$ from $$a to$$ $$b$$?

Accepted Answer

$$\int_a^b f(x) dx$$

Question 14

A medication's concentration in the bloodstream is modeled by $$C(t) = 10e$$^{-0.2t}$$. What is the rate at which the concentration is decreasing at t=5$ hours?

Accepted Answer

$$C'(5) = -0.2 	imes 10e$$^{-0.2 	imes 5}$$

Question 15

Which of the following is NOT a property of the definite integral?

Accepted Answer

$$\int_a^b f(x) dx = \int_a^b f'(x) dx$$

Question 16

Find $$\frac{d}{dx}(\ln(x^2$ + 1)).$$

Accepted Answer

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

Question 17

A company’s profit is modeled by $$P(x) = 200x - x^2$, where x$ is the number of units sold. What is the marginal profit when x=50$?

Accepted Answer

$$P'(50) = 200 - 2 	imes 50 = 100$$

Question 18

Which of the following is the correct application of the product rule for $$f(x) = x^2$ e^x$$?

Accepted Answer

$$f'(x) = 2x e^x + x^2$ e^x$$

Question 19

Evaluate $$\$$int_0$$^2 (3x^2$) dx.$$

Accepted Answer

$$12$$

Calculus I (MATH 1610) – Course Overview and Study Guide

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