Q: A medication's concentration in the bloodstream is modeled by $$C(t) = 5e$$^{-0.2t}$$, where t$ is in hours. What is the rate of change of the concentration at t = 3$ hours?

$$C'(3) = -1 \times 5 \times 0.2 \times e$$^{-0.6}$$ = -1 \times e$$^{-0.6}$$

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

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

Q: A bacteria population grows according to $$P(t) = 100e$$^{0.05t}$$. What is the instantaneous rate of growth at t = 10$ hours?

$$P'(10) = 100 \times 0.05 \times e$$^{0.5}$$

Question 1

Which of the following best describes the derivative of a function $$f(x) at a $$point $$x = a$$?

Accepted Answer

The instantaneous rate of change of $$f(x) at$$ $$x = a.$$

Question 2

Evaluate the following limit using limit laws: $$\displaystyle \lim_{x 	o 2} (3x^2$ - 5x + 4).$$

Accepted Answer

Question 3

A medication's concentration in the bloodstream is modeled by $$C(t) = 5e$$^{-0.2t}$$, where t$ is in hours. What is the rate of change of the concentration at t = 3$ hours?

Accepted Answer

$$C'(3) = -1 	imes 5 	imes 0.2 	imes e$$^{-0.6}$$ = -1 	imes e$$^{-0.6}$$

Question 4

Which of the following statements about continuity is TRUE?

Accepted Answer

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

Question 5

Suppose $$f(x) = x^3$ - 2x + 1. $$Using the limit definition, what is $$f'(2)$$?

Accepted Answer

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

Question 6

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

Accepted Answer

$$v(3) = 8 	imes 3 + 2$$

Question 7

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 8

If $$f(x) is $$differentiable and $$f'(a) = 0$$, which of the following must be true?

Accepted Answer

$$x = a is a $$critical point of $$f(x).$$

Question 9

A company finds that the cost to produce $$x$$ units is $$C(x) = 100 + 5x + 0.1x^2$. What is the marginal cost when x = 20$?

Accepted Answer

$$C'(20) = 5 + 0.1 	imes 20$$

Question 10

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

Accepted Answer

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

Question 11

Evaluate $$\displaystyle \lim_{x 	o 0} \dfrac{\sin(5x)}{x}.$$

Accepted Answer

$$5$$

Question 12

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

Accepted Answer

$$y' = 2e$$^{2x}$$ \cos(x) - e$$^{2x}$$ \sin(x)$$

Question 13

A tank is being filled with water so that the volume at time $$t is$$ $$V(t) = 3t^2$ + 2t. At $$what time is the rate of water flow $$20$$ units per minute?

Accepted Answer

$$t = 2$$

Question 14

Which of the following is NOT a requirement for a function $$f(x) to be $$continuous at $$x = a$$?

Accepted Answer

$$f'(a)$$ exists.

Question 15

If $$f(x) = x^2$ and g(x$$) = \sin(x)$$, what is the derivative of h(x) = f(x$$) \cdot g(x)$$?

Accepted Answer

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

Question 16

A bacteria population grows according to $$P(t) = 100e$$^{0.05t}$$. What is the instantaneous rate of growth at t = 10$ hours?

Accepted Answer

$$P'(10) = 100 	imes 0.05 	imes e$$^{0.5}$$

Question 17

Which of the following is the correct derivative of $$y = \arctan(x)$$?

Accepted Answer

$$y' = \dfrac{1}{1 + x^2$}$$

Question 18

Suppose $$f(x) is $$continuous on $$[a, b]$$ and differentiable on $$(a, b). $$Which theorem guarantees that there exists $$c in$$ $$(a, b)$$ such that $$f'(c) = \dfrac{f(b) - f(a)}{b - a}$$?

Accepted Answer

Mean Value Theorem

Question 19

A ball is thrown upward with velocity $$v(t) = 20 - 9.8t. At $$what time does the ball reach its maximum height?

Accepted Answer

$$t = \dfrac{20}{9.8}$$

MA 1021 Calculus 1: Course Structure, Topics, and Study Guide

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