Q: Evaluate the following limit: $$\displaystyle \lim_{x \to 3} \frac{x^2$ - 8x + 15}{x^2$ - x - 6}$$

$$\text{None of the other answers}$$

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

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

Q: Which of the following limits represents the derivative of $$f(x) = x^2$ at x = 3$?

$$\displaystyle \lim_{h \to 0} \frac{(3+h)^2$ - 9}{h}$$

Question 1

Which of the following describes the sequence of transformations needed to obtain the graph of $$y = 3 \sin(2x - \pi) + 5$$ from $$y = \sin x$$?

Accepted Answer

Shift the graph horizontally by $$\frac{\pi}{2}$$ units to the right, then compress the graph horizontally by a factor of 2, stretch it vertically by a factor of 3, and shift it up by 5 units.

Question 2

Evaluate the following limit: $$\displaystyle \lim_{x 	o 3} \frac{x^2$ - 8x + 15}{x^2$ - x - 6}$$

Accepted Answer

$$	ext{None of the other answers}$$

Question 3

An object moves along a line with position function $$s(t) = t^2$ + 4t. $$What is the average velocity of the object on the interval $$[1, 3]$$?

Accepted Answer

$$8$$

Question 4

Where is the function $$f(x) = \frac{x^2$ + 4x + 3}{x^2$ - 9}$$ not continuous?

Accepted Answer

$$x = -3$$ and $$x = 3$$

Question 5

Determine the limit: $$\displaystyle \lim_{x 	o -\infty} (e^x \cos(x) + 3)$$

Accepted Answer

$$3$$

Question 6

Suppose the domain of $$f(x) is$$ $$[-2, 8]. If$$ $$g(x) = 5f(2x - 6) - 7$$, what is the domain of $$g(x)$$?

Accepted Answer

$$[2, 7]$$

Question 7

Find all the asymptotes of $$f(x) = \frac{\sqrt{4x^4$ + 1}}{3x^2$ + 3x}. $$Which of the following is correct?

Accepted Answer

Horizontal: $$y = \frac{2}{3}$$; Vertical: $$x = 0$$, $$x = -1$$

Question 8

Suppose $$M = \log_3$ 8 - \log_3$ 2$$ and $$N = 3 \ln(x) + 2. $$Which of the following is correct?

Accepted Answer

$$M = \log_3$ 4$$ and $$N = \ln(e$$^{2}$$x^{3}$$)$

Question 9

Evaluate $$\sin$$^{-1}$$(\cos(\frac{3\pi}{4})).$$

Accepted Answer

$$-\frac{\pi}{4}$$

Question 10

Evaluate the following limit: $$\displaystyle \lim_{x 	o 0} \frac{4$$^{5+x}$$ - 4^5$}{x}$$

Accepted Answer

$$4^5$$ \ln 4$$

Question 11

Determine the limit: $$\displaystyle \lim_{x 	o 2^-} \frac{x - 5}{\sqrt{x^2$ - 5x + 6}}$$

Accepted Answer

$$-\infty$$

Question 12

A chemical reaction's temperature $$T (in$$ $$^\circC) at $$time $$t (in $$minutes) is given by $$T(t) = 2t^2$ - 3t + 10. $$What is the average rate of change of temperature from $$t = 1 to$$ $$t = 4$$?

Accepted Answer

$$13$$

Question 13

Let $$f(x) = \frac{1}{x-2}. $$Which of the following statements is true?

Accepted Answer

$$f(x) is $$not continuous at $$x = 2$$ because the denominator is zero.

Question 14

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

Accepted Answer

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

Question 15

Which of the following limits represents the derivative of $$f(x) = x^2$ at x = 3$?

Accepted Answer

$$\displaystyle \lim_{h 	o 0} \frac{(3+h)^2$ - 9}{h}$$

Question 16

A medication's concentration in the bloodstream is modeled by $$C(t) = 50e$$^{-0.1t}$$ mg/L, where t$ is in hours. What is the limit of C(t$$)$$ as t$$ 	o \infty$$?

Accepted Answer

$$0$$

Question 17

Which of the following is the correct domain for the function $$h(x) = \ln(2x - 4)$$?

Accepted Answer

$$x \u003e 2$$

Question 18

A ball is thrown upward with position $$s(t) = -16t^2$ + 64t + 80 (in $$feet, $$t in $$seconds). What is the average velocity between $$t = 1$$ and $$t = 3$$?

Accepted Answer

$$16 ft/s$$

Question 19

Which of the following statements about the function $$f(x) = \frac{1}{x^2$ - 4} is $$true?

Accepted Answer

$$f(x) is $$not continuous at $$x = 2$$ and $$x = -2.$$

MA161 Exam 1 Calculus Study Guidance

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