Q: Suppose $$f(x) is $$defined as $$f(x) = \begin{cases} x^2$, & x \u003c 3 \\ 2x + 1, & x \geq 3 \end{cases}. Is$$ $$f(x)$$ continuous at $$x = 3$$?

No, because $$f(3) \neq \lim\limits_{x \to 3} f(x)$$

Question 1

A ball is thrown straight up and its height (in meters) after $$t$$ seconds is given by $$s(t) = -16t^2$ + 160t + 144. $$What is the average velocity of the ball over the interval $$[1, 3]$$ seconds?

Accepted Answer

$$96$$

Question 2

Given the function $$f(x) = \frac{x^2$ - 9}{x - 3}$$, what is $$\lim\limits_{x 	o 3} f(x)$$?

Accepted Answer

$$6$$

Question 3

Suppose the cost to mail a letter is $$0.34 for the first ounce and 0.21 $$for each additional ounce (or fraction thereof). Which graph best represents the cost $$C(x) to $$mail a letter weighing $$x$$ ounces?

Accepted Answer

A step function that increases by $$0.21 at $$each integer value of $$x$$ after $$x=1$$

Question 4

Evaluate $$\lim\limits_{x 	o 2} \frac{\sqrt{x+7} - 3}{x-2}.$$

Accepted Answer

$$\frac{1}{6}$$

Question 5

Find the vertical asymptote(s) of the function $$f(x) = \frac{x^2$ - 9}{x^2$ - 4}.$$

Accepted Answer

$$x = 2$$ and $$x = -2$$

Question 6

Determine $$\lim\limits_{x 	o \infty} \frac{3x^2$ + 5x}{2x^2$ - x}.$$

Accepted Answer

$$\frac{3}{2}$$

Question 7

A function $$f(x) is $$defined as $$f(x) = \frac{2x^2$ - 7x + 11}{x - 3}. At $$which $$x-$$value is $$f(x)$$ discontinuous?

Accepted Answer

$$x = 3$$

Question 8

Suppose $$f(x) is $$defined as $$f(x) = \begin{cases} x^2$, & x \u003c 3 \ 2x + 1, & x \geq 3 \end{cases}. Is$$ $$f(x)$$ continuous at $$x = 3$$?

Accepted Answer

No, because $$f(3) 
eq \lim\limits_{x 	o 3} f(x)$$

Question 9

Evaluate $$\lim\limits_{x 	o 0} \frac{\sin(5x)}{x}.$$

Accepted Answer

$$5$$

Question 10

A population $$P(t)$$ approaches a steady-state value as $$t 	o \infty. If$$ $$P(t) = 3300 - 3200e$$^{-0.2t}$$, what is the steady-state value?

Accepted Answer

$$3300$$

Question 11

Given the graph of $$f(x)$$ with a local minimum at $$x = 2$$ and a local maximum at $$x = 5$$, which of the following statements is true?

Accepted Answer

$$f'(2) = 0$$ and $$f'(5) = 0$$

Question 12

Evaluate $$\lim\limits_{x 	o 0^+} \ln(x).$$

Accepted Answer

$$-\infty$$

Question 13

Which of the following best describes the end behavior of $$f(x) = \frac{2x^2$ + 3}{x^2$ - 1} as$$ $$x 	o \infty$$?

Accepted Answer

$$f(x) 	o 2$$

Question 14

Find the value of $$\lim\limits_{x 	o 4} \frac{\sqrt{x} - 2}{x - 4}.$$

Accepted Answer

$$\frac{1}{4}$$

Question 15

If $$f(x) = \sqrt{x^2$ + 1}$$, what is $$\lim\limits_{x 	o \infty} \frac{f(x)}{x}$$?

Accepted Answer

$$1$$

Question 16

Which of the following is NOT a condition for continuity of a function $$f(x) at$$ $$x = a$$?

Accepted Answer

$$f'(a)$$ exists

Question 17

Evaluate $$\lim\limits_{x 	o 0} \frac{e$$^{2x}$$ - 1}{x}.$$

Accepted Answer

$$2$$

Question 18

Suppose $$f(x) is $$continuous everywhere except at $$x = 2$$, where $$\lim\limits_{x 	o 2^-} f(x) = 3$$ and $$\lim\limits_{x 	o 2^+} f(x) = 5. $$What type of discontinuity does $$f(x)$$ have at $$x = 2$$?

Accepted Answer

Jump discontinuity

Question 19

If $$f(x) = \frac{1}{x-4}$$, which of the following statements is true?

Accepted Answer

$$f(x)$$ has a vertical asymptote at $$x = 4$$

Calculus I Study Notes: Functions, Limits, and Continuity

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