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

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

Q: Given the graph of $$f(x)$$ has a jump at $$x = 1$$, which of the following is true?

The limit $$\displaystyle \lim_{x \to 1} f(x)$$ does not exist.

Q: If $$f(x) = \begin{cases} x^2$ & x \u003c 1 \\ 2x - 1 & x \geq 1 \end{cases}$$, is $$f(x)$$ continuous at $$x = 1$$?

Yes, because $$\displaystyle \lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) = f(1).$$

Question 1

Which of the following is the correct definition of the limit $$\displaystyle \lim_{x 	o a} f(x) = L$$?

Accepted Answer

For every $$\epsilon \u003e 0$$, there exists a $$\delta \u003e 0$$ such that if $$0 \u003c |x - a| \u003c \delta$$, then $$|f(x) - L| \u003c \epsilon.$$

Question 2

Suppose $$f(x) is $$defined as follows: $$f(x) = \begin{cases} x^2$ & x 
eq 2 \ 5 & x = 2 \end{cases}. $$Which statement is true about the discontinuity at $$x = 2$$?

Accepted Answer

There is a removable discontinuity at $$x = 2.$$

Question 3

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

Accepted Answer

$$1$$

Question 4

Given the function $$f(x) = \frac{x^2$ - 9}{x - 3}$$, find a simpler function that is equivalent to $$f(x)$$ everywhere except at $$x = 3.$$

Accepted Answer

$$g(x) = x + 3$$

Question 5

Which of the following functions is continuous everywhere on $$\mathbb{R}$$?

Accepted Answer

$$f(x) = e^x$$

Question 6

Suppose $$f(x) is $$defined on $$[1, 4]$$ and $$f(1) = -2$$, $$f(4) = 5. $$Which of the following statements is true according to the Intermediate Value Theorem?

Accepted Answer

For every $$k$$ between $$-2$$ and $$5$$, there exists $$c \in [1, 4]$$ such that $$f(c) = k.$$

Question 7

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

Accepted Answer

$$4$$

Question 8

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

Accepted Answer

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

Question 9

A ball is dropped and its position at time $$t$$ seconds is given by $$s(t) = 100 - 4.9t^2$. Using the alternative definition of the derivative, what is the velocity at t = 2$?

Accepted Answer

$$-19.6$$

Question 10

Given $$f(x) = \begin{cases} x + 2 & x \u003c 1 \ 3 & x \geq 1 \end{cases}$$, what is $$\displaystyle \lim_{x 	o 1^-} f(x)$$?

Accepted Answer

$$2$$

Question 11

Evaluate $$\displaystyle \lim_{x 	o 0} x^2$ \sin\left(\frac{1}{x}ight)$$ using the Squeeze Theorem.

Accepted Answer

$$0$$

Question 12

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

Accepted Answer

It has a non-removable discontinuity at $$x = 2.$$

Question 13

Given the graph of $$f(x)$$ has a jump at $$x = 1$$, which of the following is true?

Accepted Answer

The limit $$\displaystyle \lim_{x 	o 1} f(x)$$ does not exist.

Question 14

Suppose $$f(x) is $$continuous on $$[0, 2]$$ and $$f(0) = -1$$, $$f(2) = 3. $$What does the Intermediate Value Theorem guarantee?

Accepted Answer

There exists $$c \in [0, 2]$$ such that $$f(c) = 0.$$

Question 15

Find $$\displaystyle \lim_{x 	o 0^+} \ln x.$$

Accepted Answer

$$-\infty$$

Question 16

Given $$f(x) = |x|$$, which of the following is true about its derivative at $$x = 0$$?

Accepted Answer

The derivative does not exist at $$x = 0.$$

Question 17

If $$f(x) = \begin{cases} x^2$ & x \u003c 1 \ 2x - 1 & x \geq 1 \end{cases}$$, is $$f(x)$$ continuous at $$x = 1$$?

Accepted Answer

Yes, because $$\displaystyle \lim_{x 	o 1^-} f(x) = \lim_{x 	o 1^+} f(x) = f(1).$$

Question 18

Given $$f(x) = \sin x$$, sketch the graph of $$f'(x). $$Which of the following best describes $$f'(x)$$?

Accepted Answer

$$f'(x) = \cos x$$

Question 19

Evaluate $$\displaystyle \lim_{x 	o \infty} \frac{2x^2$ + 3}{x^2$ - 1}.$$

Accepted Answer

$$2$$

Question 20

Suppose $$f(x) is $$defined as $$f(x) = \frac{x^2$ - 1}{x - 1}$$ for $$x 
eq 1. $$What is $$\displaystyle \lim_{x 	o 1} f(x)$$?

Accepted Answer

$$2$$

Calculus I: Limits, Continuity, and Introduction to Derivatives – Study Guide for Test 1

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