Q: Which of the following is the correct definition of the derivative of a function $$f at a $$point $$a$$?

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

Q: Which of the following must be true for a function $$f to be $$continuous at $$x = a$$?

$$\lim_{x \to a} f(x)$$ exists, $$f(a) is $$defined, and $$\lim_{x \to a} f(x) = f(a)$$

Q: If $$\lim_{x \to a^-} f(x) = 2$$ and $$\lim_{x \to a^+} f(x) = 5$$, which statement is true?

$$\lim_{x \to a} f(x)$$ does not exist

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

$$f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$$

Q: Which of the following is a correct application of the Squeeze Theorem?

If $$g(x) \leq f(x) \leq h(x)$$ and $$\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$$, then $$\lim_{x \to a} f(x) = L$$

Question 1

Which of the following is the correct definition of the derivative of a function $$f at a $$point $$a$$?

Accepted Answer

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

Question 2

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

Accepted Answer

$$v(3) = 26$$

Question 3

Which of the following must be true for a function $$f to be $$continuous at $$x = a$$?

Accepted Answer

$$\lim_{x 	o a} f(x)$$ exists, $$f(a) is $$defined, and $$\lim_{x 	o a} f(x) = f(a)$$

Question 4

Evaluate $$\lim_{x 	o 2} \frac{x^2$ - 3x + 2}{x^2$ - 4}.$$

Accepted Answer

$$\frac{1}{2}$$

Question 5

Which statement about the Intermediate Value Theorem is correct?

Accepted Answer

If $$f is $$continuous on $$[a, b]$$ and $$N is $$any number between $$f(a)$$ and $$f(b)$$, then there exists $$c in$$ $$[a, b]$$ such that $$f(c) = N.$$

Question 6

Find the vertical asymptotes of $$f(x) = \frac{x-1}{x^2$ - 3x + 2}.$$

Accepted Answer

$$x = 2$$ and $$x = 3$$

Question 7

Given $$f(x) = \sqrt{x-3}$$, what is the domain of $$f$$?

Accepted Answer

$$x \geq 3$$

Question 8

If $$\lim_{x 	o a^-} f(x) = 2$$ and $$\lim_{x 	o a^+} f(x) = 5$$, which statement is true?

Accepted Answer

$$\lim_{x 	o a} f(x)$$ does not exist

Question 9

A rational function $$f(x)$$ has a numerator of degree 2 and a denominator of degree 3. What is the horizontal asymptote?

Accepted Answer

$$y = 0$$

Question 10

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

Accepted Answer

$$f'(a) = \lim_{x 	o a} \frac{f(x) - f(a)}{x - a}$$

Question 11

Suppose $$f is $$continuous on $$[1,5]$$ and $$f(1) = -3$$, $$f(5) = 2. $$Which value must $$f(x)$$ take for some $$x in$$ $$[1,5]$$?

Accepted Answer

$$f(x) = 0$$

Question 12

Evaluate $$\lim_{x 	o 3} \frac{\sqrt{x^2$ - 3x}}{x^2$ - 9}.$$

Accepted Answer

$$0$$

Question 13

Which of the following best describes the real behavior of a rational function as $$x 	o \infty if $$the degree of the numerator is less than the degree of the denominator?

Accepted Answer

The function approaches zero

Question 14

If $$f(x) is $$continuous at $$x = a$$, which of the following must be true?

Accepted Answer

$$\lim_{x 	o a} f(x) = f(a)$$

Question 15

Find the slope of the tangent line to $$y = x^2$ at x = 2$.

Accepted Answer

$$4$$

Question 16

Which of the following is NOT a requirement for a function to be differentiable at a point $$x = a$$?

Accepted Answer

The function must be increasing at $$x = a$$

Question 17

If $$f(x) = \frac{2x^2$ + 4x}{x^2$ - 1}$$, what is the horizontal asymptote?

Accepted Answer

$$y = 2$$

Question 18

Which of the following is a correct application of the Squeeze Theorem?

Accepted Answer

If $$g(x) \leq f(x) \leq h(x)$$ and $$\lim_{x 	o a} g(x) = \lim_{x 	o a} h(x) = L$$, then $$\lim_{x 	o a} f(x) = L$$

Question 19

If $$f(x) is $$continuous on $$[a, b]$$ and $$f(a) \u003c 0 \u003c f(b)$$, what does the Intermediate Value Theorem guarantee?

Accepted Answer

There exists $$c in$$ $$[a, b]$$ such that $$f(c) = 0$$

Calculus I Exam #1 Review: Limits, Continuity, and the Derivative

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