Q: Which of the following is the correct definition of the instantaneous velocity of an object at time $$t = a$$?

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

Question 1

Which of the following best describes the domain of the function $$y = \frac{x^2$ + 4}{x - 20}$$?

Accepted Answer

All real numbers except $$x = 20$$

Question 2

Given $$f(x) = x^2$ - 2x + 4$$, what is the simplified form of the difference quotient $$\frac{f(x+h) - f(x)}{h}$$?

Accepted Answer

$$2x + h - 2$$

Question 3

Suppose $$f(x) is $$defined as follows:

$$\begin{cases} x^2$ + 3, & x \u003c -1 \ x - 2, & -1 \leq x \u003c 2 \ 6, & x \geq 2 \end{cases}

Is$$ $$f(x)$$ continuous at $$x = -1$$?

Accepted Answer

No, because the left and right limits at $$x = -1$$ are not equal

Question 4

Evaluate $$\lim_{x 	o 4} \frac{x^2$ - 3x - 12}{x - 4}.$$

Accepted Answer

5

Question 5

Which of the following statements about the Intermediate Value Theorem (IVT) is correct?

Accepted Answer

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

Question 6

A ball is thrown vertically upward and its height (in meters) after $$t$$ seconds is given by $$s(t) = -4.9t^2$ + 30t + 20. $$What is the average velocity of the ball from $$t = 1 to$$ $$t = 3$$?

Accepted Answer

10.2 m/s

Question 7

Which of the following is the correct analytical evaluation of $$\lim_{x 	o 0} \frac{\sin(4x)}{x}$$?

Accepted Answer

4

Question 8

Given $$f(x) = \frac{x^2$ - 3x + 1}{6x^2$ - 2}$$, what is the domain of $$f(x)$$?

Accepted Answer

Question 9

Given $$f(x) = \frac{x^2$ - 3x + 1}{6x^2$ - 2}$$, which of the following describes the vertical asymptotes?

Accepted Answer

At $$x = \pm \frac{1}{\sqrt{3}}$$

Question 10

Which of the following is the correct definition of the instantaneous velocity of an object at time $$t = a$$?

Accepted Answer

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

Question 11

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

Accepted Answer

1

Question 12

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

Accepted Answer

$$f(x) is $$differentiable at $$x = a$$

Question 13

Suppose $$f(x) is $$continuous on $$[1, 4]$$ and $$f(1) = -2$$, $$f(4) = 5. $$According to the Intermediate Value Theorem, which of the following must be true?

Accepted Answer

There exists $$c in$$ $$[1, 4]$$ such that $$f(c) = 0$$

Question 14

Which of the following best describes the horizontal asymptote(s) of $$f(x) = \frac{x^2$ - 3x + 1}{6x^2$ - 2}$$?

Accepted Answer

$$y = \frac{1}{6}$$

Question 15

Evaluate $$\lim_{x 	o \infty} \frac{4x^3$ - 5x^2$ + 2}{5x^3$ + 4x - 1}.$$

Accepted Answer

$$\frac{4}{5}$$

Question 16

Which of the following is the correct application of the epsilon-delta definition of a limit for $$\lim_{x 	o 2} f(x) = 5$$?

Accepted Answer

For every $$\varepsilon \u003e 0$$, there exists $$\delta \u003e 0$$ such that if $$0 \u003c |x - 2| \u003c \delta$$, then $$|f(x) - 5| \u003c \varepsilon$$

Question 17

If $$f(x) = \sin x$$, which of the following is the period of the function?

Accepted Answer

$$2\pi$$

Question 18

Which of the following is the correct value of $$\sin\left(\frac{\pi}{6}\right)$$?

Accepted Answer

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

Question 19

Which of the following is a removable discontinuity for the function $$f(x) = \frac{x^2$ - 3x + 1}{6x^2$ - 2}$$?

Accepted Answer

There are no removable discontinuities

Question 20

Suppose $$f(x) = x^2$ + 4$$ and $$g(x) = \sqrt{x}. $$What is the domain of the composite function $$f \circ g$$?

Accepted Answer

All real numbers $$x \geq 0$$

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