Question 1

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

Accepted Answer

$$x 
eq \pm \sqrt{20}$$

Question 2

Given $$f(x) = \frac{2x^2$ - 3x + 4}{x}$$, what is the simplified form of the difference quotient $$\frac{f(x+h) - f(x)}{h}$$?

Accepted Answer

$$\frac{2x h + 2h^2$ - 3h}{x(x+h)}$$

Question 3

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

Accepted Answer

The derivative of the position function at time $$t$$

Question 4

Evaluate $$\lim\limits_{x 	o 2} \frac{x^2$ - 4}{x - 2}.$$

Accepted Answer

2

Question 5

A function $$f(x) is $$defined as follows:

$$ f(x) = \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 6

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

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 7

Evaluate $$\lim\limits_{x 	o 0} \frac{\sin(4x)}{x}.$$

Accepted Answer

4

Question 8

The function $$f(x) = \frac{3x^2$ - 1}{6x - 2}$$ has which of the following vertical asymptotes?

Accepted Answer

$$x = \frac{1}{3}$$

Question 9

Suppose $$f(x) = \frac{2x^2$ - 3x + 4}{x}$$ and $$g(x) = x^2$ - 1. $$What is $$(f \circ g)(x)$$?

Accepted Answer

$$\frac{2(x^2$ - 1)^2$ - 3(x^2$ - 1) + 4}{x^2$ - 1}$$

Question 10

An object is launched vertically and its height (in meters) after $$t$$ seconds is $$s(t) = -4.9t^2$ + 30t + 20. $$What is the average velocity from $$t = 1 to$$ $$t = 3$$?

Accepted Answer

10.6 m/s

Question 11

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

Accepted Answer

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

Question 12

Which of the following best describes the horizontal asymptote(s) of $$f(x) = \frac{2x^2$ - 3x + 4}{6x - 2}$$?

Accepted Answer

Question 13

Which of the following is the correct radian measure for $$90^\circ$$?

Accepted Answer

$$\frac{\pi}{2}$$

Question 14

Given $$f(x) = \frac{x^2$ - 4}{x - 2}$$, which of the following best describes the type of discontinuity at $$x = 2$$?

Accepted Answer

Removable discontinuity

Question 15

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

Accepted Answer

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

Question 16

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

Accepted Answer

There is at least one $$c in$$ $$[a, b]$$ such that $$f(c) = 0.$$

Question 17

Evaluate $$\lim\limits_{x 	o \infty} \frac{4x^3$ - 3x^2$ + 12}{5x^3$ + 4x - 3}.$$

Accepted Answer

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

Question 18

Which of the following is the correct composition $$(f \circ g)(x) if$$ $$f(x) = \sin x$$ and $$g(x) = 2x$$?

Accepted Answer

$$\sin(2x)$$

Question 19

Which of the following is the correct value of $$\cos(\pi)$$?

Accepted Answer

$$-1$$

Question 20

If $$\lim\limits_{x 	o 4} f(x) = \infty$$, what does this indicate about the graph of $$f(x) at$$ $$x = 4$$?

Accepted Answer

There is a vertical asymptote at $$x = 4$$

Calculus Exam 1 Study Guide: Step-by-Step Guidance

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