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

The limit as $$h \to 0 of $$the average velocity over $$[t, t+h].$$

Q: Suppose $$f(x)$$ has a vertical asymptote at $$x = 2. $$Which of the following statements is true?

$$\lim_{x \to 2} f(x) = \infty or$$ $$-\infty$$

Question 1

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

Accepted Answer

$$x 
eq \pm 4.472$$

Question 2

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

Accepted Answer

$$4x + 2h - 3$$

Question 3

Suppose $$f(x) is $$defined as follows: $$f(x) = \begin{cases} x^2$ + 3, & x \u003c -1 \ x - 2, & -1 \leq x \u003c 2 \ 6 - x, & x \geq 2 \end{cases}. Is$$ $$f(x)$$ 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 upward and its height after $$t$$ seconds is given by $$s(t) = -4.9t^2$ + 30t + 20. $$What is the average velocity from $$t = 1 to$$ $$t = 3$$?

Accepted Answer

10.2 m/s

Question 7

Evaluate $$\lim_{x 	o 0} \frac{\sin(4x)}{3x}.$$

Accepted Answer

$$\frac{4}{3}$$

Question 8

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

Accepted Answer

The limit as $$h 	o 0 of $$the average velocity over $$[t, t+h].$$

Question 9

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

Accepted Answer

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

Question 10

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

Accepted Answer

Question 11

Evaluate $$\lim_{x 	o 0} \frac{\sin x - 1}{\cos x}.$$

Accepted Answer

$$-1$$

Question 12

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

Accepted Answer

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

Question 13

Suppose $$f(x)$$ has a vertical asymptote at $$x = 2. $$Which of the following statements is true?

Accepted Answer

$$\lim_{x 	o 2} f(x) = \infty or$$ $$-\infty$$

Question 14

Given $$f(x) = \sin x$$, what is the period of the function?

Accepted Answer

$$2\pi$$

Question 15

If $$f(x) = x^3$ - 3x + 4 on$$ $$[-3, 1]$$, which statement best explains why the Intermediate Value Theorem guarantees a zero in this interval?

Accepted Answer

$$f(-3)$$ and $$f(1)$$ have opposite signs, and $$f is $$continuous on $$[-3, 1].$$

Question 16

Which of the following is the correct composition $$(f \circ g)(x) if$$ $$f(x) = x^2$ and g(x) = 2x + 1$?

Accepted Answer

$$(2x + 1)^2$

Question 17

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

Accepted Answer

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

Question 18

If $$f(x) = \frac{x^2$ - 4}{x - 2}$$, what type of discontinuity does $$f(x)$$ have at $$x = 2$$?

Accepted Answer

Removable discontinuity

Question 19

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

Accepted Answer

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

Calculus I Exam 1 Study Guide – Step-by-Step Guidance

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