Q: Suppose $$s(t) is $$the position of an object moving along a line at time $$t \geq 0. $$What is the correct formula for the average velocity between times $$t = a$$ and $$t = b$$?

$$\displaystyle v_{\text{avg}} = \frac{s(b) - s(a)}{b - a}$$

Q: Which of the following statements about limits is TRUE?

The value of $$\lim_{x \to a} f(x)$$ depends on values of $$f$$ near $$a$$, but it does not depend on the value of $$f(a).$$

Q: If $$f(x) = \frac{x^2$ - 4}{x - 2}$$, which of the following statements is true about $$f(2)$$ and $$\displaystyle \lim_{x \to 2} f(x)$$?

$$f(2) is $$undefined, but $$\displaystyle \lim_{x \to 2} f(x)$$ exists.

Q: Suppose $$f(x) is $$continuous at $$x = a. $$Which of the following must be true?

$$\displaystyle \lim_{x \to a} f(x) = f(a)$$

Question 1

Given the function $$f(x) = 2x + 8$$, what is the simplified form of $$f(x + h) - f(x)$$?

Accepted Answer

$$2h$$

Question 2

Suppose $$s(t) is $$the position of an object moving along a line at time $$t \geq 0. $$What is the correct formula for the average velocity between times $$t = a$$ and $$t = b$$?

Accepted Answer

$$\displaystyle v_{	ext{avg}} = \frac{s(b) - s(a)}{b - a}$$

Question 3

The position of an object is given by $$s(t) = -t^2$ + 9t + 22. $$What is the average velocity of the object over the interval $$[1, 9]$$?

Accepted Answer

$$-27$$

Question 4

For the function $$f(x) = \frac{x^2$ - 9}{x - 3}$$, what is $$\displaystyle \lim_{x 	o 3} f(x)$$?

Accepted Answer

$$6$$

Question 5

Which of the following statements about limits is TRUE?

Accepted Answer

The value of $$\lim_{x 	o a} f(x)$$ depends on values of $$f$$ near $$a$$, but it does not depend on the value of $$f(a).$$

Question 6

Given the function $$g(x) = \sqrt{x + 1} - 1$$, what is $$\displaystyle \lim_{x 	o 0} \frac{g(x)}{x}$$?

Accepted Answer

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

Question 7

A rock is thrown upward from a bridge. Its height (in meters) after $$t$$ seconds is given by $$h(t) = -4.9t^2$ + 21t + 2. $$What is the approximate velocity at which the rock strikes the water, rounded to the nearest integer?

Accepted Answer

$$-18 m/sec$$

Question 8

For the function $$f(x) = \frac{x^2$ - 4}{x - 2}$$, what is $$\displaystyle \lim_{x 	o 2} f(x)$$?

Accepted Answer

$$4$$

Question 9

Which of the following best describes the domain of the function $$f(x) = \frac{1}{x - 3}$$?

Accepted Answer

$$(-\infty, 3) \cup (3, \infty)$$

Question 10

Let $$f(x) = x^2$ - 4x + 5. At $$what $$x-$$value does the function have a tangent line with zero slope?

Accepted Answer

$$x = 2$$

Question 11

Given the function $$f(x) = \frac{x^2$ - 1}{x - 1}$$, what is $$\displaystyle \lim_{x 	o 1} f(x)$$?

Accepted Answer

$$2$$

Question 12

Suppose $$f(x) is $$defined as follows: $$f(x) = 3$$ for $$x \u003c 2$$, $$f(x) = 5$$ for $$x \geq 2. $$What is $$\displaystyle \lim_{x 	o 2^-} f(x)$$?

Accepted Answer

$$3$$

Question 13

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

Accepted Answer

$$1$$

Question 14

A car travels along a straight road such that its position at time $$t is $$given by $$s(t) = t^3$ - 6t^2$ + 9t + 2. $$What is the average velocity of the car over the interval $$[0, 3]$$?

Accepted Answer

$$2$$

Question 15

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

Accepted Answer

It is the limit of the average velocity as the time interval shrinks to zero around $$t = a.$$

Question 16

Given $$f(x) = x^2$ - 4x + 5$$, what is the slope of the tangent line at $$x = 1$$?

Accepted Answer

$$-2$$

Question 17

If $$f(x) = \frac{x^2$ - 4}{x - 2}$$, which of the following statements is true about $$f(2)$$ and $$\displaystyle \lim_{x 	o 2} f(x)$$?

Accepted Answer

$$f(2) is $$undefined, but $$\displaystyle \lim_{x 	o 2} f(x)$$ exists.

Question 18

For the function $$f(x) = \frac{x^2$ - 9}{x - 3}$$, what is the value of $$f(3)$$?

Accepted Answer

Undefined

Question 19

Suppose $$f(x) is $$continuous at $$x = a. $$Which of the following must be true?

Accepted Answer

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

Limits, Average Velocity, and Secant/Tangent Lines in Calculus

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