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 $$t = b$$?

$$V_{avg} = \\frac{s(b) - s(a)}{b - a}$$

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

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

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

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

Question 1

Given the position function $$s(t) = t^2$ - 9t + 22$$, what is the average velocity of the object over the interval $$[1, 10]$$?

Accepted Answer

$$-27$$

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

$$V_{avg} = \frac{s(b) - s(a)}{b - a}$$

Question 3

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

Accepted Answer

$$2x + 2h + 8 - 2x - 8 = 2h$$

Question 4

Given the function $$g(x) = \frac{x^2$ - 9}{x - 3}$$, what is $$\lim_{x \to 3} g(x)$$?

Accepted Answer

$$6$$

Question 5

A rock is thrown into a lake and its height above the water is given by $$h(t) = -16t^2$ + 336. At $$what time does the rock strike the water?

Accepted Answer

$$t = 352 sec$$

Question 6

Which of the following statements about limits is TRUE?

Accepted Answer

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

Question 7

Given $$f(x) = \frac{x^2$ - 4}{x - 2}$$, what is $$\lim_{x \to 2} f(x)$$?

Accepted Answer

$$4$$

Question 8

For the function $$g(x) = \frac{1}{x} - 9$$, what is $$\lim_{x \to 1} g(x)$$?

Accepted Answer

$$-8$$

Question 9

If $$f(x) is $$defined by the graph below (piecewise linear), and $$f(2) = 3$$, what is $$\lim_{x \to 2} f(x)$$?

Accepted Answer

$$3$$

Question 10

Given the function $$f(x) = x^2$ - 5x + 6$$, at which $$x-$$value does the function have a tangent line with zero slope?

Accepted Answer

$$x = 2.5$$

Question 11

Consider the function $$f(x) = \frac{x^2$ - 1}{x - 1}. $$What is $$\lim_{x \to 1} f(x)$$?

Accepted Answer

$$2$$

Question 12

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

Accepted Answer

Undefined

Question 13

Suppose $$f(x) is $$defined for all $$x \neq 2$$ and $$f(2) is $$undefined. If $$\lim_{x \to 2} f(x) = 3$$, which statement is true?

Accepted Answer

The function has a removable discontinuity at $$x = 2$$

Question 14

Given the position function $$s(t) = t^2$ - 9t + 22$$, what is the instantaneous velocity at $$t = 2$$?

Accepted Answer

Question 15

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

Accepted Answer

Undefined

Question 16

Which of the following is the correct definition of the slope of the secant line between $$x = a$$ and $$x = b$$ for a function $$f(x)$$?

Accepted Answer

$$\frac{f(b) - f(a)}{b - a}$$

Question 17

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

Accepted Answer

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

Question 18

Given $$f(x) = x^2$ - 5x + 6$$, what is the domain of $$f(x)$$?

Accepted Answer

$$(-\infty, \infty)$$

Question 19

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

Accepted Answer

$$4$$

Limits, Average Velocity, and Secant/Tangent Lines: Foundational Concepts in Calculus

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