Q: A function $$f(x) is $$continuous on the closed interval $$[a, b] if$$:

It is continuous on $$(a, b)$$, $$\lim_{x \to a^+} f(x) = f(a)$$, and $$\lim_{x \to b^-} f(x) = f(b)$$

Q: Suppose $$f(x) is $$continuous on $$[a, b]. $$Which of the following must be true?

$$\lim_{x \to a^+} f(x) = f(a)$$ and $$\lim_{x \to b^-} f(x) = f(b)$$

Question 1

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

Accepted Answer

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

Question 2

Given $$f(x) = x^2$ - 3$$ for $$x \u003c 2$$ and $$f(x) = kx + k$$ for $$x \geq 2$$, what value of $$k$$ makes $$f(x)$$ continuous for all $$x$$?

Accepted Answer

$$k = -1$$

Question 3

If $$f(x) = \sqrt{x}$$, is $$f(x)$$ continuous at $$x = -2$$?

Accepted Answer

No, because $$f(-2)$$ does not exist

Question 4

A function $$f(x) is $$continuous on the closed interval $$[a, b] if$$:

Accepted Answer

It is continuous on $$(a, b)$$, $$\lim_{x 	o a^+} f(x) = f(a)$$, and $$\lim_{x 	o b^-} f(x) = f(b)$$

Question 5

Which of the following intervals is an open interval?

Accepted Answer

$$(1, 5)$$

Question 6

Suppose $$f(x) is $$defined as $$f(x) = x^2$ - 3$$ for $$x \u003c 2$$ and $$f(x) = kx + k$$ for $$x \geq 2. $$What equation must be satisfied to ensure continuity at $$x = 2$$?

Accepted Answer

$$2^2 - 3 = 2k + k$

Question 7

Find the average rate of change of $$f(x) = x^2$ from x = 1$ to x = 2$.

Accepted Answer

$$3$$

Question 8

Which formula correctly represents the average rate of change of a function $$f(x)$$ from $$x = a to$$ $$x = b$$?

Accepted Answer

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

Question 9

For $$f(x) = x^2$, what is the average rate of change from x = 1$ to x = 1 + h$?

Accepted Answer

$$2h + 1$$

Question 10

A function $$f(x) is $$continuous on an open interval $$(a, b) if$$:

Accepted Answer

It is continuous at every $$x in$$ $$(a, b)$$

Question 11

If $$f(x) is $$not defined at $$x = c$$, which condition for continuity fails?

Accepted Answer

$$f(c)$$ exists

Question 12

Suppose $$f(x) is $$continuous on $$[a, b]. $$Which of the following must be true?

Accepted Answer

$$\lim_{x 	o a^+} f(x) = f(a)$$ and $$\lim_{x 	o b^-} f(x) = f(b)$$

Question 13

If $$f(x) is $$continuous at $$x = 3$$, which of the following statements is always true?

Accepted Answer

$$\lim_{x 	o 3} f(x) = f(3)$$

Question 14

Which of the following is a correct interpretation of the average rate of change of $$f(x)$$ from $$x = a to$$ $$x = b$$?

Accepted Answer

The slope of the secant line connecting $$(a, f(a))$$ and $$(b, f(b))$$

Question 15

If $$f(x) is $$continuous on $$(a, b)$$ but not at $$a or$$ $$b$$, on which interval is $$f(x)$$ continuous?

Accepted Answer

$$(a, b)$$

Question 16

Suppose $$f(x) is $$defined as $$f(x) = x^2$ - 3$$ for $$x \u003c 2$$ and $$f(x) = kx + k$$ for $$x \geq 2. If$$ $$k = -1$$, what is $$f(2)$$?

Accepted Answer

$$-2$$

Continuity and Average Rate of Change in Business Calculus

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