Q: Which of the following functions is continuous only for $$x \u003e 0$$?

$$y = \log_a(x)$$, \text{where } a \u003e 0, a \neq 1$

Q: Which of the following is the correct limit definition for the instantaneous rate of change of $$f(x) at$$ $$x = a$$?

$$\lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$$

Question 1

Which of the following best describes a removable discontinuity in a function?

Accepted Answer

A point where the function can be made continuous by redefining its value at that point.

Question 2

Given $$f(x) = x^2$ + x + 3$$, what is $$f(x+1)$$?

Accepted Answer

$$x^2 + 2x + 4$

Question 3

For the function $$g(x) = \frac{x-2}{x^2$-4}$$, where is it NOT continuous?

Accepted Answer

At $$x = -2$$ and $$x = 2$$

Question 4

Find the slope of the line passing through the points $$(3, 2)$$ and $$(6, -4).$$

Accepted Answer

$$-2$$

Question 5

Which of the following functions is continuous for all real numbers?

Accepted Answer

$$h(x) = x^3$ - 7x^2$ + 17$$

Question 6

A company’s cost function is $$C(x) = 100 + 60x - x^2$ for 0$$ \leq x \leq 20$$. What is the marginal cost when 8 units are produced?

Accepted Answer

$$44$$

Question 7

For the piecewise function $$g(x) = \begin{cases} \sqrt{4-x} & 	ext{if } x \u003c 0 \ x^2$ + 2e^x & 	ext{if } x \geq 0 \end{cases}$$, at which value must you check for continuity?

Accepted Answer

$$x = 0$$

Question 8

If $$f(x) = x^2$ + 4x + 3$$, what is the instantaneous rate of change at $$x = 3$$?

Accepted Answer

$$13$$

Question 9

Which of the following is the correct formal definition of continuity at $$x = a$$?

Accepted Answer

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

Question 10

For the function $$h(x) = \sqrt{8-2x}$$, for which values of $$x is$$ $$h(x)$$ continuous?

Accepted Answer

$$x \leq 4$$

Question 11

The average rate of change of $$f(x) = x^2$ - x + 4$$ from $$x = 1 to$$ $$x = 4 is$$:

Accepted Answer

$$7$$

Question 12

Which of the following statements about rational functions is correct?

Accepted Answer

They are continuous everywhere except where the denominator is zero.

Question 13

For the function $$f(x) = \begin{cases} x^2$ + a & x \leq 3 \ ax - 3 & x \u003e 3 \end{cases}$$, for which value of $$a is$$ $$f(x)$$ continuous for all $$x$$?

Accepted Answer

$$a = 6$$

Question 14

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

Accepted Answer

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

Question 15

Which of the following is NOT a property of polynomials?

Accepted Answer

They can have removable discontinuities.

Question 16

If $$f(x) = x^2$ + x + 3$$, what is the instantaneous rate of change at $$x = 2$$?

Accepted Answer

$$7$$

Question 17

Which of the following functions is continuous only for $$x \u003e 0$$?

Accepted Answer

$$y = \log_a(x)$$, 	ext{where } a \u003e 0, a 
eq 1$

Question 18

A business wants to know the marginal profit at a certain production level. Which calculus concept should they use?

Accepted Answer

Instantaneous rate of change

Question 19

Which of the following is the correct limit definition for the instantaneous rate of change of $$f(x) at$$ $$x = a$$?

Accepted Answer

$$\lim_{h 	o 0} \frac{f(a + h) - f(a)}{h}$$

Question 20

A function $$f(x) is $$defined as $$f(x) = \frac{1}{2x-3} - \frac{1}{7x-5}. $$What is $$\lim_{x 	o 5} f(x)$$?

Accepted Answer

$$0$$

Continuity and Rates of Change in Business Calculus

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