Question 1

Which of the following best describes the average rate of change of a function $$f(x)$$ between $$x_1$$ and $$x_2$$?

Accepted Answer

It is the slope of the secant line connecting $$(x_1$, f(x_1$))$$ and $$(x_2$, f(x_2$))$$, given by $$\frac{f(x_2$) - f(x_1$)}{x_2$ - x_1$}.$$

Question 2

A company models its cost of producing $$x$$ units as $$C(x) = -0.05x^2$ + 50x. $$What does the expression $$\frac{C(301) - C(300)}{301 - 300}$$ represent?

Accepted Answer

The approximate cost of producing the 301st unit.

Question 3

Given $$f(x) = x^2$ + 3x - 1$$, compute and simplify the difference quotient $$\frac{f(x+h) - f(x)}{h}.$$

Accepted Answer

$$2x + h + 3$$

Question 4

A credit card balance grows according to $$A(t) = 5000(1.14)^t. $$What does $$\frac{A(3) - A(2)}{3 - 2}$$ represent?

Accepted Answer

The average annual increase in the amount owed between year 2 and year 3.

Question 5

If $$f(x) = 3x + 1$$, what is the simplified form of the difference quotient $$\frac{f(x+h) - f(x)}{h}$$?

Accepted Answer

$$3$$

Question 6

Suppose the utility function for a product is $$U(x)$$, and the average rate of change from $$x = 1 to$$ $$x = 2 is $$less than from $$x = 0 to$$ $$x = 1. $$What economic principle does this illustrate?

Accepted Answer

Diminishing marginal utility

Question 7

Given $$f(x) = \sqrt{2x + 3}$$, what is the difference quotient $$\frac{f(x+h) - f(x)}{h}$$?

Accepted Answer

$$\frac{\sqrt{2x + 2h + 3} - \sqrt{2x + 3}}{h}$$

Question 8

A company’s revenue is given by $$R(x) = x(0.03x + 100). $$What does $$\frac{R(101) - R(100)}{101 - 100}$$ represent?

Accepted Answer

The approximate revenue from selling the 101st unit.

Question 9

Given $$f(x) = \frac{1}{x - 3}$$, what is the difference quotient $$\frac{f(x+h) - f(x)}{h}$$ simplified?

Accepted Answer

$$\frac{-1}{(x - 3)(x + h - 3)}$$

Question 10

If the unemployment rate increases from 3.8% in October to 4.4% in February (over 4 months), what is the average monthly rate of change?

Accepted Answer

$$0.15\%$$

Question 11

For the function $$f(x) = x^2$ + 3x - 1$$, what is the difference quotient $$\frac{f(x+h) - f(x)}{h}$$ when $$h = 2$$?

Accepted Answer

$$2x + 5$$

Question 12

A utility function $$U(x)$$ has values $$U(0) = 0$$, $$U(1) = 70$$, $$U(2) = 109$$, $$U(3) = 138$$, $$U(4) = 161. $$What is the average rate of change from $$x = 2 to$$ $$x = 3$$?

Accepted Answer

$$29$$

Question 13

If $$A(t) = 3000(1.01$$)^{12t}$$, what does $$\frac{A(4) - A(2)}{4 - 2}$$ represent in a real-world context?

Accepted Answer

The average annual increase in the amount owed between year 2 and year 4.

Question 14

If the average rate of change of a function is positive between two points, what does this indicate about the function over that interval?

Accepted Answer

The function is increasing.

Question 15

Given the utility function $$U(x)$$ with $$U(0) = 0$$, $$U(1) = 50$$, $$U(2) = 120$$, $$U(3) = 133$$, $$U(4) = 150$$, what is the average rate of change from $$x = 0 to$$ $$x = 2$$?

Accepted Answer

$$60$$

Question 16

Which of the following is NOT a correct interpretation of the difference quotient $$\frac{f(x+h) - f(x)}{h}$$?

Accepted Answer

It is the instantaneous rate of change at $$x.$$

Question 17

If a function $$f(x)$$ models the percentage of new employees in education, and the average rate of change from 2000 to 2005 is higher than from 2005 to 2009, what does this suggest?

Accepted Answer

The percentage increased faster in the first interval.

Question 18

If $$f(x) is a $$function and $$h 
eq 0$$, which of the following is the correct formula for the difference quotient?

Accepted Answer

$$\frac{f(x+h) - f(x)}{h}$$

Average Rate of Change and Difference Quotients in Business Calculus

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