Question 1

A company tracks the number of products sold over a week. On Monday, they sold 120 units, and on Friday, they sold 200 units. What is the average rate of change in units sold per day from Monday to Friday?

Accepted Answer

$$\frac{200 - 120}{5 - 1} = 20$$ units/day

Question 2

Given $$f(x) = x^2$, what is the average rate of change of f(x$$)$$ as x$ changes from 2$ to 5$?

Accepted Answer

$$\frac{f(5) - f(2)}{5 - 2} = \frac{25 - 4}{3} = 7$$

Question 3

A water tank is filled from 10 gallons to 40 gallons over 6 hours. What is the average rate of change in gallons per hour?

Accepted Answer

$$\frac{40 - 10}{6} = 5$$ gallons/hour

Question 4

For $$f(x) = x^3$, find the simplified difference quotient $$\frac{f(x+h) - f(x)}{h}$$.

Accepted Answer

$$3x^2 + 3xh$$ + $$h^2$$

Question 5

A car travels 480 miles using 16 gallons of gas. What is the average rate of change in miles per gallon?

Accepted Answer

$$\frac{480}{16} = 30$$ miles/gallon

Question 6

Given $$f(x) = x^2$, calculate the difference quotient $$\frac{f(7) - f(5)}{2}$$.

Accepted Answer

$$\frac{49 - 25}{2} = 12$$

Question 7

For $$f(x) = x^3$ + 5$$, what is the average rate of change from $$x = 2 to$$ $$x = 5$$?

Accepted Answer

$$\frac{f(5) - f(2)}{5 - 2} = \frac{130 - 13}{3} = 39$$

Question 8

A patient’s temperature drops from 101°F at 8 a.m. to 97°F at 12 p.m. What is the average rate of change in temperature per hour?

Accepted Answer

$$\frac{97 - 101}{12 - 8} = \frac{-4}{4} = -1$$ °F/hour

Question 9

For $$f(x) = x^2$, find the difference quotient when x = 4$ and h = 2$.

Accepted Answer

$$\frac{f(6) - f(4)}{2} = \frac{36 - 16}{2} = 10$$

Question 10

A business’s profit increases from $$\2$$,000$$ to $$\$$5,500$$ over 7 months. What is the average rate of change in profit per month?

Accepted Answer

$$\frac{5500 - 2000}{7} = 500$$ dollars/month

Question 11

For $$f(x) = x^3$, use the simplified difference quotient formula to find the slope of the secant line at x = 1$ for h = 0.2$.

Accepted Answer

$$3(1)^2 + 3(1)(0.2$$) + $$(0.2)^2 = 3 + 0.6 + 0.04 = 3.64$

Question 12

Which of the following best describes the difference quotient for a function $$f(x)$$?

Accepted Answer

It is the average rate of change of $$f(x)$$ over an interval of length $$h.$$

Question 13

For $$f(x) = x^2$, what is the difference quotient $$\frac{f(x+h) - f(x)}{h}$$ in terms of x$ and h$?

Accepted Answer

$$2x + h$$

Question 14

A factory’s output increases from 500 units to 800 units over 6 days. What is the average rate of change in units per day?

Accepted Answer

$$\frac{800 - 500}{6} = 50$$ units/day

Question 15

For $$f(x) = x^3$, use the simplified difference quotient to find the slope of the secant line at x = 3$ for h = 0.01$.

Accepted Answer

$$3(3)^2 + 3(3)(0.01$$) + $$(0.01)^2 = 27 + 0.09 + 0.0001 = 27.0901$

Question 16

If $$f(x) = x^2$, what is the difference quotient when x = 2$ and h = 0.5$?

Accepted Answer

$$\frac{f(2.5) - f(2)}{0.5} = \frac{6.25 - 4}{0.5} = 4.5$$

Question 17

A chemical reaction’s temperature rises from 20°C to 50°C in 10 minutes. What is the average rate of change in temperature per minute?

Accepted Answer

$$\frac{50 - 20}{10} = 3$$ °C/min

Average Rates of Change and the Difference Quotient in Business Calculus

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