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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