Q: Given the function $$f(x) = 3x^2$ + 2x$$, what is the average rate of change from $$x = 1 to$$ $$x = 4$$?

$$\frac{f(4) - f(1)}{4 - 1} = \frac{(3 \times 16 + 8) - (3 \times 1 + 2)}{3} = \frac{56 - 5}{3} = 17$$

Q: A business’s profit function is $$P(x) = 2x^2$ - 4x + 10. $$What is the average rate of change of profit as production increases from $$x = 3 to$$ $$x = 7$$?

$$\frac{P(7) - P(3)}{7 - 3} = \frac{(2 \times 49 - 28 + 10) - (2 \times 9 - 12 + 10)}{4} = \frac{(98 - 28 + 10) - (18 - 12 + 10)}{4} = \frac{80 - 16}{4} = 16$$

Q: A company’s cost function is $$C(x) = 50x + 200. If $$the company increases production from 100 units to 120 units, what is the average rate of change of cost?

$$\frac{C(120) - C(100)}{120 - 100} = \frac{(50 \times 120 + 200) - (50 \times 100 + 200)}{20} = \frac{6200 - 5200}{20} = 50$$

Question 1

Which of the following best describes the concept of the average rate of change of a function $$f(x)$$ over the interval $$[a, b]$$?

Accepted Answer

It is the change in $$f(x)$$ divided by the change in $$x$$ over $$[a, b]$$, given by $$\frac{f(b) - f(a)}{b - a}.$$

Question 2

A company tracks its revenue $$R(t)$$ over time $$t in $$months. If $$R(2) = 5000$$ and $$R(5) = 8000$$, what is the average rate of change of revenue between month 2 and month 5?

Accepted Answer

$$\frac{8000 - 5000}{5 - 2} = \frac{3000}{3} = 1000$$

Question 3

Which scenario best illustrates the use of the average rate of change in a real-world business context?

Accepted Answer

Estimating the increase in sales per month over a quarter.

Question 4

Given the function $$f(x) = 3x^2$ + 2x$$, what is the average rate of change from $$x = 1 to$$ $$x = 4$$?

Accepted Answer

$$\frac{f(4) - f(1)}{4 - 1} = \frac{(3 	imes 16 + 8) - (3 	imes 1 + 2)}{3} = \frac{56 - 5}{3} = 17$$

Question 5

If the cost function for producing $$x$$ units is $$C(x) = 100 + 5x$$, what does the average rate of change of $$C(x)$$ from $$x = 10 to$$ $$x = 20$$ represent?

Accepted Answer

The average additional cost per unit produced between 10 and 20 units.

Question 6

A business’s profit function is $$P(x) = 2x^2$ - 4x + 10. $$What is the average rate of change of profit as production increases from $$x = 3 to$$ $$x = 7$$?

Accepted Answer

$$\frac{P(7) - P(3)}{7 - 3} = \frac{(2 	imes 49 - 28 + 10) - (2 	imes 9 - 12 + 10)}{4} = \frac{(98 - 28 + 10) - (18 - 12 + 10)}{4} = \frac{80 - 16}{4} = 16$$

Question 7

Which of the following is NOT a correct interpretation of the average rate of change of a function $$f(x)$$ over $$[a, b]$$?

Accepted Answer

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

Question 8

Suppose the demand for a product is modeled by $$D(p) = 100 - 2p$$, where $$p is $$the price. What is the average rate of change in demand as price increases from $$p = 10 to$$ $$p = 20$$?

Accepted Answer

$$\frac{D(20) - D(10)}{20 - 10} = \frac{(100 - 40) - (100 - 20)}{10} = \frac{60 - 80}{10} = -2$$

Question 9

A company’s cost function is $$C(x) = 50x + 200. If $$the company increases production from 100 units to 120 units, what is the average rate of change of cost?

Accepted Answer

$$\frac{C(120) - C(100)}{120 - 100} = \frac{(50 	imes 120 + 200) - (50 	imes 100 + 200)}{20} = \frac{6200 - 5200}{20} = 50$$

Question 10

If $$f(x)$$ represents the number of customers served by a restaurant per day, and $$f(5) = 120$$, $$f(8) = 180$$, what does the average rate of change from day 5 to day 8 tell the manager?

Accepted Answer

The average number of additional customers served per day between day 5 and day 8.

Question 11

Which algebraic manipulation is used to simplify the difference quotient $$\frac{f(x+h) - f(x)}{h}$$ for a linear function $$f(x) = mx + b$$?

Accepted Answer

Substitute $$f(x+h) = m(x+h) + b$$ and $$f(x) = mx + b$$, then simplify numerator.

Question 12

A company’s revenue function is $$R(x) = 10x^2$ + 5x. $$What is the average rate of change in revenue as production increases from $$x = 2 to$$ $$x = 6$$?

Accepted Answer

$$\frac{R(6) - R(2)}{6 - 2} = \frac{(10 	imes 36 + 30) - (10 	imes 4 + 10)}{4} = \frac{390 - 50}{4} = 85$$

Question 13

Why is the average rate of change important for business decision-making?

Accepted Answer

It helps estimate trends and forecast future values based on past data.

Question 14

If $$f(x) = x^2$, what is the average rate of change from x = 1$ to x = 5$?

Accepted Answer

$$\frac{f(5) - f(1)}{5 - 1} = \frac{25 - 1}{4} = 6$$

Question 15

A business wants to analyze the change in cost per unit as production increases. Which mathematical concept should they use?

Accepted Answer

Average rate of change of the cost function.

Question 16

If the function $$f(x) is $$nonlinear, what does the average rate of change over an interval $$[a, b]$$ represent?

Accepted Answer

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

Rates of Change and the Difference Quotient in Business Calculus

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