Average Rates of Change and the Difference Quotient in Business Calculus

Chapter 1: Differentiation

1.3 Average Rates of Change

This section introduces the concept of average rate of change, a foundational idea in calculus that measures how a quantity changes over an interval. It also covers the difference quotient, which is essential for understanding the slope of secant lines and the basis of derivatives.

Definition: Average Rate of Change

• Average rate of change of y or f(x) with respect to x as x changes from x_1 to x_2 is the ratio of the change in output to the change in input.

• Formula: $\frac{y_2 - y_1}{x_2 - x_1}$ or $\frac{f(x_2) - f(x_1)}{x_2 - x_1}$, where $x_2 \neq x_1$

• Units are important and should be included in applications (e.g., inches/hour, miles/gallon).

Examples and Applications

• Rainfall Example: If it rained 4 inches over 8 hours: $\frac{4\text{ in} - 0\text{ in}}{8\text{ hr} - 0\text{ hr}} = \frac{4\text{ in}}{8\text{ hr}} = 0.5\text{ in/hr}$ Interpretation: The average rate of rainfall was 0.5 inches per hour.

• Fuel Efficiency Example: A car travels 350 miles on 20 gallons of gas: $\frac{350\text{ mi} - 0\text{ mi}}{20\text{ gal} - 0\text{ gal}} = \frac{350\text{ mi}}{20\text{ gal}} = 17.5\text{ mi/gal}$ Interpretation: The car gets 17.5 miles per gallon.

• Temperature Change Example: Temperature drops from 82°F at 2 p.m. to 76°F at 5 p.m.: $\frac{82 - 76}{5 - 2} = \frac{6}{3} = 2\text{ degrees/hour}$ Interpretation: The temperature dropped 2 degrees every hour.

Function Examples

• For $f(x) = x^2$:

  • From $x=1$ to $x=3$: $f(1) = 1^2 = 1$, $f(3) = 3^2 = 9$ $\frac{9 - 1}{3 - 1} = \frac{8}{2} = 4$

  • From $x=1$ to $x=2$: $f(1) = 1$, $f(2) = 4$ $\frac{4 - 1}{2 - 1} = \frac{3}{1} = 3$

  • From $x=2$ to $x=3$: $f(2) = 4$, $f(3) = 9$ $\frac{9 - 4}{3 - 2} = \frac{5}{1} = 5$

• For $f(x) = x^3 + 5$:

  • From $x=1$ to $x=4$: $f(1) = 1^3 + 5 = 6$, $f(4) = 4^3 + 5 = 69$ $\frac{69 - 6}{4 - 1} = \frac{63}{3} = 21$

  • From $x=1$ to $x=2$: $f(1) = 6$, $f(2) = 2^3 + 5 = 13$ $\frac{13 - 6}{2 - 1} = \frac{7}{1} = 7$

Difference Quotient

The difference quotient is a central concept in calculus, representing the average rate of change of a function over an interval of length h. It is the foundation for the definition of the derivative.

• Definition: The difference quotient of f(x) is: $\frac{f(x+h) - f(x)}{h}$, where $h \neq 0$

• The difference quotient equals the slope of the secant line passing through the points $(x, f(x))$ and $(x+h, f(x+h))$.

Examples

• For $f(x) = x^2$:

  • When $x=5$, $h=3$: $\frac{f(5+3) - f(5)}{3} = \frac{f(8) - f(5)}{3} = \frac{64 - 25}{3} = \frac{39}{3} = 13$

  • When $x=5$, $h=0.1$: $\frac{f(5+0.1) - f(5)}{0.1} = \frac{f(5.1) - f(5)}{0.1} = \frac{26.01 - 25}{0.1} = \frac{1.01}{0.1} = 10.1$

• For $f(x) = x^3$:

  • Simplified form: $\frac{(x+h)^3 - x^3}{h}$ Expand $(x+h)^3$: $= \frac{x^3 + 3x^2h + 3xh^2 + h^3 - x^3}{h}$ $= \frac{3x^2h + 3xh^2 + h^3}{h}$ $= 3x^2 + 3xh + h^2$, $h \neq 0$

Application: Slope of Secant Line

• To find the slope of the secant line (average rate of change) at $x=2$ for $h=0.1$ and $h=0.01$ using $f(x) = x^3$:

  • For $h=0.1$: $3x^2 + 3xh + h^2$ $3(2)^2 + 3(2)(0.1) + (0.1)^2 = 12 + 0.6 + 0.01 = 12.61$

  • For $h=0.01$: $3(2)^2 + 3(2)(0.01) + (0.01)^2 = 12 + 0.06 + 0.0001 = 12.0601$

Summary Table: Average Rate of Change Examples

Situation	Change in Output	Change in Input	Average Rate of Change	Units
Rainfall	4 in	8 hr	0.5	in/hr
Car travel	350 mi	20 gal	17.5	mi/gal
Temperature	-6 deg	3 hr	-2	deg/hr

Key Takeaways:

• The average rate of change measures how a quantity changes per unit of another variable.

• The difference quotient is a formula that calculates the average rate of change over an interval and is foundational for the derivative.

• Units are essential for interpreting the meaning of rates of change in real-world contexts.

Additional info: The difference quotient is the basis for the definition of the derivative, which measures instantaneous rate of change. As h approaches zero, the difference quotient approaches the derivative.