Rates of Change - AP PreCalculus Study Guide

The Core Idea: Rates of Change

The central concept of this topic is to measure how a function's output values change in relation to a change in its input values over a specific interval. Instead of looking at a function's value at a single point, we are interested in its overall behavior across a span. This is quantified by the average rate of change, which provides a single numerical value describing the function's general trend—whether it's increasing, decreasing, or remaining constant—on average, between two points.

Geometrically, the average rate of change represents the slope of a straight line, called a secant line, that connects the two endpoints of the function on the given interval. This provides a linear approximation of the function's change over that interval. Whether the function is defined by an algebraic formula or a discrete table of values, the average rate of change gives us a powerful tool to analyze its behavior over a specified domain.

Key Formulas

The primary formula for this topic is the average rate of change.

Average Rate of Change

The average rate of change of a function $f$ over a closed interval $[a,b]$ is given by the formula:

$$\text{Average Rate of Change}=\frac{f(b)-f(a)}{b-a}$$

• $f(b)$ is the value of the function at the right endpoint of the interval.

• $f(a)$ is the value of the function at the left endpoint of the interval.

• $b-a$ is the change in the input value, or the length of the interval.

• $f(b)-f(a)$ is the change in the output value over the interval.

This expression is also the formula for the slope of the secant line that passes through the two points $(a,f(a))$ and $(b,f(b))$ on the graph of the function $f$.

Understanding the Interpretation of the Result

Calculating the average rate of change yields a number, but the true understanding comes from interpreting the sign of that number. The sign tells the story of the function's overall behavior across the interval, even if the function itself fluctuates within the interval.

• Positive Average Rate of Change: If $\frac{f(b)-f(a)}{b-a}>0$, it means that, on average, the function's output values are increasing as the input values increase from $a$ to $b$. The secant line connecting the endpoints of the interval will have a positive slope.

• Negative Average Rate of Change: If $\frac{f(b)-f(a)}{b-a}<0$, it means that, on average, the function's output values are decreasing as the input values increase from $a$ to $b$. The secant line connecting the endpoints will have a negative slope.

• Zero Average Rate of Change: If $\frac{f(b)-f(a)}{b-a}=0$, it means that, on average, the function's output values are constant across the interval. This occurs when $f(b)=f(a)$, meaning the function starts and ends at the same output value. The secant line connecting the endpoints is a horizontal line with a slope of zero.

It is crucial to use the phrase "on average" because the function may not be strictly increasing, decreasing, or constant throughout the entire interval. For example, a function can have a positive average rate of change over $[a,b]$ even if it decreases for a small portion of that interval. The average rate of change only describes the net change from the starting point to the ending point.

Core Concepts & Rules

• The average rate of change of a function $f$ on the interval $[a,b]$ is calculated using the formula $\frac{f(b)-f(a)}{b-a}$.

• This calculation can be performed when the function is given as an algebraic rule (e.g., $f(x)=x^2$) or as a table of values.

• The value of the average rate of change is geometrically equivalent to the slope of the secant line connecting the points $(a,f(a))$ and $(b,f(b))$.

• A positive average rate of change indicates that the function is, on average, increasing on the interval.

• A negative average rate of change indicates that the function is, on average, decreasing on the interval.

• A zero average rate of change indicates that the function is, on average, constant on the interval, meaning its starting and ending output values are identical ($f(a)=f(b)$).

Step-by-Step Example 1: Calculating from a Function Formula

Problem: Find the average rate of change of the function $g(x)=x^3-2x+1$ on the interval $[-1,3]$.

Step 1: Identify the interval endpoints.

The given interval is $[a,b]=[-1,3]$.

So, $a=-1$ and $b=3$.

Step 2: Evaluate the function at each endpoint.

First, calculate $g(b)=g(3)$:

$$g(3)=(3{)}^3-2(3)+1=27-6+1=22$$

Next, calculate $g(a)=g(-1)$:

$$g(-1)=(-1{)}^3-2(-1)+1=-1+2+1=2$$

Step 3: Apply the average rate of change formula.

The formula is $\frac{g(b)-g(a)}{b-a}$.

Substitute the values from the previous steps:

$$\text{Average Rate of Change}=\frac{g(3)-g(-1)}{3-(-1)}$$

Step 4: Calculate the final value.

$$\text{Average Rate of Change}=\frac{22-2}{3+1}=\frac{20}{4}=5$$

Conclusion: The average rate of change of $g(x)$ on the interval $[-1,3]$ is 5. This means that, on average, the function increases by 5 units of output for every 1 unit of input across this interval.

Step-by-Step Example 2: Exam-Style Application from a Table

Problem: The temperature of a cup of coffee, $T$, in degrees Celsius, is a function of time, $t$, in minutes after it is poured. The function $T(t)$ is represented by the table of values below.

(a) Find the average rate of change of the temperature of the coffee over the interval $[2,9]$.

(b) Interpret the meaning of your answer in the context of the problem, using appropriate units.

Part (a): Calculation

Step 1: Identify the interval and corresponding function values from the table.

The interval is $[a,b]=[2,9]$.

From the table, the value at $a=2$ is $T(2)=78$.

From the table, the value at $b=9$ is $T(9)=55$.

Step 2: Apply the average rate of change formula.

$$\text{Average Rate of Change}=\frac{T(b)-T(a)}{b-a}=\frac{T(9)-T(2)}{9-2}$$

Step 3: Substitute the values and compute.

$$\text{Average Rate of Change}=\frac{55-78}{9-2}=\frac{-23}{7}$$

The average rate of change is $-\frac{23}{7}$.

Part (b): Interpretation

Step 1: Analyze the sign and value of the result.

The average rate of change is $-\frac{23}{7}$, which is approximately $-3.286$. The sign is negative.

Step 2: Connect the sign to the function's behavior.

A negative average rate of change means the function is, on average, decreasing. In this context, the temperature $T(t)$ is decreasing over time.

Step 3: Formulate a complete sentence with units.

The units for the numerator ($T(b)-T(a)$) are degrees Celsius. The units for the denominator ($b-a$) are minutes. Therefore, the units for the rate of change are degrees Celsius per minute.

Interpretation: Between 2 minutes and 9 minutes after being poured, the temperature of the coffee is decreasing, on average, at a rate of $\frac{23}{7}$ degrees Celsius per minute.

Using Your Calculator

While the average rate of change is a straightforward calculation, a graphing calculator can help prevent arithmetic errors and speed up the process, especially with complex functions.

Scenario: Find the average rate of change for $f(x)=\frac{\sqrt{x^2+9}}{x+1}$ on the interval $[1,4]$.

Method 1: Direct Calculation with Parentheses

1. On the home screen, type the expression for the average rate of change. Use parentheses carefully for the numerator and the denominator.

2. To find $f(4)$, you would calculate $\frac{\sqrt{4^2+9}}{4+1}=\frac{\sqrt{25}}{5}=1$.

3. To find $f(1)$, you would calculate $\frac{\sqrt{1^2+9}}{1+1}=\frac{\sqrt{10}}{2}$.

4. Enter the full expression into the calculator:

   $((1)-(\surd (10)/2))/(4-1)$

5. Press ENTER to get the result, approximately $-0.187$.

Method 2: Using the Y= Editor (More Efficient and Reliable)

1. Press the Y= button and enter the function into Y1:

   Y1 = (√(X^2+9))/(X+1)

2. Return to the home screen by pressing 2nd then MODE (QUIT).

3. Now, you can use the VARS menu to access Y1 and evaluate it at the required points. Type the following expression:

   (Y1(4) - Y1(1)) / (4 - 1)

   • To get Y1, press VARS, go to the Y-VARS menu, select 1:Function..., and then 1:Y1.

   • Then type `(4)to evaluate at $x=4.

4. Press ENTER. The calculator will automatically evaluate Y1(4) and Y1(1) and compute the final answer. This method reduces the chance of a copy or arithmetic error.

AP Exam Quick Hit

Common Question Types

• Calculating from a Table: You will be given a table of values for a function $f(x)$ and asked to find the average rate of change on a specific interval $[a,b]$.

  • Example: Given a table with points $(2,5)$ and $(7,-10)$, find the average rate of change of the function on $[2,7]$.

  • Solution:$\frac{-10-5}{7-2}=\frac{-15}{5}=-3$.

• Calculating from a Function Rule: You will be given a function defined by an equation, $g(x)$, and asked to find its average rate of change on an interval $[a,b]$.

  • Example: Find the average rate of change of $g(x)=2^x$ on $[1,5]$.

  • Solution:$\frac{g(5)-g(1)}{5-1}=\frac{2^5-2^1}{4}=\frac{32-2}{4}=\frac{30}{4}=7.5$.

• Interpreting the Result: You will be given a scenario (word problem) and asked to calculate and interpret the average rate of change, including its sign and units.

  • Example: If $P(t)$ is the population of a town in thousands, where $t$ is years since 2010, and the average rate of change on $[5,10]$ is $-0.4$, what does this mean?

  • Interpretation: Between 2015 and 2020, the town's population was decreasing on average at a rate of 0.4 thousand people (or 400 people) per year.

Common Mistakes

• Incorrect Formula Order: Flipping the numerator and denominator ($\frac{b-a}{f(b)-f(a)}$) or mixing up the terms ($\frac{f(a)-f(b)}{b-a}$). Always remember it's "change in output" over "change in input," just like the slope of a line.

• Sign Errors with Subtraction: Forgetting to distribute the negative sign when $a$ or $f(a)$ is negative. For example, calculating $b-a$ for the interval $[-2,3]$ as $3-2=1$ instead of the correct $3-(-2)=5$.

• Misinterpreting "On Average": Stating that because the average rate of change is positive on $[a,b]$, the function is always increasing on that interval. The function could have small segments where it decreases, but the overall trend is positive.

• Calculator Order of Operations: Typing $f(b)-f(a)/b-a$ into a calculator without parentheses. The calculator will perform the division $f(a) / bfirst. Always enclose the entire numerator and the entire denominator in parentheses:(f(b) - f(a)) / (b - a)`.

• Forgetting Units: In contextual problems (word problems), providing only the numerical answer without the correct units (e.g., "meters per second" or "dollars per year"). The units are a critical part of the interpretation.