Topic 1.2

Rates of Change

9 MIN READ10 IDEAS23 PROBLEMS11 flashcards

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01
To estimate the rate of change at $x = a$ , take the average rate of change over a small interval around $a$ .

The smaller the interval, the better the estimate.
02
To compare, estimate the rate at each point using intervals of the same width, then compare the numbers.

Different widths make the comparison meaningless.

Practice 23 problems Next: Chapter 1.3 Rates of Change in Linear and Quadratic Functions

What Is an Average Rate of Change?

The average rate of change of a function $f$ over an interval $[a, b]$ tells you how much the output changes, on average, per unit of input.

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

CONCEPT

What This Formula Means

This is exactly the slope of the line connecting $(a, f(a))$ and $(b, f(b))$

That line is called a secant line.

The secant line's slope IS the average rate of change between two points.

Quick check

$f(1) = 5$ and $f(4) = 17$ . What is the average rate of change of $f$ over $[1, 4]$ , and what does it represent on the graph?

Reading Average Rate of Change from a Graph

You don't always get an equation — sometimes you read two points directly off a graph and compute the slope between them, the same way as with a table or formula.

Reading two points off a temperature-vs-time graph.

Worked Example

Worked example

Using the graph above, find the average rate of change of temperature between hour 2 and hour 8.

01
Read the coordinates of the two points from the graph: approximately $(2, 62)$ and $(8, 76)$ .
02
Apply the formula: $(76 - 62) \div (8 - 2) = 14 \div 6 \approx 2.33$
03
The temperature rose by about 2.33°F per hour, on average, during that interval.

Real-Life Examples

REAL-LIFE EXAMPLE

Everyday Rates of Change

Speed: distance ÷ time gives your average speed, even if your actual speed varied moment to moment.

Population growth: (population now − population 5 years ago) ÷ 5 gives the average yearly growth.

Stock price: (price today − price a month ago) ÷ days gives the average daily change.

Bathtub fill rate: (final depth − starting depth) ÷ minutes gives the average fill rate.

Positive, Negative, and Zero Rates of Change

CONCEPT

What the Sign Tells You

Positive → the function increased overall on that interval.

Negative → the function decreased overall on that interval.

Zero → the function ended where it started (it may have gone up and back down in between).

Step-by-Step: From a Table of Values

Worked example

A car's odometer reads the values below. Find the average rate of change of distance with respect to time, from $t = 1$ to $t = 3$ .

t (hours)	distance (mi)
0	0
1	50
2	115
3	170

01
Identify the two relevant values: at $t=1$ , distance=50; at $t=3$ , distance=170.
02
Apply the formula:
$\frac{170 - 50}{3 - 1} = \frac{120}{2} = 60$
03
The car averaged 60 miles per hour between $t=1$ and $t=3$ — even though it may have sped up or slowed down along the way.

Step-by-Step: From an Equation

Worked example

Find the average rate of change of $f(x) = x^{2}$ over the interval $[1, 4]$ .

01
Find $f(1)$ and $f(4)$ :
$f(1) = 1^2 = 1 \qquad f(4) = 4^2 = 16$
02
Apply the formula:
$\frac{f(4) - f(1)}{4 - 1} = \frac{16 - 1}{3} = \frac{15}{3} = 5$
03
So on average, $f(x)$ increases by 5 units for every 1 unit increase in $x$ , over this interval.

COMMON MISTAKE

Don't confuse the average rate of change with the value of the function itself — it's a slope, not an output.

Don't assume the function behaves the same way at every point within the interval; average rate of change hides the details in between.

Watch your signs carefully when $a$ or $b$ is negative.

Estimating the Rate of Change AT a Point

Everything so far has been about a rate of change OVER an interval. The exam also asks for the rate of change AT a single moment — how fast something is changing right now, not on average.

You cannot compute that exactly with 1.2 tools. But you can estimate it, and the estimate is what the exam wants.

KEY RULE

To estimate the rate of change at $x = a$ , take the average rate of change over a SMALL interval around $a$ .

The smaller the interval, the better the estimate.

CONCEPT

Why a Small Interval Works

Zoom in far enough on a smooth curve and it looks like a straight line.

Over a wide interval the secant line can be nowhere near the steepness at $a$ . Over a narrow one it hugs the curve, so its slope is close to the true rate at $a$ .

Which Small Interval to Pick

If the point $a$ sits between two table values, use the interval that STRADDLES it — one value on each side. A straddling interval is usually a better estimate than a one-sided one of the same width.

Worked example

The table gives a car's distance $d$ (meters) at time $t$ (seconds): $d(2) = 14$ , $d(3) = 23$ , $d(4) = 36$ . Estimate the speed at $t = 3$ .

01
Speed is the rate of change of distance. Use an interval around $t = 3$ .
02
Straddle the point — use $[2, 4]$ :
$\frac{d(4) - d(2)}{4 - 2} = \frac{36 - 14}{2} = \frac{22}{2} = 11$
03
The speed at $t = 3$ is about $11$ meters per second.
04
Sanity-check with the two one-sided intervals: $[2, 3]$ gives $9$ and $[3, 4]$ gives $13$ . The straddling answer $11$ sits between them, which is what you expect.

COMMON MISTAKE

Calling the estimate exact. It is an ESTIMATE — write "approximately" or "about". The exam gives credit for the method and the units, not for pretending to more precision than a table can give.

Quick check

You want the rate of change of $f$ at $x = 3$ and the table has $f(2.9)$ , $f(3)$ , and $f(3.1)$ . Which interval gives the best estimate?

Comparing Rates of Change at Several Points

A very common question shows one function and asks where it is changing fastest, or whether it is speeding up or slowing down. The method is the same estimate, done more than once.

KEY RULE

To compare, estimate the rate at each point using intervals of the SAME width, then compare the numbers.

Different widths make the comparison meaningless.

Worked example

$g$ has values $g(0) = 2$ , $g(1) = 3$ , $g(2) = 6$ , $g(3) = 11$ , $g(4) = 18$ . Is $g$ changing faster near $x = 1$ or near $x = 3$ ? Is it speeding up or slowing down?

01
Straddle $x = 1$ with $[0, 2]$ :
$\frac{6 - 2}{2 - 0} = \frac{4}{2} = 2$
02
Straddle $x = 3$ with the same width, $[2, 4]$ :
$\frac{18 - 6}{4 - 2} = \frac{12}{2} = 6$
03
$6 > 2$ , so $g$ is changing about three times faster near $x = 3$ .
04
The rate went from $2$ up to $6$ — the rate itself is increasing, so $g$ is speeding up.

CONCEPT

Rate of Change of the Rate of Change

"Is it speeding up?" is a question about whether the RATES are growing — not about whether the function is growing.

A function can be falling the whole time and still be speeding up, if it falls faster and faster. Topic 1.3 picks this idea up again.

COMMON MISTAKE

Comparing $[0, 2]$ against $[2, 3]$ and concluding from the raw differences. Those are different widths, so the two numbers are not comparable until you divide by the width.

Practice Problems

Worked example

Problem 1. A plant's height is $h(t) = 2t + 3$ (cm), $t$ in weeks. Find the average rate of change from $t=2$ to $t=5$ .

01
Evaluate $h(2)$ and $h(5)$ :
$h(2) = 2(2) + 3 = 7 \qquad h(5) = 2(5) + 3 = 13$
02
Apply the formula:
$\frac{13 - 7}{5 - 2} = \frac{6}{3} = 2$
03
The plant grows at 2 cm/week on average — matches expectations since $h(t)$ is linear, so the rate never changes.

Worked example

Problem 2. For $g(x) = -x^{2} + 6$ , find the average rate of change from $x = 0$ to $x = 3$ .

01
Evaluate $g(0)$ and $g(3)$ :
$g(0) = 6 \qquad g(3) = -3$
02
Apply the formula:
$\frac{-3 - 6}{3 - 0} = \frac{-9}{3} = -3$
03
The average rate of change is −3, meaning $g(x)$ decreased overall across this interval.

Common slips

Don't confuse the average rate of change with the value of the function itself — it's a slope, not an output.

Don't assume the function behaves the same way at every point within the interval; average rate of change hides the details in between.

Watch your signs carefully when $a$ or $b$ is negative.
Calling the estimate exact. It is an estimate — write "approximately" or "about". The exam gives credit for the method and the units, not for pretending to more precision than a table can give.
Comparing $[0, 2]$ against $[2, 3]$ and concluding from the raw differences. Those are different widths, so the two numbers are not comparable until you divide by the width.

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11 cards · Rates of change, Functions and change

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Recap card

4 lines to re-read the night before.

01
Average rate of change = $(f(b) - f(a)) / (b - a)$ — the slope of the secant line.
02
Works the same way whether you start from a table, a graph, or an equation.
03
The sign tells direction (increase/decrease/no net change); the size tells how fast.
04
It describes the overall trend between two points — not what happens at any single point in between.

Practice 23 problems Next: Chapter 1.3 Rates of Change in Linear and Quadratic Functions

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Rates of Change

What Is an Average Rate of Change?

Reading Average Rate of Change from a Graph

Worked Example

Real-Life Examples

Positive, Negative, and Zero Rates of Change

Step-by-Step: From a Table of Values

Step-by-Step: From an Equation

Estimating the Rate of Change AT a Point

Which Small Interval to Pick

Comparing Rates of Change at Several Points

Practice Problems

Recap card

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