D1.5 Compare instantaneous and average rates of change

Ontario Grade 12 Mathematics

Characteristics of Functions

Using slopes, graphs, and nearby function values

A function connects an input to an output. For example, a function might describe a moving object’s position at different times. Its rate of change tells us how quickly the output changes as the input changes. An average rate describes change across an interval. An instantaneous rate describes change at one input. Both ideas can be understood as slopes, but they answer different questions.

What you will learn

Calculate an average rate of change over an interval.
Describe an instantaneous rate of change at one input using a graph or nearby values.
Compare what average and instantaneous rates tell us, including their units.

1. Prerequisite bridge: change and slope

A rate compares a change in an output with a change in an input. If an object’s position changes by

12

metres over

3

seconds, its average rate of change is

4

metres per second. The units come from dividing output units by input units.

Recall that the slope between two points measures how much the vertical coordinate changes compared with the horizontal coordinate. For points

(x_1,y_1)

and

(x_2,y_2)

, the slope is the change in

y

divided by the change in

x

. A rate of change is this same idea applied to a function’s outputs and inputs.

The average rate depends on two input values. An instantaneous rate focuses on one input value. Since one input alone does not make a pair of points, we estimate that rate by looking at points close to the input of interest.

m=\frac{y_2-y_1}{x_2-x_1}

A rate of change is output change divided by input change.
The units of a rate are output units per input unit.
Slope gives a visual way to understand rate.

2. Average rate: change across an interval

Suppose a function is called

f

. Its average rate of change from input

a

to input

b

compares the two outputs,

f(a)

and

f(b)

, with the input change from

a

to

b

. The inputs must be different so that the interval has a nonzero width.

On a graph, mark the two points

(a,f(a))

and

(b,f(b))

. The straight line through them is a secant line. Its slope is the average rate of change over that interval. The function may curve between the points, but the secant slope summarizes the overall change from one endpoint to the other.

For a real-world situation, include the units and state the interval. An average rate over the first five seconds does not necessarily describe what happens at every moment in those five seconds. The object may speed up or slow down while having the same overall average.

\text{Average rate} = \frac{f(b)-f(a)}{b-a}

Average rate uses two endpoints.
The secant line connects the graph’s endpoint values.
A positive rate means the output increased overall; a negative rate means it decreased overall.

3. Instantaneous rate: change at one input

An instantaneous rate of change describes how quickly a function is changing at a particular input. On a graph, it is represented by the slope of the tangent line at that point. A tangent line follows the graph’s direction at the point, rather than joining two distant points.

A numerical estimate can be made by choosing an input just before the point and another just after it. Calculate the secant slope between their function values. If the chosen inputs are close to the point of interest, that slope can give a useful estimate of the instantaneous rate. It is an estimate, so report it as approximate when it comes from nearby values.

The two rates can be compared by asking what interval each describes. The average rate summarizes the whole interval. The instantaneous rate describes the graph’s local direction at one input. They may be similar, or they may differ if the graph’s steepness changes across the interval.

\text{Estimated instantaneous rate at }c \approx \frac{f(c+h)-f(c-h)}{2h}

Instantaneous rate refers to one input value.
The tangent-line slope gives its graphical meaning.
Nearby function values provide a numerical estimate.

4. Choosing and interpreting a rate

Use an average rate when a question asks for the overall change between two stated inputs. Use an instantaneous rate when it asks how quickly the output is changing at a particular input, such as at a specified time.

A graph helps you compare the rates visually. A steep secant line indicates a large average change over its interval. A steep tangent line indicates a large instantaneous change at that point. A downward slope indicates a negative rate. A horizontal line has slope zero, so the rate at that point or across that interval is zero, depending on which line is being considered.

A numerical table can show why interval size matters. Secant slopes using inputs closer to the point can help estimate the tangent-line slope. Keep the target input clear, and do not confuse the nearby estimating interval with the full interval used for an average rate.

Match the rate to the question: interval-wide or at one input.
Use the same units when comparing rates from the same function.
Label an estimate and identify the input where the instantaneous rate is being estimated.

Nearby secant slopes for estimating the rate at $t=2$

Inputs used	Secant slope	What it describes
$1$ to $3$	$13\text{ m/s}$	Average over the full interval
$1.9$ to $2.1$	$12.01\text{ m/s}$	Estimate near $t=2$

Worked example

Compare an interval average with a nearby estimate

A function

f(t)=t^3

gives an object’s position in metres after

t

seconds. Compare the average rate of change from

t=1

to

t=3

with an estimate of the instantaneous rate at

t=2

. Use nearby inputs

1.9

and

2.1

for the estimate.

Find the endpoint positions
Evaluate the function at the two endpoints of the full interval. These outputs give the total position change needed for the average rate.
$f(1)=1,\quad f(3)=27$
Calculate the average rate
Divide the position change by the time change. This rate describes the overall motion from $1$ second to $3$ seconds.
$\frac{27-1}{3-1}=13\text{ m/s}$
Find the nearby positions
Evaluate the function at inputs just before and after $2$ . Their secant slope estimates the tangent-line slope at $t=2$ .
$f(1.9)=6.859,\quad f(2.1)=9.261$
Estimate the instantaneous rate
Divide the change in position between the nearby values by the short time interval. This is an estimate at $2$ seconds, not the average over the original two-second interval.
$\frac{9.261-6.859}{2.1-1.9}=12.01\text{ m/s}$
Compare the results
The interval average is slightly greater than the nearby estimate at $2$ seconds. The two values differ because the graph does not have one constant slope throughout the interval.
$13\text{ m/s}>12.01\text{ m/s}$

Answer: The average rate from

1

to

3

seconds is

13\text{ m/s}

. The estimated instantaneous rate at

2

seconds is

12.01\text{ m/s}

.

Check: The time interval for the average is

2

seconds, and the position change is

26

metres, giving

26\div 2=13

. The nearby estimate uses a

0.2

-second interval and a

2.402

-metre change, giving

2.402\div0.2=12.01

.

Common mistakes and how to avoid them

Using only one function value to calculate an average rate.

Correction: An average rate needs two inputs and their corresponding outputs so both changes can be compared.

Calling the rate over a short interval the exact instantaneous rate.

Correction: A secant slope over nearby inputs is an estimate. Describe it as approximate unless the question provides a tangent-line slope directly.

Leaving out units or using input units as the rate units.

Correction: Divide output units by input units. For position in metres and time in seconds, the rate is in metres per second.

Lesson summary

Average rate of change is the slope of a secant line through two points on a graph.
Instantaneous rate of change is represented by the tangent-line slope at one input.
Nearby values can estimate an instantaneous rate numerically.
Compare the rates by noting whether each describes an interval or a single input, and include units.

Check your understanding

Question 1

A function has

f(2)=5

and

f(6)=17

. What is its average rate of change from

2

to

6

?

$3$
$4$
$12$
$22$

Show answer and explanation

3

The output change is

17-5=12

and the input change is

6-2=4

. The average rate is

12\div4=3

.

Question 2

Which graph feature represents the instantaneous rate at an input?

The slope of a secant line across the full graph
The slope of the tangent line at that input
The vertical coordinate at that input
The distance between two input values

Show answer and explanation

The slope of the tangent line at that input

The tangent-line slope represents how quickly the graph is changing at the selected input.

Question 3

A nearby secant slope is calculated using inputs just before and after

c

. What does it provide?

The exact average rate over every possible interval
An estimate of the instantaneous rate at $c$
The output value at $c$
The average of the two input values

Show answer and explanation

An estimate of the instantaneous rate at

c

A secant through nearby points can estimate the tangent-line slope at

c

, so it estimates the instantaneous rate there.

Key terms

Rate of change: The change in an output divided by the corresponding change in an input.
Average rate of change: The rate calculated across an interval between two input values.
Instantaneous rate of change: The rate at one input value, represented graphically by the tangent-line slope.
Secant line: A straight line through two points on a graph.
Tangent line: A line that matches the graph’s direction at a particular point.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation D1.5. It is a study resource, not an official curriculum publication.

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D1.5 · Compare instantaneous and average rates of change

Ontario Grade 12 Mathematics

Using slopes, graphs, and nearby function values

What you will learn

1. Prerequisite bridge: change and slope

2. Average rate: change across an interval

3. Instantaneous rate: change at one input

4. Choosing and interpreting a rate

Nearby secant slopes for estimating the rate at $t=2$

Compare an interval average with a nearby estimate

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MHF4U

About this lesson and its review