Approximate instantaneous rates numerically

Ontario Grade 12 Mathematics

Characteristics of Functions

Using nearby average rates to estimate how quickly a quantity is changing at one input

A car’s average speed over a trip does not tell you its speed at every moment. In the same way, an average rate of change describes what happens across an interval, while an instantaneous rate describes the change at one particular input. We can estimate that rate by calculating average rates over smaller intervals near the input. This lesson focuses on numerical and graphical estimates.

What you will learn

Distinguish an average rate of change over an interval from a rate at one input.
Use values close to a chosen input to estimate an instantaneous rate numerically.
Use a table or graph to judge whether an estimate is becoming more reliable.

1. Prerequisite bridge: average rate of change

A function pairs an input with an output. For example, a function might pair time with the distance travelled. The average rate of change tells us how much the output changes, on average, for each one-unit change in the input over an interval.

To find it, subtract the two output values and divide by the difference between their inputs. The order must match: subtract outputs in the same order as the inputs. On a graph, this rate is the slope of the straight line joining the two points. That line is called a secant line.

An instantaneous rate of change is the rate at one input, rather than across a wide interval. It is not usually found from just one function value. Instead, we estimate it using average rates across intervals close to the input of interest.

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

Average rate uses two inputs and the outputs at those inputs.
The units are output units per input unit.
An interval’s average rate is the slope of its secant line.

2. Plain language: use intervals close to the input

Suppose we want the rate at input

a

. Choose another input close to

a

, such as

a+0.1

, and calculate the average rate from

a

to

a+0.1

. This gives a nearby estimate, but it uses an interval on only one side.

We can also use an input just below

a

. Comparing estimates from the left and right helps us see whether the rate near

a

is settling around one value. A small interval often gives a better estimate than a large one, but small does not automatically mean exact. Rounded measurements or a table with limited precision can affect the result.

A useful approach is to calculate rates using several intervals that get closer to the chosen input. If the estimates move toward a stable value from both sides, that value is a reasonable numerical estimate of the instantaneous rate. If they disagree greatly, the available values may not be close enough or precise enough.

\text{nearby average rates} \approx \text{instantaneous rate at } a

Use nearby inputs on both sides when possible.
Compare estimates over successively smaller intervals.
Report an estimate with units when the context provides units.

3. Reading a table or graph

Consider the function

f(x)=x^3

and estimate its rate at

x=2

. Values on either side of 2 let us calculate average rates across intervals centered at 2. The table shows that these rates approach 12 as the intervals become narrower.

A table is especially useful when the function’s output values are available but a formula is not easy to work with. A graph gives a related visual estimate: look at the slope of the curve near the chosen input. The tangent line is the straight line that follows the curve’s direction at that point; its slope represents the instantaneous rate. Here, it provides a visual interpretation of the numerical estimate, not a different calculation method.

Numerical estimates depend on the quality of the values used. If values are rounded, or if input measurements are far apart, the estimate may be less reliable. State that a value is approximate rather than presenting it as exact.

\frac{f(a+h)-f(a-h)}{2h}

Centered intervals use inputs equally far below and above the target.
A narrowing pattern in the table can support an estimate.
On a graph, the instantaneous rate corresponds to the slope of the tangent line.

4. Application: interpret and check an estimate

Rates describe change, so their signs matter. A positive rate means the output is increasing as the input increases near the chosen input. A negative rate means it is decreasing. A rate near zero means the output changes only a little over nearby input intervals.

Always connect the number to the variables. If distance is measured in metres and time in seconds, a rate has units of metres per second. If the input is not time, do not call the rate speed; use the appropriate output-per-input units instead.

When using measurements from a context, choose values close to the target input, keep the subtraction order consistent, and compare estimates from either side if the data allow it. A final answer should identify the input, give the estimated rate, include units where known, and signal that the value is approximate.

The sign indicates the local direction of change.
Units come from output units divided by input units.
Nearby numerical estimates support an approximation, not a claim of exactness.

Centered interval estimates for $f(x)=x^3$ at $x=2$

Half-width $h$	Endpoints	Average rate
0.5	1.5 and 2.5	12.25
0.1	1.9 and 2.1	12.01
0.01	1.99 and 2.01	12.0001

Worked example

Estimate the rate at an input

For

f(x)=x^3

, estimate the instantaneous rate of change at

x=2

using centered intervals with half-widths

0.5

,

0.1

, and

0.01

.

Choose nearby endpoints
For each half-width $h$ , use the inputs $2-h$ and $2+h$ . The average rate between these endpoints estimates the rate near $x=2$ .
$\frac{f(2+h)-f(2-h)}{2h}$
Calculate the rates
Evaluate the function at each pair of endpoints, subtract the lower-input output from the higher-input output, and divide by the distance between the inputs.
$\begin{aligned}h=0.5 &: \frac{2.5^3-1.5^3}{1}=12.25\\ h=0.1 &: \frac{2.1^3-1.9^3}{0.2}=12.01\\ h=0.01 &: \frac{2.01^3-1.99^3}{0.02}=12.0001\end{aligned}$
Interpret the pattern
The estimates get closer to 12 as the endpoints move closer to 2. So 12 is a reasonable numerical estimate of the instantaneous rate at that input.
$\text{rate at }x=2\approx 12$

Answer: The estimated instantaneous rate of change at

x=2

is approximately

12

units of output per unit of input.

Check: The estimates are based on intervals centered at 2, and each narrower interval gives a value closer to 12.

Common mistakes and how to avoid them

Using the change in output without dividing by the change in input.

Correction: A rate compares output change with input change. Divide the output difference by the input difference.

Treating an average rate over a wide interval as the rate at one input.

Correction: Use intervals close to the input of interest and compare estimates over narrower intervals.

Subtracting the outputs in one order and the inputs in the opposite order.

Correction: Keep the order consistent in both differences so the sign and value of the rate are correct.

Reporting a numerical estimate as exact.

Correction: Use wording such as “approximately” because the estimate comes from nearby intervals or rounded data.

Lesson summary

An average rate of change is the slope between two points on a graph.
To estimate an instantaneous rate, calculate average rates using inputs close to the target.
Compare estimates from smaller intervals, preferably on both sides of the target.
Interpret the sign and include appropriate units.

Check your understanding

Question 1

A table gives an average rate of

7.8

from an input just below

a

to

a

, and

8.1

from

a

to an input just above

a

. Which is the best conclusion?

The instantaneous rate at $a$ is exactly $7.8$ .
The instantaneous rate at $a$ is reasonably estimated near $8$ , but closer data could improve the estimate.
The instantaneous rate must be negative because the two estimates differ.
No estimate is possible unless the function is a straight line.

Show answer and explanation

The instantaneous rate at

a

is reasonably estimated near

8

, but closer data could improve the estimate.

The nearby rates are close to each other and near 8. They support an estimate near 8, but do not establish an exact value.

Question 2

A distance-versus-time graph has a downward slope near a chosen time. What does this indicate about the rate of change of distance there?

The rate is positive.
The rate is negative.
The rate is exactly zero.
The rate cannot be described using a slope.

Show answer and explanation

The rate is negative.

A downward slope means the output decreases as the input increases, so the rate is negative.

Question 3

An output is measured in litres and the input in minutes. What are the units of the rate of change?

Minutes per litre
Litres plus minutes
Litres per minute
Litres times minutes

Show answer and explanation

Litres per minute

Rate units are output units divided by input units, so the units are litres per minute.

Key terms

Average rate of change: The change in output divided by the change in input over an interval.
Instantaneous rate of change: The rate at one chosen input, estimated numerically from nearby average rates.
Secant line: A straight line joining two points on a graph; its slope gives the average rate between those inputs.
Tangent line: A straight line that follows a curve’s direction at a chosen point; its slope represents the instantaneous rate there.

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