Estimate instantaneous rate of change numerically

Instantaneous rate of change at an input is estimated by calculating average rates of change, $\frac{f(a+h)-f(a)}{h}$ , over increasingly small intervals on either side of $a$ , using a table, graph, or function values. The learner interprets the resulting value as the slope of the tangent and in appropriate units, recognizing that nearby secant slopes may approach a stable value rather than equal it exactly; formal limit proofs and more advanced derivative generalizations are not included.

Detailed Explanation: Estimate instantaneous rate of change numerically

The instantaneous rate of change at $x=a$ describes how quickly $f(x)$ is changing at exactly that input. We estimate it by finding average rates of change over intervals that get very close to $a$ .

The average rate of change from $a$ to $a+h$ is

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

where $h$ is a small positive or negative number.

A positive $h$ uses a point to the right of $a$ .
A negative $h$ uses a point to the left of $a$ .
As $h$ gets closer to $0$ , the secant slopes may approach the slope of the tangent.

Example

Estimate the instantaneous rate of change of

f(x)=x^2

at $x=3$ .

First, calculate the function value at the input:

f(3)=3^2=9.

Now use

\frac{f(3+h)-f(3)}{h}.

For several values of $h$ , calculate the average rate of change.

$h$	$f(3+h)$	Average rate $\dfrac{f(3+h)-f(3)}{h}$
$1$	$f(4)=16$	$\dfrac{16-9}{1}=7$
$0.1$	$f(3.1)=9.61$	$\dfrac{9.61-9}{0.1}=6.1$
$0.01$	$f(3.01)=9.0601$	$\dfrac{9.0601-9}{0.01}=6.01$
$-1$	$f(2)=4$	$\dfrac{4-9}{-1}=5$
$-0.1$	$f(2.9)=8.41$	$\dfrac{8.41-9}{-0.1}=5.9$
$-0.01$	$f(2.99)=8.9401$	$\dfrac{8.9401-9}{-0.01}=5.99$

From the right, the rates are approaching $6$ :

7,\ 6.1,\ 6.01 \longrightarrow 6.

From the left, they are also approaching $6$ :

5,\ 5.9,\ 5.99 \longrightarrow 6.

Therefore, the estimated instantaneous rate of change at $x=3$ is

\boxed{6}.

This means that near $x=3$ , the graph has a tangent slope of about $6$ . The nearby average rates are not exactly $6$ , but they get closer to $6$ as the interval becomes smaller.

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Instantaneous Rate of Change - Graph Tangent to Slope Approximation

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