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Relate limits to instantaneous rate of change

Instantaneous rate of change at a point is understood as the limit of the average rate of change, f(a+h)−f(a)h\frac{f(a+h)-f(a)}{h}, as the nonzero change hh approaches zero. This limit, when it exists, is the derivative f′(a)f'(a), represented geometrically by the slope of the tangent line and, for position functions, interpreted as instantaneous velocity; it is not an average rate over a zero-length interval.

Detailed Explanation: Relate limits to instantaneous rate of change

The instantaneous rate of change of a function at one point is found by taking the limit of its average rates of change over smaller and smaller intervals.

For a function f(x)f(x) at x=ax=a:

Average rate of change=f(a+h)−f(a)h,h≠0\text{Average rate of change} = \frac{f(a+h)-f(a)}{h}, \qquad h\ne 0

Here, hh is the change in the input. The instantaneous rate of change is

f′(a)=lim⁡h→0f(a+h)−f(a)h.f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}.

Although hh approaches 00, we do not set h=0h=0 in the fraction, because division by zero is undefined. Instead, we simplify first and then find the limit.

Example: Find the instantaneous rate of change of f(x)=x2f(x)=x^2 at x=3x=3.

Step 1: Write the average rate of change.

Here, a=3a=3, so

f(3+h)−f(3)h.\frac{f(3+h)-f(3)}{h}.

Step 2: Evaluate the function values.

Since f(x)=x2f(x)=x^2,

f(3+h)=(3+h)2f(3+h)=(3+h)^2

and

f(3)=32=9.f(3)=3^2=9.

Therefore,

f(3+h)−f(3)h=(3+h)2−9h.\frac{f(3+h)-f(3)}{h} = \frac{(3+h)^2-9}{h}.

Step 3: Simplify.

Expand and factor:

(3+h)2−9h=9+6h+h2−9h=6h+h2h=6+h,\frac{(3+h)^2-9}{h} = \frac{9+6h+h^2-9}{h} = \frac{6h+h^2}{h} = 6+h,

where h≠0h\ne 0.

Step 4: Let hh approach 00.

f′(3)=lim⁡h→0(6+h)=6.f'(3)=\lim_{h\to 0}(6+h)=6.

So, the instantaneous rate of change of f(x)=x2f(x)=x^2 at x=3x=3 is

6.\boxed{6}.

Geometrically, 66 is the slope of the tangent line to the graph at x=3x=3. If ff represented a position function, then 66 would be the object’s instantaneous velocity at that moment.

This is not an average rate over an interval of length zero. It is the value approached by average rates of change as the interval becomes smaller and smaller.

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