Math101 · Independent Ontario learning library
Math101Instantaneous Rate of Change
Instantaneous rate of change measures a function's rate at one input and equals the slope of its tangent line.
An instantaneous rate is the value approached by average rates over intervals shrinking toward one point.
From an interval to an instant
Average rate of change needs two distinct inputs. To estimate a rate at $x=a$, compare $a$ with nearby inputs $a+h$:
As $h$ becomes closer to zero from both sides, the secant slopes may approach one number. That limit is the instantaneous rate of change.
Limit definition
When the limit exists,
The notation $f'(a)$ is read “$f$ prime at $a$.” The fraction itself is undefined at $h=0$, but the limit asks what values approach—not what happens after substituting zero directly.
Tangent-line meaning
Geometrically, secant lines through two graph points approach a tangent line at $(a,f(a))$. The tangent slope is $f'(a)$.
A tangent is not always a line that touches only once. It is the line giving the graph's local direction and best linear approximation near the point.
Worked example from first principles
The tangent line at $(3,9)$ has slope $6$.
Units and interpretation
An instantaneous rate has the same output-per-input units as average rate of change. If $s(t)$ measures metres and $t$ measures seconds, then $s'(4)$ is instantaneous velocity in metres per second at $t=4$.
The input $a$, output $f(a)$, and rate $f'(a)$ are three different quantities with different meanings and often different units.
Common mistakes
Substituting $h=0$ before simplifying. The quotient then divides by zero.
Using only a large interval. Instantaneous rate requires a limiting process.
Confusing $f(a)$ with $f'(a)$. One is an output; the other is a slope.
Assuming every visible point has a tangent. Check left and right behaviour.
Using curve points to measure a drawn tangent slope incorrectly. Measure the tangent line itself.
