AP® Calculus AB · Unit 2 · Topic 2.1
Defining Average and Instantaneous Rates of Change
Average rate of change is total change divided by total change in input, which equals the slope of the secant line between two points on a graph. Shrinking the interval to a single point turns that secant slope into the instantaneous rate of change.
A road trip's average speed tells you almost nothing about the drive itself. Fifty miles in one hour could mean a steady cruise, or it could mean an hour of traffic followed by a sprint down an empty highway. Average rate of change only sees the start and the end: total change in output over total change in input. It erases everything that happened in between.
Graphically, that average rate of change is the slope of the secant line, the straight line connecting two points on a curve. For a function f between x = a and x = b, the secant slope is (f(b) - f(a)) / (b - a), the same rise-over-run formula from linear slope, just applied to a curve that bends. Because the curve bends, this one number can't capture the changing steepness along the way.
The fix is to shrink the interval. Hold one endpoint fixed and slide the other one closer and closer to it. The secant line pivots and settles into the tangent line at that fixed point, and its slope becomes the instantaneous rate of change there. This is exactly the limit process behind the difference quotient (f(x) - f(a)) / (x - a) as x approaches a, and it's the idea that the rest of the unit turns into the formal definition of the derivative.
Key terms
- Average rate of change
- Total change in a function's output divided by the corresponding change in its input over an interval.
- Secant line
- A straight line connecting two points on a curve, whose slope equals the average rate of change between them.
- Tangent line
- The line that touches a curve at one point and matches its exact steepness there.
- Difference quotient
- The expression (f(x) - f(a)) / (x - a), whose limit as x approaches a gives the instantaneous rate of change.
- Instantaneous rate of change
- The rate of change at a single exact input value, found by shrinking the interval in a difference quotient to zero.
Watch it explained
- How it worksHow does the slope of a secant line measure an average rate of change?
- Worked exampleFind the average rate of change of f(x) = x2 + 3x on the interval from 1 to 4.
- Common mistakeWhy isn't the average of the starting and ending rates the same as the average rate of change?
- Ties togetherHow does shrinking the interval in a difference quotient turn a secant line into a tangent line?
Check yourself
End of topic.