Lesson 1: Average and Instantaneous Rates of Change
Estimated time: 30-40 minutes
Learning Objectives
By the end of this lesson, you will be able to:
- Calculate the average rate of change of a function over an interval
- Interpret the average rate of change as the slope of a secant line
- Understand the difference quotient and its role in calculus
- Transition from average to instantaneous rate of change using limits
- Interpret instantaneous rate of change as the slope of a tangent line
Average Rate of Change
The average rate of change of a function f over the interval [a, b] measures how much the output changes per unit change in input, on average.
Average Rate of Change
The average rate of change of f on [a, b] is: [f(b) − f(a)] / (b − a)
Geometrically, this is the slope of the secant line connecting the points (a, f(a)) and (b, f(b)).
Example 1: Average Velocity
A ball is dropped from a tower. Its height after t seconds is h(t) = 100 − 16t² feet. Find the average velocity from t = 1 to t = 2.
Step 1: h(1) = 100 − 16(1) = 84 ft. h(2) = 100 − 16(4) = 36 ft.
Step 2: Average velocity = [h(2) − h(1)]/(2 − 1) = (36 − 84)/1 = −48 ft/s.
The negative sign indicates the ball is falling (height is decreasing).
Example 2: Average Rate from a Formula
Find the average rate of change of f(x) = x³ from x = 1 to x = 3.
Solution: [f(3) − f(1)]/(3 − 1) = (27 − 1)/2 = 26/2 = 13.
The Difference Quotient
To make the average rate of change more flexible, we use a general expression called the difference quotient.
Difference Quotient
For a function f and a small increment h: [f(a + h) − f(a)] / h
This gives the average rate of change of f over the interval [a, a + h].
Example 3: Computing a Difference Quotient
For f(x) = x², compute the difference quotient at x = a.
Step 1: f(a + h) = (a + h)² = a² + 2ah + h²
Step 2: f(a + h) − f(a) = a² + 2ah + h² − a² = 2ah + h²
Step 3: Divide by h: (2ah + h²)/h = 2a + h
From Average to Instantaneous
The average rate of change over [a, a+h] gives the slope of the secant line through two points. As h → 0, this secant line rotates toward the tangent line at x = a. The slope of the tangent line is the instantaneous rate of change.
Instantaneous Rate of Change
The instantaneous rate of change of f at x = a is:
limh→0 [f(a + h) − f(a)] / h
provided this limit exists. This is the slope of the tangent line to y = f(x) at the point (a, f(a)).
Example 4: Instantaneous Rate of Change
Find the instantaneous rate of change of f(x) = x² at x = 3.
Step 1: From Example 3, the difference quotient for f(x) = x² is 2a + h.
Step 2: At a = 3: difference quotient = 2(3) + h = 6 + h.
Step 3: Take the limit: limh→0 (6 + h) = 6.
The slope of the tangent line to y = x² at x = 3 is 6.
Example 5: Instantaneous Velocity
Using h(t) = 100 − 16t², find the instantaneous velocity at t = 1.
Step 1: [h(1+h) − h(1)]/h = [100 − 16(1+h)² − 84]/h
Step 2: = [100 − 16(1 + 2h + h²) − 84]/h = [100 − 16 − 32h − 16h² − 84]/h
Step 3: = (−32h − 16h²)/h = −32 − 16h
Step 4: limh→0 (−32 − 16h) = −32 ft/s.
The Tangent Line
Once you know the instantaneous rate of change (slope) at a point, you can write the equation of the tangent line using point-slope form.
Equation of the Tangent Line
The tangent line to y = f(x) at x = a is: y − f(a) = m(x − a)
where m = limh→0 [f(a+h) − f(a)]/h is the slope.
Example 6: Equation of a Tangent Line
Find the equation of the tangent line to f(x) = x² at x = 3.
Step 1: f(3) = 9, so the point is (3, 9).
Step 2: From Example 4, the slope m = 6.
Step 3: y − 9 = 6(x − 3), so y = 6x − 9.
Graphical Interpretation
The relationship between secant and tangent lines is fundamental:
- A secant line passes through two points on a curve. Its slope equals the average rate of change.
- A tangent line touches the curve at one point and has the same direction as the curve at that point. Its slope equals the instantaneous rate of change.
- As the two points on the secant line get closer together, the secant line approaches the tangent line.
Key Takeaways
- Average rate of change = [f(b) − f(a)]/(b − a) = slope of the secant line.
- The difference quotient [f(a+h) − f(a)]/h generalizes this with a variable step size h.
- Instantaneous rate of change = limh→0 [f(a+h) − f(a)]/h = slope of the tangent line.
- The tangent line equation is y − f(a) = m(x − a) where m is the instantaneous rate.
- This limit of the difference quotient is exactly what we will call the derivative in the next lesson.
Check Your Understanding
1. Find the average rate of change of f(x) = 2x² + 1 from x = 1 to x = 4.
2. Compute the difference quotient for f(x) = 3x − x² at x = a.
3. Find the instantaneous rate of change of f(x) = 3x − x² at x = 1.