1. 2.2: Formal Definition of the Derivative
Part I: Instantaneous Rate of Change

2. I. Average vs. Instantaneous Rates of Change
Example 1: The position of an object in free fall is given by 𝑓(𝑡)=−16 𝑡 , where f is measured in feet and t is measured in seconds.
Find the average rate of change (velocity) from t = 1 to t = 3.
𝑟.𝑜.𝑐.= Δ𝑦 Δ𝑥 = 𝑓 3 −𝑓(1) 3−1 = 56−184 2 =−64 𝑓𝑡 𝑠𝑒𝑐

3. I. Average vs. Instantaneous Rate of Change
b. Find the velocity at t = 3 (instantaneous). To find a slope, we choose a point just beyond 3 Δ𝑦 Δ𝑥 = 𝑓 3+ℎ −𝑓(3) 3+ℎ −3 Δ𝑦 Δ𝑥 = −16 3+ℎ − − ℎ Δ𝑦 Δ𝑥 = 56−96ℎ−16 ℎ 2 −56 ℎ Δ𝑦 Δ𝑥 = −96ℎ−16 ℎ 2 ℎ Δ𝑦 Δ𝑥 =−96−16ℎ When h=0, Δ𝑦 Δ𝑥 =−96 𝑓𝑡 𝑠𝑒𝑐

4. II. Instantaneous Slope
Example 2: Find the slope of f(x)= 𝑥 2 at 𝑥=4. This is the derivative at 4 or f’(4)
Δ𝑦 Δ𝑥 = 𝑓 4+ℎ −𝑓(4) ℎ
Δ𝑦 Δ𝑥 = 4+ℎ 2 − ℎ
Δ𝑦 Δ𝑥 = 16+8ℎ+ ℎ 2 −16 ℎ
Δ𝑦 Δ𝑥 = 8ℎ+ ℎ 2 ℎ
Δ𝑦 Δ𝑥 =8+ℎ
When h = 0
𝑓 ′ 4 = Δ𝑦 Δ𝑥 =8

5. II. Instantaneous Slope
B. Example 3: Find the slope of g 𝑥 = 𝑥 2 −4𝑥+1 at x = 3. Δ𝑦 Δ𝑥 = 𝑔 3+ℎ −𝑔(3) ℎ Δ𝑦 Δ𝑥 = 9+6ℎ+ ℎ 2 −12−4ℎ+1− −2 ℎ Δ𝑦 Δ𝑥 = ℎ 2 +2ℎ ℎ Δ𝑦 Δ𝑥 =ℎ+2 When h = 0 𝑓′(3)=2

6. III. Slope Definition
The slope of the curve y = f(x) at x = a is: 𝑚= Δ𝑦 Δ𝑥 = 𝑓 ′ 𝑎 = lim ℎ→0 𝑓 𝑎+ℎ −𝑓(𝑎) ℎ