1. Rate of Change

2. What is it?
A slope is the rate at which the y changes as the x changes
Velocity is the rate the position of an object changes as time changes, therefore it is the slope of a position versus time graph

3. Instantaneous Rate of Change
Average Rate of Change
Graphically, the average rate of change over the interval a ≤ x ≤ b is the slope of the secant line connecting (a,f(a)) with (b,f(b))
Instantaneous Rate of Change
Instantaneous rate of change at a is the slope of the line TANGENT to the curve at a single point, (a,f(a))

4. What is the average velocity from 0 – 4 sec
What is the average velocity from 0 – 4 sec? What is the instantaneous velocity at 1 sec?

5. The Tangent Line
If you have two points on a curve, P and Q, as Q moves closer and closer to P, the slope of the secant line between the two becomes closer and closer to being the slope of the tangent at point P

7. Example
Find the equation of the line tangent to the parabola y=x2 at point P slope = m Q
(x, x2) P
(1, 1)

8. Example (cont)
As Q gets closer and closer to P, the slope of secant PQ gets closer and closer to the slope of the tangent at P

9. Another Form

10. Example
Find the slope of the tangent line to the function at a point a What is the value of this slope at the following points (1,1) , (4, 2) , (9, 3)

11. Example
Find the slope of the tangent line to the function f(x)= x2 + 3x at the point (1, 4)

12. Example
Suppose that a ball is dropped from a tower. By Galileo's law the distance fallen by any freely falling body is expressed by the equation s(t) =16t2 where s(t) is in feet and t is in seconds.
(a) Find the average velocity between t = 1and t = 2.
(b) Find the instantaneous velocity at time t = 1.