Rate of Change.

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Presentation transcript:

Rate of Change

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

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))

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?

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

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

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

Another Form

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)

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

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.