3.3: Rates of change

Objectives:• To find the average rate of change over an interval• To find the instantaneous rate of change!!

For f(x)=x2+4x-5, find

Warm Up

0,)()(

hh

xfhxf

(this is the slope of a line drawn between 2 points on the graph of the function)

AVERAGE RATE OF CHANGE OF f(x) WITH RESPECT TO x FOR A FUNCTION f AS x CHANGES FROM a TO b:

ab

afbf

)()(

http://www.coolmath.com/graphit/

Find the average rate of change of f(x)=2x3-x over the interval [1,3].

What if we wanted to know the exact speed of a car at an instant?Assume the car’s position is given by s(t)=3t2 for 0< t < 15

What is the car’s speed at EXACTLY 5 seconds??

Take shorter and shorter intervals near t=5, and find avg rate of change over the intervals. This should zoom in (Get it??? Anyone?? Anyone??) the instantaneous rate of change!!

t=5 to t=5.1: t=5 to t = 5.01: t=5 to t=5.001:

INSTANTANEOUS RATE OF CHANGE(YEAH CALCULUS!!!!!!)

Let’s call this quantity we add to 5 “h”

So we are going to find the rate of change from t=5 to t=(5+h):

We took smaller and smaller intervals each time. We added a smaller and smaller quantity to 5 each time.

We added smaller and smaller values of h so we have:

Bring back the limit!!!!

h

shsh

)5()5(lim

0

Let a be a specific x value Let h be a small number that represents the

distance between the 2 values of x

PROVIDED THE LIMIT EXISTS!!!!!

http://www.ima.umn.edu/~arnold/calculus/secants/secants1/secants-g.html

DEFINITION: INSTANTANEOUS RATE OF CHANGE

h

afhafh

)()(lim

0

Velocity is the same as instantaneous rate of change of a function that gives the position with respect to time

Velocity has direction, it can be positive or negative

Speed = | velocity |

A few notes…..

If the position function is s(t)=t2+3t-4, find the instantaneous velocity at t=1, 3, and 5.

a.) Find the average rate of change from 3 to 5 seconds.

b.) Find the instantaneous velocity at any time, t

c.) What is the instantaneous velocity at t=7?

The distance in feet of an object is given by h(t)=2t2-3t+2

Can use if you are given a specific x value

Instantaneous rate of change for a function when x=a can be written as:

PROVIDED THE LIMIT EXISTS

(b is the second x value getting closer and closer to a)

ALTERNATE FORM

ab

afbfab

)()(lim

Find the instantaneous rate of change for s(t)=-4t2-6 at t=2.

Using Alternate Form

Marginal Cost: Instantaneous rate of change of the cost

function The rate that the cost is changing when

producing one additional item

Example:The cost in dollars to manufacture x cases of the DVD “Calculus is my Life” is given by C(x)=100+15x-x2 , 0< x < 7. Find the marginal cost with respect to the number of cases produced when only 2 cases have been manufactured.

Business Application