Lecture 12

Average Rate of ChangeThe Derivative

Last Lecture’s Summary

Covered Sections 15.1 and 15.2: Limits and Continuity

• Limits• Properties of Limits, and• Continuity

Today

We will cover sections 15.3 and 15.4: Average Rate of Change and The Derivative

• Average rate of change• The difference quotient• Instantaneous rate of change• Slope of a curve• The derivative• Using and interpreting the derivative

The slope of a straight line can be determined by the two-point formula

Average Rate of Change and the Slope

12

12

xx

yy

x

ym

The following figure illustrates the graph of a linear function. With linear functions the slope is constant over the domain if the function. The slope provides an exact measure of the rate of change in the value of y with respect to a change in the value of x.

y

x

B

A

If the function in the above figure represents a cost function and x equals the number of units produced, the slope indicates the rate at which total cost increases with respect to changes in the level of output.

With nonlinear functions the rate of change in the value of y with respect to a change in x is not constant. However, one way of describing nonlinear functions is by the average rate of change over some interval.

ExampleAssume that a person takes an automobile trip and that the distance travelled d can be estimated as a function of time t by the nonlinear function

where d is measured in miles, t in hours, and

During this 5-hour journey the speed of the car may change continuously (e.g., because of traffic lights, rest stops, etc.).

tttfd 88)( 2

50 t

After 1 hour, the total distance traveled is f(1) = 8(1)2 + 8(1)

= 16 miles

The average rate of change in the distance traveled with respect to a change in time during a time interval (better known as average velocity) is computed as

Distance traveled Time traveled

For the first hour of this trip, the average velocity equals. Δd f(1) – f(0) 16 – 0

Δt 1 – 0 1

------ = ------------- = ---------- = 16 mph

The distance traveled at the end of 2 hours is f(2) = 8(2)2 + 8(2)

= 32 + 16 = 48 miles

The distance traveled during the second hours is Δd = f(2) – f(1)

= 48 – 16 = 32 miles

The average velocity for the second hour equals Δd 32

Δt 1----- = ------ = 32 mph

The average velocity for the second hour is different compared with that for the first hour.

The average velocity during the first 2 hours is the total distance traveled during that period divided by the time traveled, or

Δd f(2) – f(0) 48 – 0

Δt 2 – 0 2----- = -------------- = --------- = 24 mph

Consider two points A and B in the following Fig. The straight line connecting these two point on f is referred to as a secant line.

In moving from point A to point B, the change in the value of x is (x + Δx) – x, or Δx. The associated change in the value of y is

Δy = f(x + Δx) – f(x)

The ratio of these changes is.Δy f(x + Δx) – f(x)

Δx Δx

The above equation is sometimes referred to as the difference quotient.

----- = ---------------------

DEFINITION: THE DIFFERENCE QUOTIENTGiven any two points on a function f having coordinates [x, f(x)] and [(x + Δx), f(x + Δx)], the difference quotient provides a general expression which represents.I- the average rate of change in the value of y

with respect to the change in x while moving from

[x, f(x)] to [(x + Δx), f(x + Δx)]

II- The slope of the secant line connecting the two points.

Example(a) Find the general expression for the difference

quotient of the function y = f(x) = x2.

(b) Find the slope of the line connecting (-2, 4) and (3, 9) using the two-point formula.

(c) Find the slope in part b using the expression for the difference quotient found in part a.

SOLUTION(a) Given two points on the function f(x) = x2

which have coordinates (x, f(x)) and (x + Δx, f(x + Δx)), we

have

x

xxx

x

xxxxx

x

xxxxxxx

x

xxx

x

xfxxf

x

y

2

222

222

22

)()(2

)()(2[

])()()([

)()()(

Factoring Δx from each term in the numerator and canceling with Δx in the denominator, we get

Δy f(x + Δx) – f(x) Δx (2x + Δx)

Δx Δx Δx----- = --------------------- = -------------------

THE DERIVATIVEInstantaneous Rate of Change

A distinction needs to be made between the concepts of average rate of change and instantaneous rate of change. Example 25 discussed a situation in which the distance traveled d was estimated as a function of time by the function

d = f(t) = 8t2 + 8t where 0 ≤ t ≤ 5

Suppose that we are interested in determining how fast the car is moving at the instant that t=1. We might determine this instantaneous velocity by examining the average velocity during time intervals near t = 1.

For instance, the average velocity during the second hour (between t = 1 and t = 2 can be determined as

The average velocity between t = 1 and t = 1.5 can be determined as

The average velocity between t = 1 and t = 1.1 can be determined as

The average velocity between t = 1 and t = 1.01 can be determined as

As shown in above figures, these computations have been determining the average velocity over shorter and shorter time intervals measured from t = 1. As the time interval becomes shorter (or as the second value of t is chosen closer and closer to 1), the average velocity Δd/Δt is approaching a limiting value, The instantaneous velocity at t = 1 can be defined as this limiting value. To determine this limiting value, we could compute.

Thus, the instantaneous velocity of the automobile at t = 1 is 24 miles per hour. Note that the average velocity is measured over a time interval and the instantaneous velocity is defined for a particular point in time. The instantaneous velocity is a “snapshot” of what is happening at a particular instant.

GEOMETRIC REPRESENTATION OF INSTANTANEOUS RATE OF CHANGE

The instantaneous rate of change of a smooth, continuous function can be represented geometrically by the slope of the line drawn tangent to the curve at the point of interest.

Let’s first determine the meaning of tangent line. The tangent line at A is the limiting position of the secant line AB as point B comes closer and closer to A.

DEFINITION: SLOPE OF CURVEThe slope of a curve at x = a is the slope of the tangent line at x = a.

DEFINITION: THE DERIVATIVEGiven a function of the form y = f(x), the derivative of the function is

provide this limit exists.

x

xfxxf

dx

dyx

)()(lim

0

The following points should be made regarding this definitionCOMMENTS ABOUT THE DERIVATIVEI- The above equation is the general expression for

the derivative of the function f.II- The derivative represents the instantaneous rate

of change in the dependent variable given a change in the independent variable. The notation dy/dx is used to represent the instantaneous rate of change in y with respect to a change in x. This notation is distinguished from Δy/Δx which represents the average rate of change.

III-The derivative is a general expression for the slope of the graph of f at any point x in the domain.

IV-If the limit in the above figure does not exist, the derivative does not exist.

FINDING THE DERIVATIVE (LIMIT APPROACH)

Step 1 Determine the difference quotient for / using Eq.

Step 2 Find the limit of the difference quotient as Δx 0 using Eq.

ExampleFind the derivative .95)( xxf

ExampleFind the derivative .2)( xxf

USING AND INTERPRETING THE DERIVATIVE

To determine the instantaneous rate of change (or equivalently, the slope) at any point on the graph of a function f, substitute the value of the independent variable into the expression for dy/dx. The derivative, evaluated at x = c, can be denoted by , which is read “the derivative

of y with respect to x evaluated at x = c”.cxdx

dy

ReviewCovered sections 15.3 and 15.4:• Average rate of change• The difference quotient• Instantaneous rate of change• Slope of a curve• The derivative• Using and interpreting the derivativeNext, we’ll continue with the Differentiation