1.8 The Derivative as a Rate of Change

An important interpretation of the slope of a function at a point is as a rate of change.

Consider a function y = f(x) defined on the interval bxa

The average rate of change of f(x) over this interval is the change in f(x) divided by the length of the interval.

abafbf

xy

bxaervaltheoverxfofchangeofrateaverage

)()(int

)(

In the special case where b is a+h, the value of b – a is (a + h) – a or h, and the average rate of change of the function over the interval is the difference quotient

hafhaf )()(

Geometrically, this quotient is the slope of the secant line.

Recall that as h approaches 0, the slope of the secant line approaches the slope of the tangent line.

So, the average rate of change approaches f’(a). For this reason, f’(a) is called the instantaneous rate of change of f(x) exactly at x = a.

The derivative f’(a) measures the (instantaneous) rate of change of f(x) at x = a.

Velocity and Acceleration

An everyday illustration of rate of change is given by the velocity of a moving object.

Suppose we are driving a car along a straight road and at each time t we let s(t) be our position on the road measured from a reference point.

At any instant, the car’s speedometer tells us how fast we are moving…how fast our position s(t) is changing.

The speedometer reading is related to our calculus concept of a derivative.

Let’s examine what is happening at t = 2.

Consider a short duration h from t = 2 to t = 2 + h.

Our car will move from the position s(2) to s(2 + h), a distance of s(2 + h) – s(2).

The average velocity from t = 2 to t = 2 + h is

hshs

elapsedtimetraveledcedis )2()2(tan

If the car is traveling at a constant speed from t = 2 to t = 2 + h, then the speedometer reading will equal the average velocity.

Recall that this ratio

will approach s’(2) as h approaches zero. For this reason, we call s’(2) the instantaneous velocity at t = 2.

This number will agree with the speedometer reading at t = 2 because when h is very small, the car’s speed will be nearly steady over the time interval from t = 2 to t = 2 + h.

In general, we have that

If s(t) denotes the position function of an object moving in a straight line, then the velocity v(t) of an object at time t is given by

v(t) = s’(t)

If the car moves in the opposite (negative) direction the average velocity ratio and the limiting value s’(2) will be negative.

We interpret negative velocity as movement in the negative (opposite) direction along the road.

If we take the derivative of the velocity function v(t), we get what is called the acceleration function

a(t) = v’(t) = s’’(t)

v’(t) measures the rate of change of velocity, and the use of the word acceleration agrees with our usage in connection with automobiles.

Approximating the Change in a Function

Consider the function f(x) near x = a.

We know that

)(')()(

afh

afhaf

For small h, the average rate of change over a small interval is approximately equal to the instantaneous rate of change at x = a.

Multiplying both sides of the approximation

by h yields

)(')()( afhafhaf

)(')()(

afh

afhaf

Or geometrically, this is represented as

When h is small, h*f’(a) is a good approximation to the change in f(x).

In applications, h*f’(a) is calculated and used to estimate f(a+h) – f(a).

The Marginal Concept in Economics

Economists often use the adjective marginal to denote a derivative.

If C(x) is a cost function, then the value of the derivative C’(a) is called the marginal cost at production level a.

The number C’(a) gives the rate at which costs are increasing with respect to the level of production when the production is currently at level a.

Suppose we want to know how much the cost will increase if we produce an additional unit above a.

We have h = 1 and use our approximationC(a+h) – C(a) = h * C’(a):

C(a+1) – C(a) = 1 * C’(a)

The quantity C(a+1) – C(a) is the amount the cost rises when the production is increased from a units to a + 1 units.

Economists interpret the previous graph by saying that the marginal cost is approximately the increase in cost incurred when the production level is raised by one unit.