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12.4 Derivatives  (Page 2/18)

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Finding the average rate of change

Find the average rate of change connecting the points   ( 2 , −6 )   and   ( −1 , 5 ) .

We know the average rate of change connecting two points may be given by

AROC = f ( a + h ) − f ( a ) h .

If one point is   ( 2 , − 6 ) , or   ( 2 , f ( 2 ) ) , then   f ( 2 ) = −6.

The value   h   is the displacement from   2   to   − 1 , which equals   − 1 − 2 = −3.

For the other point,   f ( a + h )   is the y -coordinate at   a + h , which is   2 + ( −3 )   or   −1 , so   f ( a + h ) = f ( −1 ) = 5.

AROC = f ( a + h ) − f ( a ) h             = 5 − ( − 6 ) − 3             = 11 − 3             = − 11 3
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Find the average rate of change connecting the points   ( − 5 , 1.5 )   and   ( − 2.5 , 9 ) .

3

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Understanding the instantaneous rate of change

Now that we can find the average rate of change, suppose we make   h   in [link] smaller and smaller. Then   a + h   will approach   a   as   h   gets smaller, getting closer and closer to 0. Likewise, the second point   ( a + h , f ( a + h ) )   will approach the first point,   ( a , f ( a ) ) .   As a consequence, the connecting line between the two points, called the secant line, will get closer and closer to being a tangent to the function at   x = a , and the slope of the secant line will get closer and closer to the slope of the tangent at   x = a .   See [link] .

Graph of an increasing function that contains a point, P, at (a, f(a)). At the point, there is a tangent line and two secant lines where one secant line is connected to Q1 and another secant line is connected to Q2.
The connecting line between two points moves closer to being a tangent line at   x = a .

Because we are looking for the slope of the tangent at   x = a , we can think of the measure of the slope of the curve of a function   f   at a given point as the rate of change at a particular instant. We call this slope the instantaneous rate of change , or the derivative of the function at   x = a .   Both can be found by finding the limit of the slope of a line connecting the point at   x = a   with a second point infinitesimally close along the curve. For a function   f   both the instantaneous rate of change of the function and the derivative of the function at   x = a   are written as   f ' ( a ) , and we can define them as a two-sided limit    that has the same value whether approached from the left or the right.

f ′ ( a ) = lim h → 0 f ( a + h ) − f ( a ) h

The expression by which the limit is found is known as the difference quotient .

Definition of instantaneous rate of change and derivative

The derivative    , or instantaneous rate of change    , of a function   f   at   x = a , is given by

f ' ( a ) = lim h → 0 f ( a + h ) − f ( a ) h

The expression   f ( a + h ) − f ( a ) h   is called the difference quotient.

We use the difference quotient to evaluate the limit of the rate of change of the function as   h   approaches 0.

Derivatives: interpretations and notation

The derivative    of a function can be interpreted in different ways. It can be observed as the behavior of a graph of the function or calculated as a numerical rate of change of the function.

  • The derivative of a function   f ( x )   at a point   x = a   is the slope of the tangent line to the curve   f ( x )   at   x = a .   The derivative of   f ( x )   at   x = a   is written   f ′ ( a ) .
  • The derivative   f ′ ( a )   measures how the curve changes at the point   ( a , f ( a ) ) .
  • The derivative   f ′ ( a )   may be thought of as the instantaneous rate of change of the function   f ( x )   at   x = a .
  • If a function measures distance as a function of time, then the derivative measures the instantaneous velocity at time   t = a .

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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