1.2, The ARC and IRC

From the previous section we see that the first step to finding the linearization of a function at a point is the find the slope of the function there.  How do we do that?

Delta f, the slope, and the ARC of a function

Suppose we have a function  y = f(x)  and two points P(x,f(x)) and Q(x+h, f(x+h)).  What is the slope of the line connecting these two points?

The slope of a line connecting P and Q is the difference of y-values divided by the difference of x-values, that is,

m = (f(x+h) - f(x))/(x+h - x).

We may simplify the denominator and write

m = (f(x+h) - f(x))/(h).

Phrased in slightly different notation, the average rate of change (ARC) of a function describes how it changes across an interval from  x = x1 to  x = x2.   We compute the change in the function value:

Delta y := y2-y1

 which is the same as  f(x2)-f(x1)  and divide it by the change in  x, Delta x := x2–x1.  So the ARC is

ARC := (Delta y)/(Delta x) = (f(x2)-f(x1))/(x2-x1)

There is other notation:  if our first point if P(x, f(x)), (for a particular x value) then we might write our second point as Q(x+h, f(x+h)).   In this case, the ARC is

ARC :=  (Delta y)/(Delta x)  = (f(x+h)-f(x))/h.

Note that the ARC is a slope.  It is the slope of the secant line connecting the two points
 (x, f(x)) and (x+h, f(x+h)).

The Instantaneous Rate of Change

We are really after the instantaneous rate of change (IRC) of a function f(x) at a point  P(x0,f(x0)). The IRC of a function at a point is the slope of the tangent line at that point.

We are really interested in the IRC, not the ARC!  But we only know one point on the tangent line so we cannot directly compute the slope of the tangent line.

Instead we view the tangent line as a limit of secant lines as the point Q gets closer to the point P.

This means that we can estimate the IRC by computing the ARC where the second point Q is very close to the point P.

We can make this precise by introducing the idea of limits, in the next lecture.

Homework over this material

Do Worksheet 1.2 on the ARC & IRC. That worksheet, along with video podcasts on this material, is available here.

Continue reading in Section 2.1 of the Open Stax textbook.