14.8. Instantaneous Rate of Change http://www.ck12.org
Because you are interested in the slope as the “run” approaches zero, this is a limit question. One of the main reasons
that you study limits in calculus is so that you can determine the slope of a curve at a point (the slope of a tangent
line).
Example A
Estimate the slope of the following function at -3, -2, -1, 0, 1, 2, 3. Organize the slopes in a table.
Solution:By mentally drawing a tangent line at the followingxvalues you can estimate the following slopes.
TABLE14.12:
x slope
-3 0
-2 0
-1 -1
0 -1
1 2
2 0
3 0If you graph these points you will produce a graph of what’s known as the derivative of the original function.
Example B
Estimate the slope of the functionf(x) =√xat the point( 1 , 1 )by calculating 4 successively close secant lines.