Calculus: The Language of Change

Exercise Set 5.2
A View in the Microscope

by Wyatt Hadley

2)

You are told that a certain function y=f[x] has a derivative for all values of x, f'[x].  At the point x=1, we know that f'[1]=-2/3.  Sketch what you would see in a very powerful microscope focused on the graph y=f[x] above the point x=1.

Solution

Since the function f[x] has a defined derivative for all values (particularly for x=1) a powerful microscope would depict an image of a straight line with the slope equal to the derivative at that point.
In this case, f'[1]=-2/3, so at highly magnified views, one will see a line like that shown below with a slope of -2/3 (f'[1]) at x=1.

\(\(
f[x_] := 2/3 - 2/3\ x; 

    Plot[f[x], {x,  .9999995, 1.0000005}]\)\)