But Wait, It's A Little Off...
But wait a second, the breadcrumb trail doesn't exactly match
f'(x). Here's a zoomed in view to make this easier for you to see. Move A, and notice that the breadcrumb is systematically too high.Don't worry. We haven't done anything wrong. We just haven't applied what we learned from our previous study of Limits.
The problem is, in order to exactly match
f(x) at A, we must use the limit of the slope of g as h approaches 0. In this activity, h is set to .01. We need the limit of the slope secant lines as h approach 0. Any specific reading on g isn't good enough. We need the line at "end of the trend".
Recall from our first limit in the previous chapter, when A=(1,3), the line at the end of the trend had slope 4. But this number will change when A moves to other points on the graph of f(x).
How do we deal with this? Move forward to see one idea.