Section Summary: 3.1

1. Definitions

   derivative of a function f at a - denoted by ,

   

   provided the limit exists (that's right - this is exactly the same as the slope of the tangent line from section 2.6!). So the derivative of a function f at a point a is the slope of the tangent line to the curve at P(a,f(a)).

   Alternatively, it can be considered the instantaneous rate of change of y=f(x) with respect to x when x=a.

2. Theorems

   None appeared to my eyes.

3. Properties/Tricks/Hints/Etc.

   Sometimes the derivative is merely estimated from data, using average rates of change, or by a visual approximation based on a graph.

4. Summary

   This section is an easy extension of section 2.6: the big picture is that the dreaded derivative, one of the fundamental concepts of calculus, is actually just the same as the slope of a tangent line to a curve. It can also be considered an instantaneous rate of change.

Problems to consider:

pp. 134-135, #1, 2, 6, 8, 16, 20, 25, 32, 34; Board: 5, 8, 16, 25, 32, 33