Section Summary: 3.2

  1. Definitions

  2. Theorems

    If f is differentiable at a, then f is continuous at a.

    For the derivative to exist, the function must be defined at a (so f(a) exists), then the limit of f(x) must exist and approach the value f(a) at a. This is the essence of continuity, however: hence, differentiability implies continuity.

    Note: it is not true that if f is continuous at a, then f is differentiable at a. For example, continuous function with a corner at x=a is not differentiable there.

  3. Properties/Tricks/Hints/Etc.

  4. Summary

    In this case we take the derivative as defined in section 3.1 one step further: if for each value of a there is something called tex2html_wrap_inline186 , then we could write

    displaymath154

    which means that with every value a of the domain, there is associated a value tex2html_wrap_inline186 of the range. Hence we can think of tex2html_wrap_inline176 as a function (the slope function, which gives the slope of the curve at any point of the graph - where defined).

    The slope might not be defined for a number of reasons: the graph may be discontinuous (have a hole, or jump, or infinite discontinuity); the graph of a continuous function may have a corner (where the tangent line is not defined); or a smooth, continuous function may have a tangent line with infinite slope.

    Note that in the definition of the derivative function we simply replace the value of a with x: we've been thinking of a as a fixed number, but now that we want to think of a as varying, we replace it with x (to make you think of it as a variable).

Problems:

Assignment #10: Problems pp. 144-147, #3, 4, 5, 6, 10, 12, 14, 18, 21, 32; Together: 2, 4, 31, 7, 21





LONG ANDREW E
Thu Feb 8 09:51:26 EST 2001