Published on: Wed Mar 03 2010
I found out the answer to why there are two very similar definitions of the derivative. One has x-a on the bottom, with a limit x → a. The other one has h on the bottom, with a limit h → 0. So these two things are the same, just different ways of thinking about the same concept. h represents the difference between x and a, so the smaller the difference gets the closer h is to 0. The official definition is the one with h. A function F is differentiable on an interval if F is differentiable for every point a. Key Properties of Differentiability 1. F is differentiable if F is continuous 2. F is differentiable if F is smooth. For |x| write it in bracket form!
Q1: Is f(x) continuous? Yes because a limit exists for all point a.
Q2: Is f(x) differentiable? Use the definition of derivative (h → 0) and see if the same from left and right. This derivative from the left and the right is not the same, it is -1 and 1. F is not differentiable at 0.
Rule
Tan, always break apart.
Property – Write this on a card
1. If f’(x) < 0 is decreasing near x and inverse: a. f’’(x) > 0 f is concave up.
b. f’’(x) < 0 f is concave down.