2.9 Derivative as a Function. From yesterday: the definition of a derivative: The derivative of a function f at a number a, denoted by is: if this limit.

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Presentation transcript:

2.9 Derivative as a Function

From yesterday: the definition of a derivative: The derivative of a function f at a number a, denoted by is: if this limit exists Another useful form of the derivative occurs if we write x = a + h, then h = x – a, and h approaches zero as x approaches a

A function f is differentiable at a if f(a) exists. It is differentiable on an open interval (a,b) IF it is differentiable at EVERY NUMBER in the interval. Theorem: If f is differentiable at a, then f is continuous at a. The converse of this theorem is false. There are many functions that are continuous but not differentiable.

If, find a formula for & graph it on the next page

If find the derivative of f. State the domain of f

Given, graph its derivative:

Other notations for the derivative: A function f is differentiable at a if fa exists. It is differentiable on an open interval or IF it is differentiable at EVERY number in the interval.

When is differentiable????

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

Read p. 165 – 173 Work p. 173 # 1, 5, 14, 21, 22, 23, 33, 37, 44