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2.9 Derivative as a Function

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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

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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.

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

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

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Given, graph its derivative:

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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.

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When is differentiable????

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

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

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If f(x) is a continuous function on a closed interval x ∈ [a,b], then f(x) will have both an Absolute Maximum value and an Absolute Minimum value in the.

If f(x) is a continuous function on a closed interval x ∈ [a,b], then f(x) will have both an Absolute Maximum value and an Absolute Minimum value in the.

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