3.2. Definition of Derivative.

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Presentation transcript:

3.2. Definition of Derivative. Differentiability on the closed interval and at the point. Corners, cusps, vertical tangent. Domain of the derivative. Rita Korsunsky

Definition of Derivative

Example 1

Example 2

Applications of Derivative Notations of Derivative Why quotient ? Next slide… Not common Higher derivatives

Example 3

Differentiability on a Closed Interval A function f is differentiable on a closed interval [a,b] if f is differentiable on the open interval (a,b) and if the following limits exist:

Example 4

P(5,0) P(-5,0) l1 l2 y = f(x) P(a, f(a)) l y

Example 5 -5

Theorem Proof: f is continuous at a If a function f is differentiable at a, then f is continuous at a. Proof: f is continuous at a