Sec 2.8: The Derivative as a Function

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Presentation transcript:

Sec 2.8: The Derivative as a Function Def: The derivative of a function ƒ at a point x0 DEFINITION If ƒ’(a) exists, we say that ƒ is differentiable at a. (has a derivative at a) ƒ is differentiable at 2

Sec 2.8: The Derivative as a Function DEFINITION Notations 3) is ƒ differentiable at 3 ?

Sec 2.8: The Derivative as a Function DEFINITION Right-hand derivative at a exist DEFINITION Left-hand derivative at a exist 2) Find the right-hand derivative at 0 3) Find the left-hand derivative at 0

Sec 2.8: The Derivative as a Function DEFINITION DEFINITION If ƒ’(a) exists, we say that ƒ is differentiable at a. (has a derivative at a) A function f is differentiable on an open interval (a, b) if it is differentiable at every number in the interval. Example: ƒ is differentiable on (3, 4) Example: ƒ is differentiable on Example: ƒ is not differentiable on

Slopes (negative, positive, zero) Slope is +1 Slope is -1 Slope is zero Slope is negative Slope is positive

Sec 2.8: The Derivative as a Function Find: When: Positive or Negative: When it is positive:

Sec 2.8: The Derivative as a Function

Sec 2.8: The Derivative as a Function Sketch the Graph of the derivative of the function

Sec 2.8: The Derivative as a Function

Sec 2.8: The Derivative as a Function

Theorem: Sec 2.8: The Derivative as a Function Proof: Differentiability Continuity Both continuity and differentiability are properties for a function. The following theorem shows how these properties are related. Theorem: Differentiable at Continuous at Proof:

Theorem: Sec 2.8: The Derivative as a Function Remark: Remark: Remark: Differentiable at Continuous at Remark: f cont. at a f diff. at a Remark: f discont. at a f not diff. at a Remark: f not diff. at a f discont. at a

Sec 2.8: The Derivative as a Function f cont. at a f diff. at a f discont. at a f not diff. at a f not diff. at a f discont. at a Example:

Sec 2.8: The Derivative as a Function HOW CAN A FUNCTION FAIL TO BE DIFFERENTIABLE? corner discontinuity vertical tangent kink corner discontinuity discontinuity vertical tangent, oscillates

Sec 2.8: The Derivative as a Function

Sec 2.8: The Derivative as a Function Differentiable at exists Continuous at

Sec 2.8: The Derivative as a Function TERM-122 Exam-2

Sec 2.8: The Derivative as a Function Higher Derivative Note: Example: velocity acceleration jerk

Sec 2.8: The Derivative as a Function

Sec 2.8: The Derivative as a Function