Sec 3.1: Tangents and the Derivative at a Point

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Presentation transcript:

Sec 3.1: Tangents and the Derivative at a Point Def: Def: The derivative of a function ƒ at a point x0 The difference quotient of ƒ at x0 with increment h. Example: Example: Find the difference quotient of ƒ at x0=2 with increment h. Find the derivative of ƒ at x0=2

Sec 3.1: Tangents and the Derivative at a Point Example: Find the derivative of ƒ at x0=2 Def: The derivative of a function ƒ at a point x0 Example: Find

Sec 3.1: Tangents and the Derivative at a Point RATES OF CHANGE Def: The average rate of change of ƒ with respect to x over the interval [a, b] Chane in x = Chane in y = Term 102

Sec 3.1: Tangents and the Derivative at a Point Def: Def: The average rate of change of ƒ with respect to x over the interval [a, b] The rate of change of ƒ with respect to x at x0 The instantaneous rate of change of ƒ with respect to x at x0 Term 102

Sec 3.1: Tangents and the Derivative at a Point quotient The slope of the tangent to the curve The slope of the secant The slope of the curve instantaneous rate of change Average rate of change

Sec 3.1: Tangents and the Derivative at a Point quotient The slope of the tangent to the curve The slope of the secant The slope of the curve instantaneous rate of change Average rate of change

Sec 3.1: Tangents and the Derivative at a Point

Sec 3.1: Tangents and the Derivative at a Point Slopes : 0 + -

Sec 3.1: Tangents and the Derivative at a Point

Sec 3.1: Tangents and the Derivative at a Point Vertical Tangents <-2 >2 2 -2 1 -1

Sec 3.1: Tangents and the Derivative at a Point Example: Example: has a vertical tangent at x = 0. has a vertical tangent at x = 0.

Sec 3.1: Tangents and the Derivative at a Point Example: has no vertical tangent at x = 0.

Sec 3.1: Tangents and the Derivative at a Point

Sec 3.1: Tangents and the Derivative at a Point

Sec 3.1: Tangents and the Derivative at a Point

Sec 3.1: Tangents and the Derivative at a Point