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Sec 3.1: Tangents and the Derivative at a Point The difference quotient of ƒ at x 0 with increment h. Example: Find the difference quotient of ƒ at x0=2.

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Presentation on theme: "Sec 3.1: Tangents and the Derivative at a Point The difference quotient of ƒ at x 0 with increment h. Example: Find the difference quotient of ƒ at x0=2."— Presentation transcript:

1 Sec 3.1: Tangents and the Derivative at a Point The difference quotient of ƒ at x 0 with increment h. Example: Find the difference quotient of ƒ at x0=2 with increment h. Def: The derivative of a function ƒ at a point x 0 Example: Find the derivative of ƒ at x0=2

2 Sec 3.1: Tangents and the Derivative at a Point Def: The derivative of a function ƒ at a point x 0 Example: Find the derivative of ƒ at x0=2 Example: Find

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

4 Sec 3.1: Tangents and the Derivative at a Point Def: The average rate of change of ƒ with respect to x over the interval [a, b] Def: The rate of change of ƒ with respect to x at x 0 The instantaneous rate of change of ƒ with respect to x at x 0 Term 102

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

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

7 Sec 3.1: Tangents and the Derivative at a Point

8 Slopes : Sec 3.1: Tangents and the Derivative at a Point

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10 0 1 2 >2 Vertical Tangents 0 -2 <-2

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

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

13 Sec 3.1: Tangents and the Derivative at a Point

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