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**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

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**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

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**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

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**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

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**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

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**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

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**Sec 3.1: Tangents and the Derivative at a Point**

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**Sec 3.1: Tangents and the Derivative at a Point**

Slopes :

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**Sec 3.1: Tangents and the Derivative at a Point**

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**Sec 3.1: Tangents and the Derivative at a Point**

Vertical Tangents <-2 >2 2 -2 1 -1

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**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.

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**Sec 3.1: Tangents and the Derivative at a Point**

Example: has no vertical tangent at x = 0.

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**Sec 3.1: Tangents and the Derivative at a Point**

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**Sec 3.1: Tangents and the Derivative at a Point**

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**Sec 3.1: Tangents and the Derivative at a Point**

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**Sec 3.1: Tangents and the Derivative at a Point**

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Tangent lines Recall: tangent line is the limit of secant line The tangent line to the curve y=f(x) at the point P(a,f(a)) is the line through P with slope.

Tangent lines Recall: tangent line is the limit of secant line The tangent line to the curve y=f(x) at the point P(a,f(a)) is the line through P with slope.

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