lecture10 - 2.7 Tangents and Derivatives at a Point Finding...

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2.7 Tangents and Derivatives at a Point Finding a Tangent to the Graph of a Function DEFINITIONS The slope of the curve y = f ( x ) at the point P ( x 0 , f ( x 0 )) is the number (provided the limit exists). The tangent line to the curve at P is the line through P with this slope. Example 1 f ( x ) = 1 x a) Slope at y at x = a = 0 1
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b) Where is the slope equal to - 1 4 ? 2
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c) What happens to the tangent to the curve at the point ( a, 1 a ) as a changes? 3
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Rates of Change: Derivative at a Point The derivative of a function f at a point x 0 , denoted f ( x 0 ), is provided this limit exists. Example 2 A rock is falling freely from rest near the surface of the earth. The rock fell y = 16 t 2 feet during the first t seconds. What was the rock’s speed at t = 1? 4
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Summary The following are all the same! 1. The slope of y = f ( x ) at x = x 0 . 2. 3. 4. 5. 5
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3 Differentiation 3.1 The Derivative as a Function We investigate the derivative as a function derived from f by considering the limit at each point x in the domain of f .
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