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Engineering Calculus Notes 236

Engineering Calculus Notes 236 - 224 CHAPTER 3 REAL-VALUED...

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224 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION 3.2 Linear and Affine Functions So far we have seen the derivative in two settings. For a real-valued function f ( x ) of one variable, the derivative f ( x 0 ) at a point x 0 first comes up as a number, which turns out to be the slope of the tangent line. This in turn is the line which best approximates the graph y = f ( x ) near the point, in the sense that it is the graph of the polynomial of degree one, T x 0 f = f ( x 0 ) + f ( x 0 ) ( x x 0 ), which has first-order contact with the curve at the point ( x 0 ,f ( x 0 )): vextendsingle vextendsingle f ( x ) T x 0 f ( x ) vextendsingle vextendsingle = o ( | x x 0 | ) or vextendsingle vextendsingle f ( x ) T x 0 f ( x ) vextendsingle vextendsingle | x x 0 | 0 as x x 0 . If we look back at the construction of the derivative vector p ( t ) of a vector-valued function −→ p : R R 3 in § 2.3 , we see a similar phenomenon: vector p ( t 0 ) = −→ v ( t 0 ) is the direction vector for a parametrization of the tangent line, and the resulting vector-valued function, T t 0
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