Engineering Calculus Notes 236

Engineering - 224 CHAPTER 3 REAL-VALUED FUNCTIONS DIFFERENTIATION 3.2 Linear and Affine Functions So far we have seen the derivative in two

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Unformatted text preview: 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 ) at a point x 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 f = f ( x ) + f ′ ( x ) ( x − x ), which has first-order contact with the curve at the point ( x ,f ( x )): vextendsingle vextendsingle f ( x ) − T x f ( x ) vextendsingle vextendsingle = o ( | x − x | ) or vextendsingle vextendsingle f ( x ) − T x f ( x ) vextendsingle vextendsingle | x − x | → 0 as x → x . 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 ) = −→ v ( t ) is the direction vector for a parametrization of the tangent line, and the...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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