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Engineering Calculus Notes 181

# Engineering Calculus Notes 181 - 2.3 CALCULUS OF...

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2.3. CALCULUS OF VECTOR-VALUED FUNCTIONS 169 so that (using the first statement) ρ ( t 0 ) = 1 2 ρ ( t 0 ) d dt vextendsingle vextendsingle vextendsingle vextendsingle t = t 0 bracketleftbig ρ 2 ( t ) bracketrightbig = −→ f ( t 0 ) ρ ( t 0 ) · vector f ( t 0 ) = −→ u · vector f ( t 0 ) where −→ u = −→ f ( t 0 ) vextenddouble vextenddouble vextenddouble −→ f ( t 0 ) vextenddouble vextenddouble vextenddouble is the unit vector in the direction of −→ f ( t 0 ). Linearization of Vector-Valued Functions In single-variable calculus, an important application of the derivative of a function f ( x ) is to define its linearization or degree-one Taylor polynomial at a point x = a : T a f ( x ) := f ( a ) + f ( a ) ( x a ) . This function is the affine function ( e.g. , polynomial of degree one) which best approximates f ( x ) when x takes values near x = a ; one formulation of this is that the linearization has first-order contact with f ( x ) at x = a : lim x a | f ( x ) T a f ( x ) | | x a | = 0; or, using “little-oh” notation,
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