Engineering Calculus Notes 181

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

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Unformatted text preview: 2.3. CALCULUS OF VECTOR-VALUED FUNCTIONS 169 so that (using the first statement) 1 d ρ2 (t) 2ρ(t0 ) dt t=t0 − → f (t0 ) ′ = · f (t0 ) ρ(t0 ) → = − · f ′ (t ) u ρ′ ( t 0 ) = 0 where → − → − = f (t0 ) u → − f (t0 ) → − is the unit vector in the direction of f (t0 ). 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: Ta 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) − Ta f (x)| = 0; |x − a| or, using “little-oh” notation, f (x) − Ta f (x) = o(x − a). This means that the closer x is to a, the smaller is the discrepancy between the easily calculated affine function Ta f (x) and the (often more complicated) function f (x), even when we measure the discrepancy as a percentage of the value. The graph of Ta f (x) is the line tangent to the graph of f (x) at the point corresponding to x = a. The linearization has a straightforward analogue for vector-valued functions: Definition 2.3.17. The linearization of a differentiable vector-valued → function − (t) at t = t0 is the vector-valued function p → → Tt0 − (t) = − (t0 ) + tp ′ (t0 ) p p ...
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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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