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Unformatted text preview: 2.3. CALCULUS OF VECTOR-VALUED FUNCTIONS 169 so that (using the ﬁrst statement)
2ρ(t0 ) dt t=t0
f (t0 ) ′
· f (t0 )
= − · f ′ (t )
u ρ′ ( t 0 ) = 0 where
− = f (t0 )
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)|
|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
Deﬁnition 2.3.17. The linearization of a diﬀerentiable vector-valued
function − (t) at t = t0 is the vector-valued function
Tt0 − (t) = − (t0 ) + tp ′ (t0 )
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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.
- Fall '08