Engineering Calculus Notes 180

# Engineering Calculus Notes 180 - 168 CHAPTER 2. CURVES then...

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Unformatted text preview: 168 CHAPTER 2. CURVES then using Proposition 2.3.12 we see that d dt t=t0 → − (t + h) − − (t ) → g0 g0 h → − → − f (t0 + h) − f (t0 ) → · − ( t0 ) g h → − → − → f (t) · − (t) = lim f (t0 + h) · g h→ 0 + lim h→ 0 → − → = f (t0 ) · g′ (t0 ) + f ′ (t0 ) · − (t0 ) . g An interesting and useful corollary of this is → − Corollary 2.3.16. Suppose f : R → R3 is diﬀerentiable, and let ρ(t) := → − f (t) . Then ρ2 (t) is diﬀerentiable, and 1. → − d2 ρ (t) = 2 f (t) · f ′ (t). dt → − 2. ρ(t) is constant precisely if f (t) is always perpendicular to its derivative. 3. If ρ(t0 ) = 0, then ρ(t) is diﬀerentiable at t = t0 , and ρ′ (t0 ) equals the → − component of f ′ (t0 ) in the direction of f (t0 ): ρ′ (t0 ) = → − f ′ (t0 ) · f (t0 ) . → − f (t0 ) (2.23) → − → − Proof. Since ρ2 (t) = f (t) · f (t), the ﬁrst statement is a special case of the Product Rule for dot products. To see the second statement, note that ρ(t) is constant precisely if ρ2 (t) is constant, and this occurs precisely if the right-hand product in the ﬁrst statement is zero. Finally, the third statement follows from the Chain Rule applied to ρ(t) = ρ2 (t) ...
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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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