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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