Engineering Calculus Notes 183

Engineering Calculus Notes 183 - 171 2.3. CALCULUS OF...

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Unformatted text preview: 171 2.3. CALCULUS OF VECTOR-VALUED FUNCTIONS each of the component functions fi (t), i = 1, 2, 3 is integrable over [a, b], and then b a f3 (t) dt . f2 (t) dt, f1 (t) dt, a b b b → − f (t) dt = a a A direct consequence of this and the Fundamental Theorem of Calculus is that the integral of (vector) velocity is the net (vector) displacement: → → ˙ p Lemma 2.3.19. If − (t) = − (t) is continuous on [a, b], then v b a → − (t) dt = − (b) − − (a) . → → v p p The proof of this is outlined in Exercise 11. In Appendix C, we discuss the way in which the calculus of vector-valued functions can be used to reproduce Newton’s derivation of the Law of Universal Gravitation from Kepler’s Laws of Planetary Motion. Exercises for § 2.3 Practice problems: → 1. For each sequence {− n } below, find the limit, or show that none p exists. (a) (b) (c) (d) (e) 1 n , n n+1 π nπ (cos( ), sin( )) n n+1 1 (sin( ), cos(n)) n −n (e , n1/n ) n n 2n , , n + 1 2n + 1 n + 1 n n n2 ,2 , n+1 n +1 n+1 nπ nπ nπ , cos , tan ) (g) (sin n+1 n+1 n+1 1 1 1 (h) ( ln n, √ , ln n2 + 1) n n2 + 1 n (f) (i) (x1 , y1 , z1 ) = (1, 0, 0), (xn+1 , yn+1 , zn+1 ) = (yn , zn , 1 − xn ) n ...
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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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