ISM_T11_C06_C - Section 6.3 Lengths of Plane Curves 17(a dy...

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Section 6.3 Lengths of Plane Curves 381 17. (a) 2x 4x dy dy dx dx œÊ œ Š‹ # # L 1 dx Êœ ± ' 1 2 Ê Š‹ dy dx # 14 x d x œ± ' 1 2 È # (c) L 6.13 ¸ (b) 18. (a) sec x sec x dy dy dx dx œ #% # L 1 sec x dx ± ' Î 1 3 0 È % (c) L 2.06 ¸ (b) 19. (a) cos y cos y dx dx dy dy œ # # L 1 cos y dy ± ' 0 1 È # (c) L 3.82 ¸ (b) 20. (a) dx dx dy dy 1 y yy 1y œ² Ê œ È ± # ± # # # L 1 dy dy ± œ '' Î Î ÎÎ 12 É É y # ## ab ±± " 1 y dy ' Î Î # ±"Î# (c) L 1.05 ¸ (b) 21. (a) 2y 2 2 (y 1) ±œ Ê œ ± dx dx dy dy # # L 1 (y 1) dy ' 1 3 È # (c) L 9.29 ¸ (b)
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382 Chapter 6 Applications of Definite Integrals 22. (a) cos x - cos x + x sin x x sin x dy dy dx dx œÊ œ Š‹ # ## L 1 x sin x dx Êœ ± ' 0 1 È (c) L 4.70 ¸ (b) 23. (a) tan x tan x dy dy dx dx œ # # L 1 tan x dx dx ± œ '' 00 66 11 ÎÎ È É # ± sin x cos x cos x # sec x dx œœ dx cos x (c) L 0.55 ¸ (b) 24. (a) sec y 1 sec y 1 dx dx dy dy œ² Ê œ ² È # # # L 1 sec y 1 dy ± ² ' Î Î 1 1 3 4 È ab # sec y dy sec y dy Î Î 33 44 kk (c) L 2.20 ¸ (b) 25. 2 x 1 dt, x 0 2 1 1 y f(x) x C where C is any ÈÈ ÊÊ œ±   Ê œ ±Ê œ Ê œ œ ± ' 0 x dy dy dy dt dx dx real number. 26. (a) From the accompanying figure and definition of the differential (change along the tangent line) we see that dy f (x ) x length of kth tangent fin is œ˜ Ê w k1 k (x ) ( d y ) ) [ f ( x)x ] . ˜± œ˜± ˜ k 1 k # w # (b) Length of curve lim (length of kth tangent fin) ( x ) [f (x ) x ] ˜ ± ˜ nn Ä_ !! È 1 k oeoe #w # 1 [f (x )] x 1 [f (x)] dx ˜ œ ± n Ä _ ! n k a b oe w# w # '
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Section 6.3 Lengths of Plane Curves 383 27. (a) correspondes to here, so take as . Then y x C and since ( ) lies on the curve, C 0. Š‹ È dy dy dx 4x dx x # "" # È œ± " ß " œ So y x from ( ) to (4 2). œ" ß " ß È (b) Only one. We know the derivative of the function and the value of the function at one value of x. 28. (a) correspondes to here, so take as . Then x C and, since ( ) lies on the curve, C 1 dx dy dx y y y dy # " % # œ² ± ! ß" œ So y . œ " x (b) Only one. We know the derivative of the function and the value of the function at one value of x. 29. (a) 2 sin 2t and 2 cos 2t ( 2 sin 2t) (2 cos 2t) 2 dx dx dt dt dt dt dy dy œ² œ Ê ± œ ² ± œ Ê ˆ‰ È # # ## Length 2 dt 2t Êœ œ œ ' 0 2 1 Î cd 1 Î# ! 1 (b) cos t and sin t ( cos t) ( sin t) dx dx dt dt dt dt dy dy œœ ² œ ± œ ± ² œ 11 1 Ê È # # Length dt t ' Î Î 12 1 "Î# ±"Î# 30. x a( sin ) a(1 cos ) a 1 2 cos cos and y a(1 cos ) Êœ² Ê œ ² ± )) ) ) ) ) dx dx dd ab # a sin a sin Length d 2a (1 cos ) d Ê œ Ê œ ± œ ² dy dy dy d d dx ) ) ) Ê È # # '' 00 22 a 2 2 d 2a sin d 2a sin d 4a cos 8a œ œ ² œ ÈÈ É ¸¸ ± ' 0 2 1 1c o s 2 ± # # ! ))) ) 1 ) 31-36. Example CAS commands: : Maple with( plots ); with( Student[Calculus1] ); with( student ); f := x -> sqrt(1-x^2);a := -1; b := 1; N := [2, 4, 8 ]; for n in N do xx := [seq( a+i*(b-a)/n, i=0. .n )]; pts := [seq([x,f(x)],x=xx)]; L := simplify(add( distance(pts[i+1],pts[i]), i=1. .n )); # (b) T := sprintf("#31(a) (Section 6.3)\nn=%3d L=%8.5f\n", n, L ); P[n] := plot( [f(x),pts], x=a. .b, title=T ): # (a) end do: display( [seq(P[n],n=N)], insequence=true, scaling=constrained ); L := ArcLength( f(x), x=a. .b, output=integral ): L = evalf( L ); # (c) 37-40. Example CAS commands: : Maple with( plots ); with( student ); x := t -> t^3/3; y := t -> t^2/2; a := 0;
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384 Chapter 6 Applications of Definite Integrals b := 1; N := [2, 4, 8 ]; for n in N do tt := [seq( a+i*(b-a)/n, i=0. .n )]; pts := [seq([x(t),y(t)],t=tt)]; L := simplify(add( student[distance](pts[i+1],pts[i]), i=1. .n )); # (b) T := sprintf("#37(a) (Section 6.3)\nn=%3d
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This note was uploaded on 10/13/2011 for the course MATHEMATIC 103 taught by Professor Thommas during the Spring '11 term at LCC Intl University.

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ISM_T11_C06_C - Section 6.3 Lengths of Plane Curves 17(a dy...

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