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 œ Ê œ Š # # L 1 dx Ê œ ' 1 2 Ê Š dy dx # 1 4x dx œ ' 1 2 È # (c) L 6.13 ¸ (b) 18. (a) sec x sec x dy dy dx dx œ Ê œ # % # Š L 1 sec x dx Ê œ ' 3 0 È % (c) L 2.06 ¸ (b) 19. (a) cos y cos y dx dx dy dy œ Ê œ Š # # L 1 cos y dy Ê œ ' 0 È # (c) L 3.82 ¸ (b) 20. (a) dx dx dy dy 1 y y y 1 y œ Ê œ È # Š L 1 dy dy Ê œ œ ' ' 1 2 1 2 1 2 1 2 É É y 1 y 1 y a b " 1 y dy œ ' 1 2 1 2 a b # "Î# (c) L 1.05 ¸ (b) 21. (a) 2y 2 2 (y 1) œ Ê œ dx dx dy dy Š # # L 1 (y 1) dy Ê œ ' 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 œ Ê œ Š # # # L 1 x sin x dx Ê œ ' 0 È # # (c) L 4.70 ¸ (b) 23. (a) tan x tan x dy dy dx dx œ Ê œ Š # # L 1 tan x dx dx Ê œ œ ' ' 0 0 6 6 È É # sin x cos x cos x sec x dx œ œ ' ' 0 0 6 6 dx cos x (c) L 0.55 ¸ (b) 24. (a) sec y 1 sec y 1 dx dx dy dy œ Ê œ È Š # # # L 1 sec y 1 dy Ê œ ' 3 4 È a b # sec y dy sec y dy œ œ ' ' 3 3 4 4 k k (c) L 2.20 ¸ (b) 25. 2 x 1 dt, x 0 2 1 1 y f(x) x C where C is any È È Ê Ê Š Š 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 k 1 k ( x ) (dy) ( x ) [f (x ) x ] . È È ˜ œ ˜ ˜ k k k 1 k # # # w # (b) Length of curve lim (length of kth tangent fin) lim ( x ) [f (x ) x ] œ œ ˜ ˜ n n Ä _ Ä _ ! ! È n n k 1 k 1 k k 1 k # w # lim 1 [f (x )] x 1 [f (x)] dx œ ˜ œ n Ä _ ! È È n k 1 k 1 k a b 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 Š dx dy dx y y y dy # " " " œ !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 œ œ Ê œ œ Ê ˆ Š È # # # # Length 2 dt 2t Ê œ œ œ ' 0 2 c d 1 Î# ! 1 (b) cos t and sin t ( cos t) ( sin t) dx dx dt dt dt dt dy dy œ œ œ œ œ 1 1 1 1 1 1 1 1 1 Ê ˆ Š È # # # # Length dt t Ê œ œ œ ' 1 2 1 2 1 1 1 c d "Î# "Î# 30. x a( sin ) a(1 cos ) a 1 2 cos cos and y a(1 cos ) œ Ê œ Ê œ œ ) ) ) ) ) ) dx dx d d ) ) ˆ a b # # # a sin a sin Length d 2a (1 cos ) d Ê œ Ê œ Ê œ œ dy dy dy d d d d dx ) ) ) ) ) ) ) ) ) Š Š Ê ˆ È # # # # # # ' ' 0 0 2 2 a 2 2 d 2a sin d 2a sin d 4a cos 8a œ œ œ œ œ È È É ¸ ¸ ' ' ' 0 0 0 2 2 2 1 cos 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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