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CALCULUS WITH ANALYTIC GEOMETRY' Answers to OddDNumbered Exercises SelhTests / Daniel J. Fleming Department of Mathematics St. Lawrence University James J. Kaput Department of Mathematics Southeastern Massachusetts University Harper & Row, Publishers New York Hagerstown Philadelphia San Francisco London Answers to Odd-Numbered Exercises/Self-Tests to accompany Calculus with Analytic Geometry 0 1979 by Daniel J. Fleming and James J. Kaput. ANSWERS TO ODD-NUMBEItED EXERCISES / SELFlTESTS A-2 ANSWERS TO ODD-NUMBERED EXERCISES/SELF-TESTS Chapter 1 See. 1.1 1. [-11/2, -1/2] 3. x = 2 5. [-9,3] 7. (a) x = -3, -2; (b) x = 5 9. (-co, -2), [-1,2/5], and [4, +co) 11. (-00, -1) and (0,5/3) 13. (-3, -fi)an d (fi+,c o) See. 1.2 p5) \;. 1. (u ;)5 1 (6) 9 3. (T-,1lo7) -23 5. (3, 3) 7. (u) -1; ()jb)y = -x + 6 9. the line in 8(c) with slope = 4 + 11.9 - 1 = 4(x - 4) 13. x = 3 1,5. (a)y - 1 = 4(x - 4); (b)y= 4~ - 15; (6) 4~ -J - 15 = 0 17. y = -X 1 + 25. (a)y = x 1; (b) yes; (d)y = x 19- 21- 23. (I/'& 0) ~ ~ (0, -15) (0, -1) + 27. (6) 3y x - 9 = 0 29. (4, 13/2), (6,5), and (8, 7/2) 31. (a) (13, 3), (6) (-9, 13) See. 1.3 1. 3. 5. (a) parabola; (b) axis x = - 4, conc. up, ext. pt. ( - 71 7>-81 ) x = -1/6 7. (a) (x - 2)z + (y + 3)2 = 25; (6) a circle, center at (2, -3), radius 5 9. (a) parabola; (b) axis x = 3/2, ext. pt. (3/2, 15/4) conc. down 11. no graph 13. (u) parabola; (b) opening right, axisy = 1, ext. pt. (0, 1) 15. y=o 2 1. single pt. (0, 0) (-11/6, -169/12) x = -11/6 Review Exercises I.x<3/4 3.21/4>x 5.x>-9 9. -j<x<3 l l . x = - 5 13. -2>xorx>14/5 15.nosolution 1 7 . ( ~ + 2 ) ~ + 4 19- 2(x - $z - 9 21- 2(x - Jfi+) S' 23. 3(x + i)z- 25. (-3, -2) U (0, 1/2) U (3, +w) 3(2c)37 2.. x ((-a--y)W 5,; + ( b-31) 1 =( U-0 (903-,4,7$)).; U(( x6[ )l- ,2 ; + 5()md2) +) 3y (2y-9 - 4.x 6(-)-20 0 =1,0 1=+/9 0 w; 3)(e9 3).y 1+ (.x 4+( a= )2 )a2-a (+. ;((y+6) + 9( 44),)s2 )3 ;= 5( c1.) 6 -( 4a;4) 51x(. d +)c 5 e3xny t+.- (2 -y16 5,- 8= )2 ,07 ;7 = (=b )0 ;12 0x( e-)4yy3 --. S c6 =e =n {t( .x0 (;3 -, - 44)) , ANSWERS TO ODD-NUMBERED EXERCISES/SELF-TESTS A-3 r = 5 45. none of these 47. st. line, slope 3,y int. 2 49. par. opening right, axisy = - 1, ext. pt. (-5, - 1) 5 1. st. line, slope 1, y int. 8 53. par. opening up, axis x = -3, ext. pt. (-$, +y) .55. par. opening up, axis x = 2, ext. pt. at (2, 3) 57. st. line, slope 2, y int. -5 59. vert. st. line through (46.1,O) 61. none of these 63. none of these 65. a circle, cent. at (0, 2), 7 = 2 67. a par., opening right, axisy = -4, ext. pt. is (0, -4) 69. none of these + 3. (a) (2, -2); (b)y - x 4 = 0; (c)y- x = O (-l/Z, -25/4)' I + 4. (x - 6)' (y - 8)' = 100 Chapter 2 See. 2.1 x = -1; (b) all R except m(x) = -2; (c) 7- [-3,O]U[l,+o0) 9. (-00,-2)U(7,+00) 11.afunctionisgiven I 1 x( b#) Z 2-x1T -/2T ;1 +(e )2 2(x2 4+? 1(;6 )( fx)++2 x h +-+ 11 ; (+9) 2x(;x (h+) 4hx)' ;+ ( d3 ) hh2-+1-.ll (+a) 23hx 2-; ( e4) _+33 _-+_x1_1+3 _ 4+ ;2 (.b9) 3~1 -9 .4 (-u) - x3 x+ 3+ 4 1,> ;( 6()b )( 3-xx -- 4 3)1( ;x ( 6+) 24x) 2; +(d x); ( 3(xdx )- -+2 24x4)+ .* 3-1 ,' x # -4; (e) x; (f)x ; (g) 9x - 16; (h) _+_ _ 23. (a) 5x; (b) -x; (c) 6x2;( d) 2/3, x # 0; (e) 6x; (f)6 x; (g) 4x; (h) 9x 9 25. (a) x3 + 3x + 3; (b) x3 + 3x - 1; (c) 2x3 + 6x + 2; (d) x3 + + ; (e) 15; (f)2 ; (9) (x3 + 3x + 1)3 + 3(x3 + 3x + 1) + 1; (h) 2 2 27. f(x) = (g o h)(x), where g(x) = $ and h(x) = 2x + 1 29. f(x) = (g o h)(x), where g(x) = x + 1 and h(x) = 2x A-4 ANSWERS TO ODD-NUMBERED EXERCISES/SELF-TESTS See. 2.2 1. -1/2 3. does not exist 5. 1/(4 6)7. -6 9. -& 11. 96 13. the answers are the same 15. r-l-izm+ ~ x +1 2 -- +m, m-l-iz-m -x -+1 2 - -f fi 17. -l-iIr1n2 ~ 12x 1+ 11 -- + ffi 19. (a) Il-i1r-n S(X) = 0; (b) zl-i1m+ ~(x=) 1; (c) no See. 2.3 1. (a) 1944 ft/s; (b) 1836 ft/s; (c) ~1732.3f2t /s; (d) 432(4 + Ax); (e) 1728 ft/s; (f)8 64x ft/s; (g) 4665.6 ft/s; (h) 1/3 ft 3. -2 5. 10x -2 1 4 + + 7. 6 ~ 2- 9. 99 11. 2 ~ 3+ 1 3. 2 ~ 3-x2 15. - 17. - 19. (u) AV = -$3? A7 37(A7)’ AT)^); (b) r = 2, A7 = 1 3x3 2& 3 4?r gives AV = -4?r * l9 ; r’= 20, Ar = 1 gives AV = -* 1261; (c) 4 ~ 7(~d); t hey are equal-a sphere grows by spherical shells 21. (a) 144 - 32x; 3 3 (6) 4.5 s; (c) 324 ft; (d) x = 9, - 144 ft/s; (e) 324 ft; (f) f’ See. 2.4 > < < > 1- (a) f’ (b) sincef’(x) = -2x,,f(x) 0 for x 0,f(x) = 0 for x = 0,f(x) 0 for x 0; (c)y - 2 = -2(x - 1) + 3. (a) f’ (6) sincef’(x) = 3, the slope is always positive; (c)y 2 = 3(x - 1) (same as graph) (6) sincef’(x) = -32, the slope is everywhere negative except at 0; (c)y= -3(x - 1) 9. since lirn 6= 0, S-O+ + then lirn _1_ = co; thus there is a vertical tangent at x = 0 11 . (a) __-1 _ ;(b)lima=-m 17.x<3 19.(u)l;(b)l;(c)l; x-o+ 2 6 (x - 1)2 c-1 dx (d) - \/loo=: 0.05 # 1; (e) horizontal distance between points with samey coordinate is constant; (f) they are horizontally parallel, but not vertically parallel See. 2.5 l.5/12 3. fi 5.4 7. -1/25 9.4 11.lim 15. yes 17. f is disc. at x = 1 x-0 19.fis disc. at x = %2 21. fis disc. at x = 2 25. S(X) = [XI is disc. at every integer value 27. definef(x) = 0 and g(x) = 0; then + + lirn [f(x) g(x)] = lim [O] = 0 =f(0) g(0) x-0 x-0 ANSWERS TO ODD-NUMBERED EXERClSES/SELF-TESTS A-5 Review Exercises 1. (u) x2 + 3x - 1; 3; (b) 3x3 - x2; 2; (c) -3; x - 1 2; (d) 3x2 - 1; 2; (e) (3x - 1)’; 4; (f)3 (3x - 1) - I; 5; (g) (x2)2; 1 3. (a) ;1 + x + 5; 7; X2 8; (d) 3x4 + 5; 8; (e) 3x4 + 5; 8; (f) 3(3x4 + 5)4 + 5; 3*84+ 5; (9) x; 1 7. (a)x 2 - 3x + 2; 0; (6) 4(x2 - 3x - 2); -16; (c) 4 . -I; (d) 4; 4; (e) 2m; 2; cf) ,4; 4; (g) (x2 - 3x - 2)’ - 3(x2 - 3x - 2) - 2; 26 9. (u) -+ 2x2; not defined at 1; (b) 2x2-; x2 - 3x n-o t2 ’ defined at 1; (c) 7not ,defined at 1; (d) d m ;not defined at 1; (e) 2(x - 3); not defined at 1, since g(f(l)) is impossible; 2x => 2x 2 (f) d z 3 ;not defined at 1; (g) 8x4; 8 11. (a)_ x2 _-_ 4 ’ -5;(b ) ’ -1/3; (c) *x. - 2’ -3; (d) -,-2x - 3’ 2x - 3’’ 1; (f) Gx-2 ,’ T-1 ;(9) -. x + 2 -3 13. f(x) = (g o h)(x), where h(x) = x + 1 and g(x) = x3 15. h(x) = (fo g)(x), where g(x) = x2 and 2xf5’ 7 f(x) = 3x3 + 4.9 + 2~ 17. (a)y = 3(2r)’ - 2(2r) - 2; ( b )=~ 3(3t + 1)’ - 2(3t + 1) - 2; (C)Y = 3(t + At)’ - 2(t + At) - 2; (d)y = 3(m2)’ - 2(w2) - 2; (e)y = 3(3~-)~ 2 (3u) - 2 19. not a function of x 2 1. is a function of x 23. is a function of x 25. is a function of x x2 - 5x ifx#O 27. (a) h(x) = x - 5, x # 0; (b) h(x) = (-5 ifx=O 29. (u) h(x) = x + 3, x # 4; (b) h(x) = x2-x- 12 ifx # 4 (7 ifx = 4 31. (u) limg(x) = 8 35. lim (2 - 3x) = - 1 37. only number 31 39. (a) lim- x2 - 5x = -5; (6) lim x2-x- 12 = 7 x+4 x-1 2-0 x x-4 x - 4 41. lim 2(x - 5) = 2; lim 2(x - 5) = -2 45. (u) g’(x) = 1 - -1; (b)y - 3/2 = 0 47. (u)f(t)= --;2 (b)y - 4 = -2(t - 1) 2-5+ ~ Ix - 5) z-5- ~ Ix - 5) X2 t3 49. (a) -4 = -1 . (6) x = 1 is not in the domain 51. (a) g’(x) = --; 3 (b)y - 1 = -3(x - 1) 53. (u)f(u)= 3u2 - IOU; dx ~ 2 m ’ x4 + (b)y 4 = -7(u - 1) 55. (u) h’(r) = 12r; (b)y- 16 = 12(r - 1) 57. cont. everywhere 59. disc. at x = 2 x X Ix= I Self -Test 1. (u) (f- g)(x) = -2 + 5x - 1; (b) cf.g)(x) = (7x - 1)(X* + 2x); (6) (f/g)(x) = -7x +- 1 ’ (d) (fog)(x) = 7(x2 + 2x) - 1; x2 2x’ (e) (g ~f)(x=) (7x - 1)’ + 2(7x - 1); (f) (fof)(x) = 7(7x - 1) - 1 2. (a) f; (b) 1/6 3. (a) x = - 1; (6) x = 9 4. (u) 20; (6) 20; (c) 20; (d) 20 5. no 6. (a) cont. on [a, c) U (c, d) U (d, e) U (3,f) U (J; k]; (6) diff. everywhere except at b, c, d, e,J; h; (c) discont. at c, d, e,f Chapter 3 See. 3.1 1. f(x) = -2 3. g’(x) = 3.9 + lox 5. f’(x) = 0.01 7. t’(x) = (x2 + 1)(21x2 + 6x + 1) + 2x(7x3 + 3x2 + x - 3) (t3 + 3t2)(8t + 1) - (4t2 + l)(3t2 + 6t) + -1 13. -4 = (4x3 + x)( - 14~)- (7~’+ 2)(12~’+ 1) 9. f(t)= + 1 11. h’(s) = __+ _ + + +(t 3 3t2)+2 + + + (s 112 dx + + (4x3 x)2 + 15. fyx) = (x2 3)“x 1)(2x) (x2 2) * I] - (x 1)(9 2)2x 17. g’(x) = 0 - l[(x3 3x2)(7) (7x - 2)(3x2 6x)l (x2 + 3)‘ (x3 + 3X2)2(7X - 2)2 + + + + 19. n’(y) = (@‘ - 9)[(Y2- 5!)(4Y3 2) (y4 2y - 1)(23~- 5)] - (y2- 5y)(y4 2y - 1)*6 + + + (6Y - 912 21. 7’(X) = - 4x-5 - (x3 2x +1) 2x -+ XZ(3X2 2) 23. -dU = (x + l)[(x4 + x3 + 1)(6x - 7) + (3x2 - 7x)(4x3 + 32)] + (x3 2x 1)2 dx A-6 ANSWERS TO ODD-NUMBERED EXERCISES/SELF-TESTS + + dy (x2 - 3)(2) - (2x + 4)(2x) (x2 - x + 1)(6x2) - (2x3 + 1)(2x - 1) (x“ x3 1)(3x2 - 7~)(1) 25. - = - + dx (x2 - 3)2 (x2 - x 1)2 0 - l(2v - 324 +- 0 - 6(2u) 1 27. f’(o) = (v2 - u3)2 (W 3 1. (a) 44; (b)y - 33 = 44(x - 2); (c) y - 33 = - -4(x4 - 2) 33. for all x +# 0, -+ 1 - 2u’(x) -2(4x3 3x2 2x) 35. (u) h’(t) = 120 - 32t; (b) 321 ft; (6) t = 15/2 39. f ( ~= )~[ u(x)][u’(x)] 41. ~ 14413 = h‘(x) 43. f’(X) = (x4 + x3 + x2 + 3)3 19. f(z) = 6(z4+ - 32 - 2)5(4.“ - 3+) 2 I. f(y) += (y+3 + 1)+“ * 7(5y2 - y + 2)6+( lO y - 1) + +(5y 2 - y + 2)? -4(y3 + 1)3(3y2) 23. g’(x) = (x2 ZX)’/~(-~/~)(X5~~ )-~/’(5x” 5) (x5 5~)-’/~(1/5)(x~ 2x)-“I5(2x 2) + + + + + + 25. m’(t) = (t 2)1/2(1/3)(t2 t)-2/3(2t 1) (t2 t)1/3(l/2)(t 2)-1/2(1) 27. f(h)= -(- ‘I2 . (h + 1)+ * 1 - h(1) (3h2 + h)(3h2 + 1) +- (h3 + h)(6h + 1) 2 h + l (h 1)2 (3h2 h)2 29. f(t) = -1(t 4 + 1)-’”(4t3) - (t3 + t2 + t + 1)-5/3(3t2 + 2t + 1) 2 + + + + (x4 l)’l3(l/2)(x5 4)-’/2(5x4) - (x5 4)1/2(1/3)(x4 1)-2/3(4x3) 31. h’(x) = + 33. (a) dy/dx = -x/y; (6) &/dy = -y/x ~ ( ~ 14)1/3 12 + + 35. (a) dy/dx = ( 2-~ 3 Y)/(3x 3~’); (b) dx/dy = ( 3 ~ 3 y2)/(2x - 3y) 37. (a) dy/dx = -xy2/xTy = -y/x; (6) dx/dy = -x/y 39. (u) dy/dx = - (1/(21)/y21)x/21x/2-1y/-21 /2 +- y- 1xy --2 1 ; (b) dx/dy is l/(dy/dx) 41. (a) dy/dx = (x + 22)y2x 2- 2xyz ; (6) dx/dy is l/dy/dx , , ~ 43. dy/dx = dx/dy = -1 fory # -x 45. 1 = dy/dx 47. (u) y - 5 = -2(x - 5); (b) yes 49. dy/dx = 2/5 51. Au = 3, Ag = 33 See. 3.3 7. (u) dy = --dx3x4; (b) --130 9. (u) df= 2(x- - 1 4-)1 /12(0x5~ -4-( 1~ )42 ) . (2t + 3) dt; (b) d f 0.~000 000023 1 1. (a) dy/dx = 5x/2y; (b) dX/dy = 2y/5~ 13. (u) dy/dx = -(39 +y2)/(2~-~ 3~’); (b) dX/’dy = -(~xY - 3y2)/(3xZ +y2) 15. (u) dy/dx = (5x4 + 3~3~)/(5y-“ 3x57; (b) dx/dy = (5y“ - 3~5~)/(5x+“ 3x33) 17. (u) dy/dx = -(3x2 + +y1/2x-1/2 + y2)/(2xy + ij~l/2y-~/~(b)); dx/dy = -(29 + + iy1/2x-1/2 +y2) See. 3.4 1. f’(x) = 3x2 + 2x,f”(x) = 6x + 2,f”’(x) = 6 3. g’(x) = -!k3g”(,x ) = 6x-“, g”’(x) = - 2 4 ~5~. f ’(t) = ft-’/’,f’(t) = + + + f’”(t) = 7. f’(x) = 0 =f”(x) =f”(x) 9. h’(r) = 5(r 4)4, h”(r) = 20(r 4)3, h”’(r) = 60(r 4)* + (3y 1).1 -y.3 + + + + 11. m’(y) = = (3y 1)-2, m”(y) = -2(3Y 1)-3(3) = -6(3 1)-3, m”’(y) = 54(3y 1)-4, + (3Y 13. n(n - 1). . . (n - k + 15. -n(-n - 1)(-n - 2)~-%-~1 7. acc. = s”(t) = 24t2 - 8 19. acc. =f”(t) = 24t - 6 > < 2 1. y” = 2a; conc. up if a 0, con. down if a 0 23. - 32 Review Exercises. 1. y(x)= -x(l - x2)-’/’ 3. g’(t) = (t - 5)11*4(ti + 1)-’l3(7t6) + (ti + 1)1/3*11 *(t - 5)’O S.f(x) = -i(x + 4)-3/2 7. f’(x) = -1(- -)x -1-415 . (x + l -+x + l ) 9. y’ = (x2 + l)50. 10(x2 - 1)9 * 2x + (x2 - 1)1° * 50(x2 + 1)49. 2x 5 x + l (x 1)2 11 51.. fh’(’x()x ) == 44([u(xx32 ++ 4bxx +- Cl))(~x(5~ -U.+ Yx 2b +) x1 +3 .2 f)’I(3x[)( x3= + & 4lx +- (13x)( 5-x 45 -)6 )2-1x’ 2+. 61()3 +x -(x 55) -5 * x32 + x + 2)(3x2 + 4)] + + + + 17. p’(s) = f(s4 s3 s2)-1/4(4s3 3s2 2s) 19. t’(x) = 0 2 1. g’(z) = #(llz - 4~~)-~/~-( 8121) + 23. t’(w) = (1 w2)(-1) + - (1 - w)2w 25. gyy) = (y - 1)1/2.+(~+ 1)-2/3 + (y + 1)1/3-4(y - i)-1/2 27. g‘(x) = (x2 + 1)5((1~ 7+ wx)2~) U-7 (~4+ 1)6-4~3+ (x2 + 1)5(x4 + l)i1*0( x7 + X ) ~ ( ~+ X1)~ + (x7 + x)1°(x4 + l)?.5(x2 + l)4(2x) by Sec. 3.1, ANSWERS TO ODD-NUMBERED EXERCISES/SELF-TESTS A-7 Ex. 45 29. t‘(s) = *9(s - 4)8 + (s - 4)g*-(1-) s+ 1-1’2 [ s- 1 -(s+ 1) ] 3 1. f”(x) = 6x 2 s-1 (s - 1)2 33. m’(t) = 71/72 35. g’(x) = (x + l)l0(x2 + 1)”4(x3 + 1)”3x2) + (x + l)‘’(x3 + 1)4*5(x2+ l)4(2x) + (2+ 1)5(x3 + l)4-10(x+ 1)9 37. (a)y’ = (3x + 2y)/(y - x); (b)y - 2 - 2 fi=(7 4fi) *(x - 1) 39. (a) dy/dx = -(sXy2 + 14x)/(5y4 + 6x5); +Z f i (b)y - 1 = -50\/137/ 10 (X - m)4 1. dy/dx = -(2xy2 + 3~$~)/(2x+3 3x9’) 43. (a) dy/dx = (y2 -y)/-(x + 8y2); (T6) - 1 2fi-3 (b)Y - = (x - 4) 45. (a) dy/dx = -y3/x3; (b)y - 2 = 3*(x - 2/*) 47-55. In every case, dx/+ = l/dy/dx ~ 44 67. f(31) zf(32) + df= 2 - b = 1.9875 69. 0.94 7 1. 0.09833 73. 0.33 75. 195.8 gal 77. (a) vel. inc. if t < 1; vel. dec. if > < < > > < t 1; (b) moving away for 0 t 2 and t 3 79. f’(x) = 0 =f”(x) 8 1. if x 0 the slopes inc. and if x 0 the slopes dec. 83. (a) ~(t=) 144 - 32; (b) t = 9/2 S, h(9/2) = 324 ft 85. t = 96 Self-Test + + + + dy (x7 x6)(l/2)(x3 4)-’/’(3x2) - (x3 4)l/’(7x6 6x7 dy + 1. (a) - = + ; (b) - = (x2 1)4/5(8)(-~ dx (x7 x92 dx 3y - 2xy+2 2. -d2y- - (2x5 - 3~ + 2~)(3(dy/dx)- 4xy(dy/dx) - 2y2) - (+3y - 2xy2)(2x2(dy/dx) + 4xy - 3 + 2(dy/dx)) 3. -0.012 2x3 - 3x 2y dx2 (2x3 - 3x 2y2) 4. (a) x = 2; (b) -4 5. 360/t7 Chapter 4 See. 4.1 1. vert.: x = 2, obl.:y = x + 2 3. none 5. hor.:y = 3/4 7. note: no vert. at x = - 1/3 as u only defined for 1x121 hor.:y = 1/3 + 9. vert.: x = -1, hor.:y = 1 11. vert.: x = 2, obl.:y = x 2 13. vert.: x = -9/2, obl.:y = x/2 - 7/4 15. vert.: x = 0, hor.:y = 0 17. hor.:y = 0 19. vert.: t = 1/2 21. vert.: x = -1, obl.:y = x - 2 23. vert.: x = (3 i- a ) / 2 25. hor.:y = 1 See. 4.2 < < (6) C.U. on (x2, xJ, (x4, a), where x5 a x6, (x7, x9); c.d. on (x,,, xz), (u, x7); (c) rel. max. at x = xI, x4, x6, rel. min. at x = x3, x5, xs; (d) all but xq, (e) at x = x2, x7; (Aa t x = a bet. x5 and x6 5. (a) (-00, -2), (1, +00); (b) (-2, 1); (c) x = -2, l;f(-2) = 16,f(l) = -11 7. (a) inc. for all x; (6) none 9. (a) (-4,0), (4. +w); (b) (-00, -4), (0,4); (6) rel. min. x = *4,f(?4) = 0; A. max. x = 0,f(0)= 256 < 13. c.d. everywhere 15. c.d. for x 3; 3 > C.U. for x 3; p.i. x = 3 See. 4.3 I. i-:Ij13. 15. 17. t’ A-8 ANSWERS TO ODD-NUMBERED EXERCISES/SELF-TESTS t’ 31. 33. +y 27-+ 29- I x= -1 x = l See. 4.4 1. no deriv. at x = 1 3. no deriv. at x = 0 5. no deriv. at x = 0 9. c = kfi 11. c = 1 13. c = &1/(3fi) 15. c = 0 17. c can be any value in the interval since tangent line = graph Review Exercises 1. l;hor.asy.y=l 3. +w;vert.asy.x=O 5. +w;vert.asy.x=2 7. -w;obl.asy.y=jx-j 9.0;noasy. 11.incr. (-W, 1/2); decr. (1/2, +w); rel. max. x = 1/2 13. decr. for all x 15. inc. for all t 17. C.U. (-a,1); c.d. (1, +w); p.i. x = 1 19. c.d. (-1 fi,l /fi); C.U. elsewhere; pi. x = %l/fi 21. (-1606,) (2/3, - 16/27)