Calc 1 WA3 - WA 3 Calculus I CHAPTER 3 Section 3.1 f ( x) =...

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Unformatted text preview: WA 3 Calculus I CHAPTER 3 Section 3.1 f ( x) = 3x + 2 f ( x +Vx) - f ( x) x 0 Vx (3( x +Vx ) + ( x +Vx)) - (3x + 2) = Vx 14) (5 x - 5Vx) - 3 x + 2 = Vx 2 x + 5Vx + 2 = Vx = 2x + 2 Result here should be just 3 f '( x ) = lim = f ( x) = 1 x2 Vx f 0 f '( x ) = lim = f ( x +Vx) - f ( x ) Vx 1 2 x x ( x +Vx) 2 - x 2 +Vx 2 22) = 2 Vx x x +Vx = = x 2 - ( x +Vx) 2 2 xVx +Vx 2 Vx (2 x +Vx) = 2 = 2 2 2 2 Vx( x +Vx) ( x ) ( x + 2 xVx +Vx )( x )(Vx) (Vx) ( x 2 )( x 2 + 2 xVx +Vx 2 ) 2x 2 = 3 32 x + 2x2 x + 2x 43 Result here should be -2 x3 f ( x) = 4 x x 0 f ( x +Vx) - f ( x) Vx 4 4 1 1 - x + x + x x f '( x ) = x + x 1 x 1 + x x + x 4 4 1 24) - x x + x x + x x 4 = 1 1 x x + x 1 x + x4 x + x f '( x ) = lim = x - x + x 1 = = 1 1 1 1 1 x ( ( 4 x ) ( x + x ) 4 x ) ( x ) x + x + x x+ x 2 1 1 2 = ( x ) = 8x x 2 x 4 0 1 f ( x) = x 2 + 2 x + 1 ( -3, 4 ) f '( x) = lim = 0 2 f ( x +Vx) - f ( x) Vx 2 ( x - x ) + 2 ( x + x ) + 1 - ( x 2 + 2 x + 1) 26) (x = = x + 2 xx + x 2 + 2 x + 2x + 1 - x 2 + 2x - 1 x = 2x + 2 1 ) x ( ) x ( 2 x + x ) + 2 1 x m = 2(-3) + 2 = -4 y - 4 = -4( x - (-3)) y - 4 = -4 x - 12 y = -4 x - 8 f ( x) = x - 1 ( 5, 2 ) f '( x ) = lim = = = ( x 0 x + x - 1 - x 1 1 ) ( f ( x +Vx) - f ( x) Vx x - 1 x + x - 1 + x - 1 x + x - 1 + x - 1 ) x + x -1 + - x + 1 x 30) ( ( ) x + x - 1 + x - 1 ) = 1 2 x -1 m= 1 1 = 2 5 -1 4 1 y - 2 = ( x - 5) 4 1 5 y = x- +2 4 4 1 3 y = x+ 4 4 f ( x) = ( 0,1) 1 x +1 1 1 - f ( x +Vx) - f ( x) x + x + 1 x + 1 + x + 1 x + 1 x f '( x ) = lim = = x 0 Vx x x + x + 1 x + 1 32) = ( x + 1) + - x - x -1 x ( x + x + 1) ( x + 1) 1 ( )= -1 x + 2x +1 2 m= y - 1 = -1 ( x - 0 ) y = -x +1 -1 = -1 0 + ( 2) ( 0) + 1 2 Section 3.2 y = t 2 + 2t - 3 = 2t + 2 y = 5 + sin x = cos x 12) 22) 3 y = e x + 2 cos x 4 24) 3 = e x + 2sin x 4 26) y = 5 5 5 10 5 = x -2 = ( -2 x -3 ) = - 3 = - 3 2 2x 2 2 2x x f ( t) = 3- 3 ,3 5 3 5t 32) f '(t ) = - 3 t -2 = - 3 5 5t 2 3 3 2 f ' - = = -1 2 5 3 3 5 5 f ( x) = x + 40) 1 = x + x -2 2 x - 2 f ' ( x ) = 1 + ( -2 x -3 ) = 1 + 3 x 42) 2 x 2 - 3x + 1 h( x ) = = ( 2 x 2 - 3 x + 1) x -1 x h ' ( x ) = ( 4 x - 3) ( - x 3 -2 ) 4 x3 - 3 = - x2 2 x 2 - 3x - 1 h( x ) = = 2 x - 3 - x -1 x Result should be 1 h '( x) = 2 - 2 x f (t ) = t - t + 4 48) 2 2 -1 1 -3 2 1 3 f '(t ) = t - t = 1 - 2 3 3 3t 3 3t 3 2 3 1 3 1 h(t ) = sin t + et 2 1 , e 2 56) 1 h '(t ) = cos t + et 2 1 1 @ , e cos + e = 12.569 = 2 2 s (t ) = 16t 2 + vo + so vo = -22 m s (t ) = 16t 2 - 22 t ) + 220 ( s t = 3sec 94) s ' = 32t - 22 = 32(3) - 22 = 96 - 22 = 74 s (t ) - so = 108 0 = 16t 2 - 22t - 108 0 = ( t + 2 ) ( 16t - 54 ) t+2=0 t = -2 m s Section 3.3 f ( x ) = ( x 2 - 2 x + 1) ( x 3 - 1) c =1 14) f '( x ) = ( x 2 - 2 x + 1) ( 3 x 2 ) + ( x 3 - 1) ( 2 x - 2 ) = ( 3x 4 - 6 x3 + 3x 2 ) + ( 2 x 4 - 2 x - 2 x3 + 2 ) = 5 x 4 - 8 x3 + 3x 2 - 2 x + 2 f '(1) = 5 - 8 + 3 - 2 + 2 = 0 f ( x) = c= 18) sin x x 4 f '( x ) = (sin x)( x -1 ) cos x sin x cos x x 1 sin - 2 = - x x x x x = ( cos x ) ( x -1 ) + ( - x -2 ) ( sin x ) = = cos x 1 - ( 1) x x = 0 20) cos x ex f '( x ) = ( cos x ) ( e - x ) = ( - sin x ) ( e - x ) + ( cos x ) ( e - x ) f ( x) = = - sin x cos x cos x - sin x + x = ex e ex f '(0) = 1 24) y = 5 5 5 10 5 = ( x -2 ) = ( -2 x -3 ) = - 3 = - 3 2 4x 4 4 4x 2x x3 + 3x + 2 x2 + 1 (3x 2 + 3)( x 2 + 1) - ( x 3 + 3x + 2)(2 x) f '( x ) = ( x 2 + 1) 2 f ( x) = 4 2 2 4 2 28) = (3 x + 3 x + 3 x + 3) - (2 x + 6 x + 4 x) ( x 2 + 1) 2 = ( x 4 - 4 x + 3) ( x 2 + 1) 2 = x (1/3) ( x1/2 + 3) f ( x ) = 3 x ( x + 3) 1 1 f '( x ) = x -2/3 x1/2 + 3) + ( x1/3 ) x -1/2 ( 3 2 32) = ( x +3 3 x 3 )+ x 2 x 3 52) y ' = (1) - (sin x) 2 + (1)(cos x ) 2 y = -1 h(t ) = et sin t y = x cos x + sin x y ' = (1)(- sin x )(sin x) + (1)(cos x)(cos x) h '(t ) = (et )(cos t ) + (et )(sin t ) 104) h ''(t ) = (et )(- sin t )(et )(sin t ) + (et )(cos t )(et )(cos t ) = (et ) 2 ( -1) ( sin t ) + ( et ) 2 2 ( cos t ) 2 Section 3.4 y = (2 x 3 + 1) 2 y ' = 2 ( 2 x3 + 1) ( 6 x 2 ) 10) = ( 4 x 3 + 2 ) (6 x 2 ) = 2(6 x 5 + 3 x 2 ) = 12 x5 + 6 x 2 = 2((2 x 3 + 1)(3 x 2 )) g ( x) = (5 - 3x ) g '( x) = (5 - 3 x)1/2 1/2 16) = ( 1/ 2 ) (5 - 3 x) = -3 2 (5 - 3x ) 22) 1 t + 3t - 1 -2 = 1( t + 3t -1 - 1) s (t ) = 2 s ' t = (-2t ) -3 (-3t -2 ) 1 1 = 3 = 5 2 t 3t 2 6t y = (1/ 2) x 2 16 - x 2 y = (1/ 2) ( x 2 ) ( 16 - x 2 ) 1/2 y ' = ((1/ 2) x 2 )(1/ 2)(16 - x 2 )(-2 x) + (16 - x 2 )1/2 ( 1/ 2 ) ( 2 ) ( x ) 30) ( 1 x 2 )( -2 x) = 2 + 2(16 - x 2 ) = ( 16 - x 2 ) ( x) - x3 + ( 16 - x 2 )( x) 2(16 - x 2 ) g (t ) = 5cos3 t 60) = (5 cos)3 t g ' = 3(5cos)(5)(3) - (sin 2 )( ) g ' = (15cos)( -15sin 2 )( ) y = e- x 2 66) y ' = (e - x )(-2 x ) 1 x-2 = ( x - 2) -1 f ( x) = f '( x ) = -1( x - 2) -2 2 1 102) f '( x ) = - ( x - 2) 2 f '( x ) = -( x - 2) -2 f ''( x) = -(-2( x - 2) -3 ) f ''( x) = 2 ( x - 2)3 f ( x) = 1 ( x - 3 x) 2 2 = (1)( x 2 - 3 x) -2 110) f '( x ) = -2( x 2 - 3x ) -3 ( 2 x - 3) = 2x - 3 -2( x 2 - 3 x)3 (2)(4) - 3 5 5 5 = = = 3 2 3 3 -2(4 - 3(4)) -2(16 - 12) -128 -2 ( 4 ) f '(4) = 1 1 y = ( cos12t ) - ( sin12t ) 3 4 position 1 12 1 12 y = ( cos ) - ( sin ) 3 8 4 8 1 1 y = (0) - ( )(-1) 3 4 1 y = ft 4 1 1 160) y ' = sin12t - cos12t 3 4 velocity... 1 12 1 12 y ' = ( )(- sin)( ) - ( )(cos) 3 8 4 8 1 = -0 3 = 1 ft 3 sec Section 3.5 6) x 2 y + y 2 x = -3 d 2 d d d ( x )( y ) + ( y )( x 2 ) + ( y 2 ) ( x ) + ( x )( y 2 ) = 0 dx dx dx dx 2 2 2 xy + x y '+ 2 yy ' x + y = 0 x 2 y '+ 2 yy ' x = 2 xy - y 2 y' = ( 2 xy - y 2 ) x 2 + 2 yx cot y = x - y d d cot y = ( x - y ) dx dx d d - sin y = ( x ) - ( y ) dx dx 16) - sin y = 1 - 1( y ') - sin y - 1 = y' -1 y ' = sin y + 1 ln xy + 5 x = 30 1 dy dy ( )( ) + 5( ) = 0 xy dx dx 20) ( dy )( 1 + 5) = 0 dx xy ( dy 1 ) = xy + dx 5 1 d ( xy ) + 5 = 0 xy dx 1 (1 y + xy ') = 5 xy Result should be 1 y ' + =5 x y y + xy ' = 5 xy 5 xy - y y' = x x3 - y 2 = 0 ( 1,1) 3 x 2 - 2 yy ' = 0 -2 yy ' = -3 x 2 26) 3x 2 y'= 2y y'= ( 3(1) ) = 2 2 ( 1) 3 2 y 2 = ln x 2 yy ' = ( e,1) 1 x 1 y'= 2 yx 34) y 1 y'= 2 ( 1) ( e ) y'= 1 5.437 x2 y 2 - 2x = 3 2 xy 2 + x 2 2 yy '- 2 = 0 x 2 2 yy ' = 2 - 2 xy 2 y'= 2 - 2 xy 2 x2 2 y ( 2 )(1) 2 x y2 y'= 2 - x ( 2 )y x2 2 y y'= 52) 1 y - 2 x y x y ' = x -2 y -1 - yx -1 y '' = ( -2 x -3 )( y -1 ) + ( x -2 )( - y -2 y ') - ( y ')( x -1 ) + ( y )( - x -2 ) y '' = ( -2 x -3 )( y -1 ) + ( x -2 )( - y -2 )( x -2 y -1 - yx -1 ) - ( x -2 y -1 - yx -1 )( x -1 ) + ( y )( - x -2 ) 1 y '' = ( 3 ) - ( x -4 y -3 ) - ( x -3 y -1 ) - ( x -3 y -1 ) - ( yx -2 ) + ( y )( - x -2 ) 2x y 1 1 1 1 y y y '' = -1( 3 + 4 3 + 3 + 3 + 2 + 2 ) 2x y x y x y x y x x y '' = -( 1 1 2 2y + 4 3+ 3 + 2) 3 2x y x y x y x Section 3.7 y = 2 ( x 2 - 3x ) = 2 x 2 - 6 x dy dx = ( 4x - 6) dx dt x = 3& dx =2 dt 2) dy = ( 4 ( 3) - 6 ) ( 2 ) = 12 dt x = 1& dy =5 dt dy dy 5 5 = / ( 4x - 6) = = - dt dt ( 4 ( 1) - 6 ) 2 1 = 1-1 + x -2 2 1+ x dy dy dy 1 = ( 0 + -2 x -3 ) 3 = dt dt dt 2x y= x = -1 dy 1 cm 1 cm cm = 2 = 2 = - -1 3 dt 2 ( -1) sec 2 sec sec x=0 dy 1 cm = 2 = undefined dt 2 ( 0 ) 3 sec x =1 dy 1 cm 1 cm cm = 2 = 2 = 1 dt 2 ( 1) 3 sec 2 sec sec 6) 4 dr in v = r3 & = 2 3 dt min dV dr = 4 r 2 dt dt r =6 2 in 4 ( 6 ) 2 = 904.78 min 18a) r = 24 2 in 4 ( 24 ) 2 = 14, 476.46 min V = l *h*w& dl cm =3 dh sec 3 l = h = w V = l dV dl = 3l 2 dt dt l = 1cm dV cm3 2 cm = 3( 1cm ) 2 = 6 dt sec sec l = 10cm dV cm3 2 cm = 3( 10cm ) 2 = 900 dt sec sec 20) 22) 1 dr in V = r 2h & = 2 3 dt min h = 3r 1 V = r 2 3r = r 3 3 dV dr = 3 r 2 dt dt r = 6 in then r = 24in dV in3 2 in = 3 ( 6in ) 2 = 678.58 dt min min dV in3 2 in = 3 ( 24in ) 2 = 10, 857.34 dt min min V= 2 r h 3 h 12 ft h = = 2.4 therefore r = r 5 ft 2.4 r = 5 ft h = 12 ft 2 h V = h 3 2.4 2 dV h h dh = 2 ( 1) 2.4 h + 2.4 dt 3 dt dV 3 h 24) ( 2 h ) - dh dt 2.4 = 2 dt h 2.4 when h = 8 dh = dt 10 ft 3 8 3 - ( 2 8 ) 9.55 - 53.33 ft min 2.4 = = 3.9 2 11.11 min 8 2.4 ds = 4 ft / sec dt dx ds 2 2 x = s dt dt 30) a) dx s 13 ft ds ft = = 4 = 4.333 dt x 12 sec dt sec 122 + x 2 = s 2 as boat gets closer s dx goes dt 122 + x 2 = s 2 dx ft =4 dt sec b) dx ds 2 2 x = s dt dt ds dx x 12 ft ft = 4 = = 3.69 dt dt s 13 sec sec as boat gets closer s ds goes dt ds m = 240 dt hr x = 8.66 s = 10miles x 2 = s 2 - 52 = 100 - 25 = 75 2 2 2 32) 5 + x = s dx ds 2 2 x = s dt dt dx s 10 m ds m = = 240 277.14 = dt x 8.66 hr dt hr dx ft = 28 dt sec 902 + x 2 = s 2 s 2 = 602 + 902 = 3600 + 8100 = 11700 s = 108.17 34) dx ds 2 x = 2s dt dt ds x 60 ft dx ft = = 28 = 15.53 dt s 108.17 sec dt sec x = 25 dx ft =1 dt sec a 2 = x 2 - 102 = 252 - 102 = 525 a = 22.91 10 tan = x 44) d 10 dx sec 2 = - 2 dt x dt d 10 dx 22.91 10 = - 2 cos 2 - 2 = ( 1) = -.013 dt x dt 25 25 change is 0.013 radians sec 2 ...
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This note was uploaded on 06/17/2008 for the course MATH 251 taught by Professor All during the Spring '08 term at Thomas Edison State.

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