Calc 1 WA 4 - WA 4 Calculus I Chapter 4 Section 4.1 f ( x )...

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Unformatted text preview: WA 4 Calculus I Chapter 4 Section 4.1 f ( x ) = x2 + 2x - 4 0 = 2 x + 2 = 2 ( -1) + 2 x = -1 CriticalPoint & minimum ( -1, -5 ) maximum = ( 1, -1) f '( x) = 2x + 2 [ -1,1] 24) g ( x) = 3 x g ( x) = x3 1 [ -1,1] 1 -2 g '( x) = x 3 no critical point 28) 3 1 min = 1, - - 3 1 max = 1, 3 Section 4.4 f ( x ) = 2 x 3 - 3 x 2 - 12 x + 5 f ' ( x ) = 6 x 2 - 6 x - 12 f '' ( x ) = 12 x - 6 concavity changesx = 1 2 1 points of inflection , -1.5 12) 2 0.25 - 0.75 - 6 + 5 = -1.5 1 concave up , 2 1 concave down , - 2 f ( x ) = 2 x4 - 8x + 3 f ' ( x ) = 8x3 - 8 14) concavity changes @ x = 0 point of inflection ( 0,3) concave up ( 0, concave down ( - , 0 ) f '' ( x ) = 24 x 2 ) f ( x ) = x3 ( x - 2 ) = x 4 - 2 x3 f ' ( x ) = 4 x3 - 6 x 2 f '' ( x ) = 12 x 2 - 12 x 16) concavity changes @ x = 1 concave up ( 1, point of inflection ( 1, -1) concave down ( - ,1) ) Section 4.5 2 3+ 3 3 x3 + 2 3 1 x = = = 22) lim 3 2 xx 9x - 2x + 7 9 - 2 + 7 9 3 x x3 1 1 3- -3 x + 1 x = x = 3 = -3 = 30) lim 2 2 xx 1 - 1 x +x x +x -1 - x - x2 -3 + x-3 x=3 x-2 3 intercepts ( 3, 0 ) 0, 2 y= 0= 64) y= - x - 3 -3 - 3 6 = = - no symmetry - x - 2 -3 - 2 5 x-3 x-2 Horizontal Asymtote = 1 Vertical Asymtote = 2 68) x2 x2 0= 2 x2 - 9 x -9 intercept = ( 0, 0 ) y= symmetry y - axis Horizontal Asymtote = 1 Vertical Asymtote = 3 and - 3 74) 2x 1 - x2 intercept ( 0, 0 ) y= 0= 2x 1 - x2 symmetry @ origin Horizontal asymtope = 0 vertical asymtope = 1 and - 1 Section 4.6 16) f ( x) = x3 x2 - 4 0= x3 x=0 x2 - 4 intercept ( 0, 0 ) 3 Relative Extrema ( 0, 0 ) 2 2 2 critical number f ' ( x ) = 0 when x = 0 vertical asymtope = 2 horizontal Asymtope = none f '' ( x ) = 5 x 4 - 12 x 2 ( x ) ( x - 4 ) = ( 3x ) ( x - 4 ) + ( x ) ( 2 x ) = 5x - 12 x f '( x) = ( x - 4) ( x - 4) ( x - 4) 2 2 3 4 2 2 2 2 2 point of inflection = ( 0, 0 ) (x 2 - 4) 2 (x = 2 - 4 ) ( 5 x 4 - 12 x 2 ) (x 2 - 4) 2 = ( 2 x ) ( 5 x 4 - 12 x 2 ) + ( x 2 - 4 ) ( 20 x3 - 24 x ) (x 2 - 4) 3 1 y = x 3 - 3 x + 2 ) - ( 3 1 intercepts : 0 = - ( x 3 - 3x + 2 ) 3 2 0, ( 1, 0 ) & - 3 relative extrema : 1 2 f ' ( x ) = - x3 + x - = - x 2 + 1 3 3 26) critical numbers 1& - 1 1 - , ( 4 ) 3 4 = max 1 - = min 3 no asymtopes f '' ( x ) = -2 x x=0 2 point of inflection - 0, 3 y = 3x 4 - 6 x 2 + intercepts : ( 0, 0 ) ; ( 1, 0 ) ; ( -1, 0 ) 32) max = ( 1, 0 ) critical numbers : 0,1, - 1 min = ( -1, 0 ) points of in flection : ( 0.577, 0 ) ; ( -0.577, 0 ) no asymtopes f '' ( x ) = 36 x 2 - 12 5 3 3 f ' ( x ) = 12 x - 12 x intercepts : ( 0,5) ( 1, 0 ) y = 0 when x = 1 f ( x ) = x 4 - 8 x 3 + 18 x 2 - 16 x + 5 f ' ( x ) = 4 x 3 - 24 x 2 + 36 x - 16 34) f '' ( x ) = 12 x 2 - 48 x + 36 point of inflection (1, 0) no asymtopes Section 4.7 Height 1 2 3 4 2) a) 5 6 7 8 9 10 Length 24 - 2 ( 1) 24 - 2 ( 2 ) 24 - 2 ( 3) 24 - 2 ( 4 ) 24 - 2 ( 5 ) 24 - 2 ( 6 ) 24 - 2 ( 7 ) 24 - 2 ( 8 ) 24 - 2 ( 9 ) 24 - 2 ( 10 ) Volume 1 - 2 ( 1) = 484 24 2 2 - 2 ( 2 ) = 800 24 2 2 3 - 2 ( 3) = 972 24 4 - 2 ( 4 ) = 1024 24 2 5 - 2 ( 5 ) = 980 24 2 6 - 2 ( 6 ) = 864 24 2 7 4 - 2 ( 7 ) = 700 2 2 8 - 2 ( 8 ) = 512 24 2 9 - 2 ( 9 ) = 324 24 2 10 - 2 ( 10 ) = 160 24 2 Guess max volume is 1050in3 b) f ( V ) = x ( 24 - 2 x ) 2 = 4 x ( 144 - 24 x + x 2 ) = 4 ( 144 - 24 x + x 2 ) + 4 x ( -24 + 2 x ) 2 c) = 576 - 192 x + 12 x 0 = 576 - 192 x + 12 x 2 f ' ( v ) = x ( ( 24 - 2 x ) ( 24 - 2 x ) ) = x ( 576 - 96 x + 4 x 2 ) = 576 x - 96 x 2 + 4 x 3 f ' ( v ) = 0 when x = 4,12 maximum value = 1024 x 2 + y = 27 P = xy = x ( 27 - x 2 ) = 27 x - x 3 8) dP = 27 - 3 x 2 dx 0 = 27 - 3 x 2 x=3 y = 18 y = 27 - x 2 f ( x) = x - 8 d= y= d= ( 2, 0 ) 2 ( x - 2) ( x - 8) ( x - 2) 2 + ( y - 0) 2 + 14) f ( x ) = x 2 - 3 x - 4 f '( x) = 2x - 3 3 2 x= (( x -8 -0 ) ) 2 = x 2 - 4 x + 4 + x - 8 = x 2 - 3x - 4 3 3 closest point = , 2 2 3/2 is not in the domain of the unction. You have to look at the graph in this case. The closet point will be (8,0) 4 x + 2 y = 200 y = 100 - 2 x A = xy A = x ( 100 - 2 x ) = 100 x - 2 x 2 20) dA = 100 - 4 x dx 0 = 100 - 4 x x = 25 y = 50 A = ( x + 3) ( y + 3 ) 36 = xy 36 y= x 36 108 A = ( x + 3) + 3 36 + 3 x + = + 9 = 45 + 3 x + 108 x -1 x x dA 108 30) dx = 3 - x 2 108 -3 = - 2 x 2 -3 x = -108 x 2 = 36 x=6 y=6 ...
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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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