hw3.ms (2119B)
1 .TL 2 Calculus assignment 3 3 .AU 4 Lucas Standen 5 .AI 6 Aberystwyth University 7 .2C 8 9 .EQ 10 delim @@ 11 .EN 12 .EQ 13 delim @# 14 .EN 15 16 .LP 17 2) The number of bee's after 15 days 18 19 4) 20 21 @ int f''(x) dx # 22 23 @ int x sup {-3 over 2} dx # 24 25 @ 2 over sqrt x + C # 26 27 @ C = 3 # 28 29 @ int 2x sup {-1 over 2} + 3 dx # 30 31 @ 4 sqrt x + 3x + C = 0 # 32 33 @ C = 0 # 34 35 @ y = 4 sqrt x + 3x # 36 37 5)d) 38 39 @ int from 0 to a x sqrt {a sup 2 - x sup 2} dx# 40 41 let @ u = a sup 2 - x sup 2 # 42 43 @ du over dx = - 2x # 44 45 @ - 1 over {2x} du = dx # 46 47 @ u = a sup 2 - a sup 2 # Adjust the bounds 48 49 @ u = 0 # 50 51 @ int from 0 to 0 sqrt u du # 52 53 @ 0 # Any integral from 0 to 0 is 0 54 55 5)f) 56 57 @ int 1 over {a sup 2 + x sup 2} # 58 59 let @ u = x over a # 60 61 @ du over dx = 1 over a # 62 63 @ 1 over a int 1 over {a sup 2 + 1} du # 64 65 @ 1 over a ln(u sup 2 + 1) 2u # 66 67 @ { ln({x over a + 1}) } over a {2x over { a sup 2 }} # 68 69 6)b) 70 71 @ int 8x ln(x) dx # 72 73 @ u = ln(x) # 74 75 @ u' = 1 over x # 76 77 @ v' = 8x # 78 79 @ v = 4x sup 2 # 80 81 @ 4x sup 2 ln(x) - int 1 over x 4x sup 2 dx # 82 83 @ 4x sup 2 ln(x) - int 1 4x dx # 84 85 @ 4x sup 2 ln(x) - 2x sup 2 + C # 86 87 6)e) 88 89 @ int from 1 to 4 e sup {sqrt x} dx # 90 91 let @ u = sqrt x # 92 93 @ du over dx = 1 over 2 x sup { - 1 over 2 } # 94 95 @ du over dx = 1 over { 2 sqrt x } # 96 97 @ 2 int from {1 over 2} to {1 over 4} u e sup 2 du # 98 99 @ u = u # 100 101 @ u' = 1 # 102 103 @ v' = e sup u # 104 105 @ v = e sup u # 106 107 @ 2 (u e sup u - int from {1 over 2} to {1 over 4} e sup u du) # 108 109 @ 2 e sup {sqrt x} ( sqrt x - 1 ) # 110 111 @ [ from {1 over 2} to {1 over 4} 2 e sup {sqrt x} ( sqrt x - 1 ) ] # 112 113 @ - sqrt e - e sup { { sqrt 2} over 2 } ( - 2 + sqrt 2 ) # 114 115 7)a) 116 117 @ y = tan sup -1 x # 118 119 @ x = tan y # 120 121 @ tan(x) = tan(tan sup -1 x) # 122 123 @ d over dx tan y = d over dx x # 124 125 @ sec sup 2 y dy over dx = 1 # 126 127 @ 1 over { sec sup 2 y} = dy over dx # 128 129 @ 1 over { 1 + tan sup 2 y } = dy over dx # 130 131 @ 1 over {1 + x sup 2} = dy over dx # 132 133 7)b) 134 135 @ int from 0 to 1 tan sup -1 x dx # 136 137 @ u = tan sup -1 x # 138 139 @ u' = 1 over {1 + x sup 2} # 140 141 @ v' = 1 # 142 143 @ v = x # 144 145 @ x tan sup -1 x - int from 0 to 1 x over {1 + x sup 2 } dx # 146 147 @ x tan sup -1 x - [ from 0 to 1 x tan sup -1 x ] # 148 149 @ [ from 0 to 1 x tan sup -1 x - 1 over {4 pi} ] # 150 151 @ 1 over {4 pi} - 1 over {4 pi} # 152 153 @ 0 #