limits

limits - The Quadratic a/2 b c s a s = t2 2 4.9 0 0 350 300...

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Unformatted text preview: The Quadratic a/2 b c s a s = t2 2 4.9 0 0 350 300 250 200 150 s 100 50 0 -50 -100 -150 0 1 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 3 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 8 t SecantLine Tangent 0 -68.6 0.049 -64.19 0.196 -59.78 0.441 -55.37 0.784 -50.96 1.225 -46.55 1.764 -42.14 2.401 -37.73 3.136 -33.32 3.969 -28.91 4.9 -24.5 5.929 -20.09 7.056 -15.68 8.281 -11.27 9.604 -6.86 11.025 -2.45 12.544 1.96 14.161 5 6.37 4 6 15.876 10.78 17.689 15.19 19.6 19.6 21.609 24.01 t 23.716 28.42 25.921 32.83 28.224 37.24 30.625 41.65 33.124 46.06 35.721 50.47 38.416 54.88 41.209 59.29 44.1 63.7 47.089 68.11 50.176 72.52 53.361 76.93 56.644 81.34 60.025 85.75 63.504 90.16 67.081 94.57 70.756 98.98 74.529 103.39 78.4 107.8 82.369 112.21 86.436 116.62 90.601 121.03 94.864 125.44 99.225 129.85 103.684 134.26 108.241 138.67 112.896 143.08 117.649 147.49 122.5 151.9 127.449 156.31 132.496 160.72 137.641 165.13 142.884 169.54 148.225 173.95 153.664 178.36 159.201 182.77 164.836 187.18 170.569 191.59 176.4 196 182.329 200.41 188.356 204.82 194.481 209.23 200.704 213.64 207.025 218.05 213.444 222.46 219.961 226.87 226.576 231.28 233.289 235.69 240.1 240.1 247.009 244.51 254.016 248.92 261.121 253.33 268.324 257.74 275.625 262.15 283.024 266.56 290.521 270.97 298.116 275.38 305.809 279.79 313.6 284.2 -99.225 -94.815 -90.405 -85.995 -81.585 -77.175 -72.765 -68.355 -63.945 -59.535 -55.125 -50.715 -46.305 -41.895 -37.485 -33.075 -28.665 -24.255 8 7 -19.845 -15.435 -11.025 -6.615 -2.205 2.205 6.615 11.025 15.435 19.845 24.255 28.665 33.075 37.485 41.895 46.305 50.715 55.125 59.535 63.945 68.355 72.765 77.175 81.585 85.995 90.405 94.815 99.225 103.635 108.045 112.455 116.865 121.275 125.685 130.095 134.505 138.915 143.325 147.735 152.145 156.555 160.965 165.375 169.785 174.195 178.605 183.015 187.425 191.835 196.245 200.655 205.065 209.475 213.885 218.295 222.705 227.115 231.525 235.935 240.345 244.755 249.165 253.575 What is the slope of the secant between two points on this curve? Specify the x and y points for which you want to determine the slope. Column D Column E 99.225 Column F 19.6 240.1 9 s1 2 s2 7 Secant Slope Secant Intercept 44.1 -68.6 tangent slope = at s t 4.5 99.225 Tangent Slope 44.1 Tangent Intercept -99.225 t1 19.6 t2 240.1 The Quadratic y = ax 2 + bx + c a b c Y -3 -2 X -2 1 -1.9 0.5 -1.8 1 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -1 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 Y SecantLine Tangent 4 3.66 3.34 3.04 2.76 2.5 15 2.26 2.04 1.84 10 1.66 1.5 1.36 1.24 5 1.14 1.06 1 0 0.96 0 1 0.94 0.94 -5 0.96 1 1.06 -10 1.14 X 1.24 1.36 1.5 1.66 1.84 2.04 2.26 2.5 2.76 3.04 3.34 3.66 4 4.36 4.74 5.14 5.56 6 6.46 6.94 7.44 7.96 8.5 9.06 9.64 10.24 10.86 11.5 -6.125 -7.390625 -5.8 -7.065625 -5.475 -6.740625 -5.15 -6.415625 -4.825 -6.090625 -4.5 -5.765625 -4.175 -5.440625 -3.85 -5.115625 -3.525 -4.790625 -3.2 -4.465625 -2.875 -4.140625 -2.55 -3.815625 -2.225 -3.490625 -1.9 -3.165625 -1.575 -2.840625 -1.25 -2.515625 -0.925 2 -2.190625 3 -0.6 -1.865625 -0.275 -1.540625 0.05 -1.215625 0.375 -0.890625 0.7 -0.565625 1.025 -0.240625 1.35 0.084375 1.675 0.409375 2 0.734375 2.325 1.059375 2.65 1.384375 2.975 1.709375 3.3 2.034375 3.625 2.359375 3.95 2.684375 4.275 3.009375 4.6 3.334375 4.925 3.659375 5.25 3.984375 5.575 4.309375 5.9 4.634375 6.225 4.959375 6.55 5.284375 6.875 5.609375 7.2 5.934375 7.525 6.259375 7.85 6.584375 8.175 6.909375 8.5 7.234375 8.825 7.559375 9.15 7.884375 9.475 8.209375 9.8 8.534375 10.125 8.859375 What is the slope of the secant between two points on this curve? Specify the x and y points for which you want to determine the slope. X1 0.25 Column D Column E 3.578125 Column F 4 X2 2.5 Secant Slope Secant Intercept 3.25 0.375 Y1 1.1875 tangent slope = 2ax+b x y 1.375 3.578125 Tangent Slope 3.25 Tangent Intercept -0.890625 Y2 8.5 The Polynomial X y = ax 3 + bx 2 + cx + d a b c d -5 -2.5 30 80 180 160 140 120 y 100 80 60 40 20 0 -3 -2 -1 -2 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0 0.2 x 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 Y y SecantLine Tangent 50 32.5 24.6875 48.27 34.5 27.46875 47.06 36.5 30.25 46.34 38.5 33.03125 46.08 40.5 35.8125 46.25 42.5 38.59375 41.375 =46.82 + bx 244.5cx + d ax 3 + 47.76 46.5 44.15625 49.04 48.5 46.9375 50.63 50.5 49.71875 52.5 52.5 52.5 54.62 54.5 55.28125 Y 56.96 56.5 58.0625Column F 59.49 58.5 60.84375 Column G 62.18 60.5 63.625 87.265625 65 62.5 66.40625 52.5 67.92 64.5 69.1875102.5 70.91 66.5 71.96875 73.94 68.5 74.75 76.98 70.5 77.53125 80 72.5 80.3125 82.97 74.5 83.09375 1 2 3 4 85.86 76.5 85.875 88.64 78.5 88.65625 91.28 80.5 91.4375 93.75 82.5 94.21875 96.02 84.5 97 98.06 86.5 99.78125 99.84 88.5 102.5625 101.33 90.5 105.34375 102.5 92.5 108.125 103.32 94.5 110.90625 103.76 96.5 113.6875 103.79 98.5 116.46875 103.38 100.5 119.25 102.5 102.5 122.03125 101.12 104.5 124.8125 99.21 106.5 127.59375 96.74 108.5 130.375 93.68 110.5 133.15625 90 112.5 135.9375 85.67 114.5 138.71875 80.66 116.5 141.5 74.94 118.5 144.28125 68.48 120.5 147.0625 61.25 122.5 149.84375 53.22 124.5 152.625 44.36 126.5 155.40625 34.64 128.5 158.1875 24.03 130.5 160.96875 12.5 132.5 163.75 What is the slope of the secant between two points on this curve? Specify the x and y points for which you want to determine the slope. X1 -1 X2 1.5 Secant Slope Secant Intercept 20 72.5 Y1 52.5 tangent slope = 3ax 2 +2bx+c x y 0.25 87.265625 Tangent Slope 27.8125 Tangent Intercept 80.3125 20 Y2 102.5 a mp l i t u d e θ (degrees) θ (radians) cos( θ ) -sin( θ ) Secant Line Points used to define a secant line θ1 θ2 cos( θ 1 ) cos( θ 2 ) -90 -1.570796 6.1E-017 1 2.219718 -89 -1.553343 0.017452 0.999848 2.208513 35 45 0.819152 0.707107 -88 -1.53589 0.034899 0.999391 2.197309 slope = -0.64197 -0.0112 -87 -1.518436 0.052336 0.99863 2.186104 Intercept = 29.59585 1.21131 -86 -1.500983 0.069756 0.997564 2.1749 40 -0.64279 -85 -1.48353 0.087156 0.996195 2.163695 -84 -1.466077 he cosine and it s derivat ive T 0.104528 0.994522 2.152491 -831.5 -1.448623 0.121869 0.992546 2.141286 -82 -1.43117 0.139173 0.990268 2.130082 -81 1 -1.413717 0.156434 0.987688 2.118877 -80 -1.396263 0.173648 0.984808 2.107673 co s( q ) -790 .5 -1.37881 0.190809 0.981627 2.096468 C o lumn D -78 -1.361357 0.207912 0.978148 2.085264 0 .8 1915 20 44 3 -77 -1.343904 0.224951 0.97437 2.074059 0 0 .70 710 6 78 12 -76 -1.32645 0.241922 0.970296 2.062854 C o lumn E -75 -1.308997 0.258819 0.965926 2.05165 -0 .5 -0 .6 4 278 76 09 7 -74 -1.291544 0.275637 0.961262 2.040445 -73 -1.27409 0.292372 0.956305 2.029241 -1 -72 -1.256637 0.309017 0.951057 2.018036 -71 -1.239184 0.325568 0.945519 2.006832 -1.5 -70 -1.22173 0.34202 0.939693 1.995627 0 90 180 270 -69 -9 0 -1.204277 0.358368 0.93358 1.984423 36 0 the t a -68 -1.186824 0.374607 0.927184 1.973218 -67 -1.169371 0.390731 0.920505 1.962014 -66 -1.151917 0.406737 0.913545 1.950809 -65 -1.134464 0.422618 0.906308 1.939605 -64 -1.117011 0.438371 0.898794 1.9284 -63 -1.099557 0.45399 0.891007 1.917196 -62 -1.082104 0.469472 0.882948 1.905991 -61 -1.064651 0.48481 0.87462 1.894787 -60 -1.047198 0.5 0.866025 1.883582 -59 -1.029744 0.515038 0.857167 1.872378 -58 -1.012291 0.529919 0.848048 1.861173 -57 -0.994838 0.544639 0.838671 1.849968 -56 -0.977384 0.559193 0.829038 1.838764 -55 -0.959931 0.573576 0.819152 1.827559 -54 -0.942478 0.587785 0.809017 1.816355 -53 -0.925025 0.601815 0.798636 1.80515 -52 -0.907571 0.615661 0.788011 1.793946 -51 -0.890118 0.62932 0.777146 1.782741 -50 -0.872665 0.642788 0.766044 1.771537 -49 -0.855211 0.656059 0.75471 1.760332 -48 -0.837758 0.669131 0.743145 1.749128 -47 -0.820305 0.681998 0.731354 1.737923 -46 -0.802851 0.694658 0.71934 1.726719 -45 -0.785398 0.707107 0.707107 1.715514 -44 -0.767945 0.71934 0.694658 1.70431 -43 -0.750492 0.731354 0.681998 1.693105 -42 -0.733038 0.743145 0.669131 1.681901 -41 -0.715585 0.75471 0.656059 1.670696 -40 -0.698132 0.766044 0.642788 1.659492 -39 -0.680678 0.777146 0.62932 1.648287 -38 -0.663225 0.788011 0.615661 1.637082 -37 -0.645772 0.798636 0.601815 1.625878 -36 -0.628319 0.809017 0.587785 1.614673 -35 -0.610865 0.819152 0.573576 1.603469 -34 -0.593412 0.829038 0.559193 1.592264 -33 -0.575959 0.838671 0.544639 1.58106 -32 -0.558505 0.848048 0.529919 1.569855 -31 -0.541052 0.857167 0.515038 1.558651 -30 -0.523599 0.866025 0.5 1.547446 -29 -0.506145 0.87462 0.48481 1.536242 -28 -0.488692 0.882948 0.469472 1.525037 -27 -0.471239 0.891007 0.45399 1.513833 -26 -0.453786 0.898794 0.438371 1.502628 -25 -0.436332 0.906308 0.422618 1.491424 -24 -0.418879 0.913545 0.406737 1.480219 -23 -0.401426 0.920505 0.390731 1.469015 -22 -0.383972 0.927184 0.374607 1.45781 -21 -0.366519 0.93358 0.358368 1.446606 -20 -0.349066 0.939693 0.34202 1.435401 -19 -0.331613 0.945519 0.325568 1.424196 -18 -0.314159 0.951057 0.309017 1.412992 -17 -0.296706 0.956305 0.292372 1.401787 -16 -0.279253 0.961262 0.275637 1.390583 -15 -0.261799 0.965926 0.258819 1.379378 -14 -0.244346 0.970296 0.241922 1.368174 -13 -0.226893 0.97437 0.224951 1.356969 -12 -0.20944 0.978148 0.207912 1.345765 -11 -0.191986 0.981627 0.190809 1.33456 -10 -0.174533 0.984808 0.173648 1.323356 -9 -0.15708 0.987688 0.156434 1.312151 -8 -0.139626 0.990268 0.139173 1.300947 -7 -0.122173 0.992546 0.121869 1.289742 -6 -0.10472 0.994522 0.104528 1.278538 -5 -0.087266 0.996195 0.087156 1.267333 -4 -0.069813 0.997564 0.069756 1.256129 -3 -0.05236 0.99863 0.052336 1.244924 -2 -0.034907 0.999391 0.034899 1.23372 -1 -0.017453 0.999848 0.017452 1.222515 0 0 1 -0 1.21131 1 0.0174533 0.999848 -0.01745 1.200106 2 0.0349066 0.999391 -0.0349 1.188901 3 0.0523599 0.99863 -0.05234 1.177697 4 0.0698132 0.997564 -0.06976 1.166492 5 0.0872665 0.996195 -0.08716 1.155288 6 0.1047198 0.994522 -0.10453 1.144083 7 0.122173 0.992546 -0.12187 1.132879 8 0.1396263 0.990268 -0.13917 1.121674 9 0.1570796 0.987688 -0.15643 1.11047 10 0.1745329 0.984808 -0.17365 1.099265 11 0.1919862 0.981627 -0.19081 1.088061 12 0.2094395 0.978148 -0.20791 1.076856 13 0.2268928 0.97437 -0.22495 1.065652 14 0.2443461 0.970296 -0.24192 1.054447 15 0.2617994 0.965926 -0.25882 1.043243 16 0.2792527 0.961262 -0.27564 1.032038 17 0.296706 0.956305 -0.29237 1.020834 18 0.3141593 0.951057 -0.30902 1.009629 19 0.3316126 0.945519 -0.32557 0.998424 20 0.3490659 0.939693 -0.34202 0.98722 21 0.3665191 0.93358 -0.35837 0.976015 22 0.3839724 0.927184 -0.37461 0.964811 23 0.4014257 0.920505 -0.39073 0.953606 24 0.418879 0.913545 -0.40674 0.942402 25 0.4363323 0.906308 -0.42262 0.931197 26 0.4537856 0.898794 -0.43837 0.919993 27 0.4712389 0.891007 -0.45399 0.908788 28 0.4886922 0.882948 -0.46947 0.897584 29 0.5061455 0.87462 -0.48481 0.886379 30 0.5235988 0.866025 -0.5 0.875175 31 0.5410521 0.857167 -0.51504 0.86397 32 0.5585054 0.848048 -0.52992 0.852766 33 0.5759587 0.838671 -0.54464 0.841561 34 0.5934119 0.829038 -0.55919 0.830357 35 0.6108652 0.819152 -0.57358 0.819152 36 0.6283185 0.809017 -0.58779 0.807948 37 0.6457718 0.798636 -0.60182 0.796743 38 0.6632251 0.788011 -0.61566 0.785538 39 0.6806784 0.777146 -0.62932 0.774334 40 0.6981317 0.766044 -0.64279 0.763129 41 0.715585 0.75471 -0.65606 0.751925 42 0.7330383 0.743145 -0.66913 0.74072 43 0.7504916 0.731354 -0.682 0.729516 44 0.7679449 0.71934 -0.69466 0.718311 45 0.7853982 0.707107 -0.70711 0.707107 46 0.8028515 0.694658 -0.71934 0.695902 47 0.8203047 0.681998 -0.73135 0.684698 48 0.837758 0.669131 -0.74314 0.673493 49 0.8552113 0.656059 -0.75471 0.662289 50 0.8726646 0.642788 -0.76604 0.651084 51 0.8901179 0.62932 -0.77715 0.63988 52 0.9075712 0.615661 -0.78801 0.628675 53 0.9250245 0.601815 -0.79864 0.617471 54 0.9424778 0.587785 -0.80902 0.606266 55 0.9599311 0.573576 -0.81915 0.595062 56 0.9773844 0.559193 -0.82904 0.583857 57 0.9948377 0.544639 -0.83867 0.572652 58 1.012291 0.529919 -0.84805 0.561448 59 1.0297443 0.515038 -0.85717 0.550243 60 1.0471976 0.5 -0.86603 0.539039 61 1.0646508 0.48481 -0.87462 0.527834 62 1.0821041 0.469472 -0.88295 0.51663 63 1.0995574 0.45399 -0.89101 0.505425 64 1.1170107 0.438371 -0.89879 0.494221 65 1.134464 0.422618 -0.90631 0.483016 66 1.1519173 0.406737 -0.91355 0.471812 67 1.1693706 0.390731 -0.9205 0.460607 68 1.1868239 0.374607 -0.92718 0.449403 69 1.2042772 0.358368 -0.93358 0.438198 70 1.2217305 0.34202 -0.93969 0.426994 71 1.2391838 0.325568 -0.94552 0.415789 72 1.2566371 0.309017 -0.95106 0.404585 73 1.2740904 0.292372 -0.9563 0.39338 74 1.2915436 0.275637 -0.96126 0.382176 75 1.3089969 0.258819 -0.96593 0.370971 76 1.3264502 0.241922 -0.9703 0.359766 77 1.3439035 0.224951 -0.97437 0.348562 78 1.3613568 0.207912 -0.97815 0.337357 79 1.3788101 0.190809 -0.98163 0.326153 80 1.3962634 0.173648 -0.98481 0.314948 81 1.4137167 0.156434 -0.98769 0.303744 82 1.43117 0.139173 -0.99027 0.292539 83 1.4486233 0.121869 -0.99255 0.281335 84 1.4660766 0.104528 -0.99452 0.27013 85 1.4835299 0.087156 -0.99619 0.258926 86 1.5009832 0.069756 -0.99756 0.247721 87 1.5184364 0.052336 -0.99863 0.236517 88 1.5358897 0.034899 -0.99939 0.225312 89 1.553343 0.017452 -0.99985 0.214108 90 1.5707963 6.1E-017 -1 0.202903 91 1.5882496 -0.01745 -0.99985 0.191699 92 1.6057029 -0.0349 -0.99939 0.180494 93 1.6231562 -0.05234 -0.99863 0.16929 94 1.6406095 -0.06976 -0.99756 0.158085 95 1.6580628 -0.08716 -0.99619 0.14688 96 1.6755161 -0.10453 -0.99452 0.135676 97 1.6929694 -0.12187 -0.99255 0.124471 98 1.7104227 -0.13917 -0.99027 0.113267 99 1.727876 -0.15643 -0.98769 0.102062 100 1.7453293 -0.17365 -0.98481 0.090858 101 1.7627825 -0.19081 -0.98163 0.079653 102 1.7802358 -0.20791 -0.97815 0.068449 103 1.7976891 -0.22495 -0.97437 0.057244 104 1.8151424 -0.24192 -0.9703 0.04604 105 1.8325957 -0.25882 -0.96593 0.034835 106 1.850049 -0.27564 -0.96126 0.023631 107 1.8675023 -0.29237 -0.9563 0.012426 108 1.8849556 -0.30902 -0.95106 0.001222 109 1.9024089 -0.32557 -0.94552 -0.00998 110 1.9198622 -0.34202 -0.93969 -0.02119 111 1.9373155 -0.35837 -0.93358 -0.03239 112 1.9547688 -0.37461 -0.92718 -0.0436 113 1.9722221 -0.39073 -0.9205 -0.0548 114 1.9896753 -0.40674 -0.91355 -0.06601 115 2.0071286 -0.42262 -0.90631 -0.07721 116 2.0245819 -0.43837 -0.89879 -0.08841 117 2.0420352 -0.45399 -0.89101 -0.09962 118 2.0594885 -0.46947 -0.88295 -0.11082 119 2.0769418 -0.48481 -0.87462 -0.12203 120 2.0943951 -0.5 -0.86603 -0.13323 121 2.1118484 -0.51504 -0.85717 -0.14444 122 2.1293017 -0.52992 -0.84805 -0.15564 123 2.14675498 -0.54464 -0.83867 -0.16685 124 2.1642083 -0.55919 -0.82904 -0.17805 125 2.1816616 -0.57358 -0.81915 -0.18926 126 2.1991149 -0.58779 -0.80902 -0.20046 127 2.21656815 -0.60182 -0.79864 -0.21166 128 2.2340214 -0.61566 -0.78801 -0.22287 129 2.2514747 -0.62932 -0.77715 -0.23407 130 2.268928 -0.64279 -0.76604 -0.24528 131 2.2863813 -0.65606 -0.75471 -0.25648 132 2.3038346 -0.66913 -0.74314 -0.26769 133 2.3212879 -0.682 -0.73135 -0.27889 134 2.3387412 -0.69466 -0.71934 -0.2901 135 2.3561945 -0.70711 -0.70711 -0.3013 136 2.3736478 -0.71934 -0.69466 -0.31251 137 2.3911011 -0.73135 -0.682 -0.32371 138 2.4085544 -0.74314 -0.66913 -0.33491 139 2.4260077 -0.75471 -0.65606 -0.34612 140 2.443461 -0.76604 -0.64279 -0.35732 141 2.4609142 -0.77715 -0.62932 -0.36853 142 2.4783675 -0.78801 -0.61566 -0.37973 143 2.4958208 -0.79864 -0.60182 -0.39094 144 2.5132741 -0.80902 -0.58779 -0.40214 145 2.5307274 -0.81915 -0.57358 -0.41335 146 2.5481807 -0.82904 -0.55919 -0.42455 147 2.565634 -0.83867 -0.54464 -0.43575 148 2.5830873 -0.84805 -0.52992 -0.44696 149 2.6005406 -0.85717 -0.51504 -0.45816 150 2.6179939 -0.86603 -0.5 -0.46937 151 2.6354472 -0.87462 -0.48481 -0.48057 152 2.6529005 -0.88295 -0.46947 -0.49178 153 2.6703538 -0.89101 -0.45399 -0.50298 154 2.687807 -0.89879 -0.43837 -0.51419 155 2.7052603 -0.90631 -0.42262 -0.52539 156 2.7227136 -0.91355 -0.40674 -0.5366 157 2.7401669 -0.9205 -0.39073 -0.5478 158 2.7576202 -0.92718 -0.37461 -0.559 159 2.7750735 -0.93358 -0.35837 -0.57021 160 2.7925268 -0.93969 -0.34202 -0.58141 161 2.8099801 -0.94552 -0.32557 -0.59262 162 2.8274334 -0.95106 -0.30902 -0.60382 163 2.8448867 -0.9563 -0.29237 -0.61503 164 2.86234 -0.96126 -0.27564 -0.62623 165 2.8797933 -0.96593 -0.25882 -0.63744 166 2.8972466 -0.9703 -0.24192 -0.64864 167 2.9146999 -0.97437 -0.22495 -0.65985 168 2.9321531 -0.97815 -0.20791 -0.67105 169 2.9496064 -0.98163 -0.19081 -0.68225 170 2.9670597 -0.98481 -0.17365 -0.69346 171 2.984513 -0.98769 -0.15643 -0.70466 172 3.0019663 -0.99027 -0.13917 -0.71587 173 3.0194196 -0.99255 -0.12187 -0.72707 174 3.0368729 -0.99452 -0.10453 -0.73828 175 3.0543262 -0.99619 -0.08716 -0.74948 176 3.0717795 -0.99756 -0.06976 -0.76069 177 3.0892328 -0.99863 -0.05234 -0.77189 178 3.1066861 -0.99939 -0.0349 -0.7831 179 3.1241394 -0.99985 -0.01745 -0.7943 180 3.1415927 -1 -1E-016 -0.8055 181 3.1590459 -0.99985 0.017452 -0.81671 182 3.1764992 -0.99939 0.034899 -0.82791 183 3.1939525 -0.99863 0.052336 -0.83912 184 3.2114058 -0.99756 0.069756 -0.85032 185 3.2288591 -0.99619 0.087156 -0.86153 186 3.2463124 -0.99452 0.104528 -0.87273 187 3.2637657 -0.99255 0.121869 -0.88394 188 3.281219 -0.99027 0.139173 -0.89514 189 3.2986723 -0.98769 0.156434 -0.90635 190 3.3161256 -0.98481 0.173648 -0.91755 191 3.3335789 -0.98163 0.190809 -0.92875 192 3.3510322 -0.97815 0.207912 -0.93996 193 3.3684855 -0.97437 0.224951 -0.95116 194 3.3859387 -0.9703 0.241922 -0.96237 195 3.403392 -0.96593 0.258819 -0.97357 196 3.4208453 -0.96126 0.275637 -0.98478 197 3.4382986 -0.9563 0.292372 -0.99598 198 3.4557519 -0.95106 0.309017 -1.00719 199 3.4732052 -0.94552 0.325568 -1.01839 200 3.4906585 -0.93969 0.34202 -1.02959 201 3.5081118 -0.93358 0.358368 -1.0408 202 3.5255651 -0.92718 0.374607 -1.052 203 3.5430184 -0.9205 0.390731 -1.06321 204 3.5604717 -0.91355 0.406737 -1.07441 205 3.577925 -0.90631 0.422618 -1.08562 206 3.5953783 -0.89879 0.438371 -1.09682 207 3.6128316 -0.89101 0.45399 -1.10803 208 3.6302848 -0.88295 0.469472 -1.11923 209 3.6477381 -0.87462 0.48481 -1.13044 210 3.6651914 -0.86603 0.5 -1.14164 211 3.6826447 -0.85717 0.515038 -1.15284 212 3.700098 -0.84805 0.529919 -1.16405 213 3.7175513 -0.83867 0.544639 -1.17525 214 3.7350046 -0.82904 0.559193 -1.18646 215 3.7524579 -0.81915 0.573576 -1.19766 216 3.7699112 -0.80902 0.587785 -1.20887 217 3.7873645 -0.79864 0.601815 -1.22007 218 3.8048178 -0.78801 0.615661 -1.23128 219 3.8222711 -0.77715 0.62932 -1.24248 220 3.8397244 -0.76604 0.642788 -1.25369 221 3.8571776 -0.75471 0.656059 -1.26489 222 3.8746309 -0.74314 0.669131 -1.27609 223 3.8920842 -0.73135 0.681998 -1.2873 224 3.9095375 -0.71934 0.694658 -1.2985 225 3.9269908 -0.70711 0.707107 -1.30971 226 3.9444441 -0.69466 0.71934 -1.32091 227 3.9618974 -0.682 0.731354 -1.33212 228 3.9793507 -0.66913 0.743145 -1.34332 229 3.996804 -0.65606 0.75471 -1.35453 230 4.0142573 -0.64279 0.766044 -1.36573 231 4.0317106 -0.62932 0.777146 -1.37694 232 4.0491639 -0.61566 0.788011 -1.38814 233 4.0666172 -0.60182 0.798636 -1.39934 234 4.0840704 -0.58779 0.809017 -1.41055 235 4.1015237 -0.57358 0.819152 -1.42175 236 4.118977 -0.55919 0.829038 -1.43296 237 4.1364303 -0.54464 0.838671 -1.44416 238 4.1538836 -0.52992 0.848048 -1.45537 239 4.1713369 -0.51504 0.857167 -1.46657 240 4.1887902 -0.5 0.866025 -1.47778 241 4.2062435 -0.48481 0.87462 -1.48898 242 4.2236968 -0.46947 0.882948 -1.50018 243 4.2411501 -0.45399 0.891007 -1.51139 244 4.2586034 -0.43837 0.898794 -1.52259 245 4.2760567 -0.42262 0.906308 -1.5338 246 4.29351 -0.40674 0.913545 -1.545 247 4.3109633 -0.39073 0.920505 -1.55621 248 4.3284165 -0.37461 0.927184 -1.56741 249 4.3458698 -0.35837 0.93358 -1.57862 250 4.36332313 -0.34202 0.939693 -1.58982 251 4.3807764 -0.32557 0.945519 -1.60103 252 4.3982297 -0.30902 0.951057 -1.61223 253 4.415683 -0.29237 0.956305 -1.62343 254 4.4331363 -0.27564 0.961262 -1.63464 255 4.4505896 -0.25882 0.965926 -1.64584 256 4.4680429 -0.24192 0.970296 -1.65705 257 4.4854962 -0.22495 0.97437 -1.66825 258 4.5029495 -0.20791 0.978148 -1.67946 259 4.5204028 -0.19081 0.981627 -1.69066 260 4.5378561 -0.17365 0.984808 -1.70187 261 4.5553093 -0.15643 0.987688 -1.71307 262 4.5727626 -0.13917 0.990268 -1.72428 263 4.5902159 -0.12187 0.992546 -1.73548 264 4.6076692 -0.10453 0.994522 -1.74668 265 4.6251225 -0.08716 0.996195 -1.75789 266 4.6425758 -0.06976 0.997564 -1.76909 267 4.6600291 -0.05234 0.99863 -1.7803 268 4.6774824 -0.0349 0.999391 -1.7915 269 4.6949357 -0.01745 0.999848 -1.80271 270 4.712389 -2E-016 1 -1.81391 271 4.7298423 0.017452 0.999848 -1.82512 272 4.7472956 0.034899 0.999391 -1.83632 273 4.7647489 0.052336 0.99863 -1.84753 274 4.7822022 0.069756 0.997564 -1.85873 275 4.7996554 0.087156 0.996195 -1.86993 276 4.8171087 0.104528 0.994522 -1.88114 277 4.834562 0.121869 0.992546 -1.89234 278 4.8520153 0.139173 0.990268 -1.90355 279 4.8694686 0.156434 0.987688 -1.91475 280 4.8869219 0.173648 0.984808 -1.92596 281 4.9043752 0.190809 0.981627 -1.93716 282 4.9218285 0.207912 0.978148 -1.94837 283 4.9392818 0.224951 0.97437 -1.95957 284 4.9567351 0.241922 0.970296 -1.97078 285 4.9741884 0.258819 0.965926 -1.98198 286 4.9916417 0.275637 0.961262 -1.99318 287 5.009095 0.292372 0.956305 -2.00439 288 5.0265482 0.309017 0.951057 -2.01559 289 5.0440015 0.325568 0.945519 -2.0268 290 5.0614548 0.34202 0.939693 -2.038 291 5.0789081 0.358368 0.93358 -2.04921 292 5.0963614 0.374607 0.927184 -2.06041 293 5.1138147 0.390731 0.920505 -2.07162 294 5.131268 0.406737 0.913545 -2.08282 295 5.1487213 0.422618 0.906308 -2.09402 296 5.1661746 0.438371 0.898794 -2.10523 297 5.1836279 0.45399 0.891007 -2.11643 298 5.2010812 0.469472 0.882948 -2.12764 299 5.2185345 0.48481 0.87462 -2.13884 300 5.2359878 0.5 0.866025 -2.15005 301 5.253441 0.515038 0.857167 -2.16125 302 5.2708943 0.529919 0.848048 -2.17246 303 5.2883476 0.544639 0.838671 -2.18366 304 5.3058009 0.559193 0.829038 -2.19487 305 5.3232542 0.573576 0.819152 -2.20607 306 5.3407075 0.587785 0.809017 -2.21727 307 5.3581608 0.601815 0.798636 -2.22848 308 5.3756141 0.615661 0.788011 -2.23968 309 5.3930674 0.62932 0.777146 -2.25089 310 5.4105207 0.642788 0.766044 -2.26209 311 5.427974 0.656059 0.75471 -2.2733 312 5.4454273 0.669131 0.743145 -2.2845 313 5.4628806 0.681998 0.731354 -2.29571 314 5.4803339 0.694658 0.71934 -2.30691 315 5.4977871 0.707107 0.707107 -2.31812 316 5.5152404 0.71934 0.694658 -2.32932 317 5.5326937 0.731354 0.681998 -2.34052 318 5.550147 0.743145 0.669131 -2.35173 319 5.5676003 0.75471 0.656059 -2.36293 320 5.5850536 0.766044 0.642788 -2.37414 321 5.6025069 0.777146 0.62932 -2.38534 322 5.6199602 0.788011 0.615661 -2.39655 323 5.6374135 0.798636 0.601815 -2.40775 324 5.6548668 0.809017 0.587785 -2.41896 325 5.6723201 0.819152 0.573576 -2.43016 326 5.6897734 0.829038 0.559193 -2.44137 327 5.7072267 0.838671 0.544639 -2.45257 328 5.7246799 0.848048 0.529919 -2.46377 329 5.7421332 0.857167 0.515038 -2.47498 330 5.7595865 0.866025 0.5 -2.48618 331 5.7770398 0.87462 0.48481 -2.49739 332 5.7944931 0.882948 0.469472 -2.50859 333 5.8119464 0.891007 0.45399 -2.5198 334 5.8293997 0.898794 0.438371 -2.531 335 5.846853 0.906308 0.422618 -2.54221 336 5.8643063 0.913545 0.406737 -2.55341 337 5.8817596 0.920505 0.390731 -2.56461 338 5.8992129 0.927184 0.374607 -2.57582 339 5.9166662 0.93358 0.358368 -2.58702 340 5.9341195 0.939693 0.34202 -2.59823 341 5.9515727 0.945519 0.325568 -2.60943 342 5.969026 0.951057 0.309017 -2.62064 343 5.9864793 0.956305 0.292372 -2.63184 344 6.0039326 0.961262 0.275637 -2.64305 345 6.0213859 0.965926 0.258819 -2.65425 346 6.0388392 0.970296 0.241922 -2.66546 347 6.0562925 0.97437 0.224951 -2.67666 348 6.0737458 0.978148 0.207912 -2.68786 349 6.0911991 0.981627 0.190809 -2.69907 350 6.1086524 0.984808 0.173648 -2.71027 351 6.1261057 0.987688 0.156434 -2.72148 352 6.143559 0.990268 0.139173 -2.73268 353 6.1610123 0.992546 0.121869 -2.74389 354 6.1784656 0.994522 0.104528 -2.75509 355 6.1959188 0.996195 0.087156 -2.7663 356 6.2133721 0.997564 0.069756 -2.7775 357 6.2308254 0.99863 0.052336 -2.78871 358 6.2482787 0.999391 0.034899 -2.79991 359 6.265732 0.999848 0.017452 -2.81111 360 6.2831853 1 2.4E-016 -2.82232 z A0 c 0.50 -0.50 A secant line tangent 0 0.5 0.242200354 0.1436487 0.1 0.4756147 0.239287323 0.1415966 0.2 0.4524187 0.236374291 0.1395445 0.3 0.430354 0.233461259 0.1374924 0.4 0.4093654 0.230548228 0.1354402 0.5 0.3894004 0.227635196 0.1333881 0.6 0.3704091 0.224722164 0.131336 0.7 0.352344 0.221809132 0.1292839 0.8 0.33516 0.218896101 0.1272317 0.9 0.3188141 0.215983069 0.1251796 1 0.3032653 0.213070037 0.1231275 1.1 0.2884749 0.210157006 0.1210754 1.2 0.2744058 0.207243974 0.1190232 1.3 0.2610229 0.204330942 0.1169711 1.4 0.2482927 0.201417911 0.114919 1.5 0.2361833 0.198504879 0.1128669 1.6 0.2246645 0.195591847 0.1108147 1.7 0.2137075 0.192678816 0.1087626 4 6 8 10 12 1.8 0.2032848 0.189765784 0.1067105 1.9 0.1933705 0.186852752 0.1046584 Z 2 0.1839397 0.183939721 0.1026062 2.1 0.1749689 0.181026689 0.1005541 2.2 0.1664355 0.178113657 0.098502 2.3 0.1583184 0.175200626 0.0964499 2.4 0.1505971 0.172287594 0.0943977 2.5 0.1432524 0.169374562 0.0923456 2.6 0.1362659 0.16646153 0.0902935 2.7 0.1296201 0.163548499 0.0882414 2.8 0.1232985 0.160635467 0.0861892 2.9 0.1172851 0.157722435 0.0841371 3 0.1115651 0.154809404 0.082085 3.1 0.106124 0.151896372 0.0800329 3.2 0.1009483 0.14898334 0.0779807 3.3 0.096025 0.146070309 0.0759286 3.4 0.0913418 0.143157277 0.0738765 3.5 0.086887 0.140244245 0.0718244 3.6 0.0826494 0.137331214 0.0697722 3.7 0.0786186 0.134418182 0.0677201 3.8 0.0747843 0.13150515 0.065668 3.9 0.071137 0.128592119 0.0636159 4 0.0676676 0.125679087 0.0615637 4.1 0.0643675 0.122766055 0.0595116 4.2 0.0612282 0.119853024 0.0574595 4.3 0.0582421 0.116939992 0.0554074 4.4 0.0554016 0.11402696 0.0533552 4.5 0.0526996 0.111113928 0.0513031 4.6 0.0501294 0.108200897 0.049251 4.7 0.0476846 0.105287865 0.0471989 4.8 0.045359 0.102374833 0.0451467 4.9 0.0431468 0.099461802 0.0430946 5 0.0410425 0.09654877 0.0410425 5.1 0.0390408 0.093635738 0.0389904 5.2 0.0371368 0.090722707 0.0369382 5.3 0.0353256 0.087809675 0.0348861 5.4 0.0336028 0.084896643 0.032834 5.5 0.0319639 0.081983612 0.0307819 5.6 0.030405 0.07907058 0.0287297 5.7 0.0289222 0.076157548 0.0266776 5.8 0.0275116 0.073244517 0.0246255 5.9 0.0261699 0.070331485 0.0225734 6 0.0248935 0.067418453 0.0205212 6.1 0.0236795 0.064505421 0.0184691 6.2 0.0225246 0.06159239 0.016417 6.3 0.0214261 0.058679358 0.0143649 6.4 0.0203811 0.055766326 0.0123127 6.5 0.0193871 0.052853295 0.0102606 6.6 0.0184416 0.049940263 0.0082085 6.7 0.0175422 0.047027231 0.0061564 6.8 0.0166866 0.0441142 0.0041042 6.9 0.0158728 0.041201168 0.0020521 7 0.0150987 0.038288136 0 7.1 0.0143623 0.035375105 -0.002052 7.2 0.0136619 0.032462073 -0.004104 7.3 0.0129956 0.029549041 -0.006156 7.4 0.0123618 0.02663601 -0.008208 7.5 0.0117589 0.023722978 -0.010261 7.6 0.0111854 0.020809946 -0.012313 7.7 0.0106399 0.017896915 -0.014365 7.8 0.010121 0.014983883 -0.016417 7.9 0.0096274 0.012070851 -0.018469 8 0.0091578 0.009157819 -0.020521 8.1 0.0087112 0.006244788 -0.022573 8.2 0.0082863 0.003331756 -0.024625 8.3 0.0078822 0.000418724 -0.026678 8.4 0.0074978 -0.00249431 -0.02873 8.5 0.0071321 -0.00540734 -0.030782 8.6 0.0067843 -0.00832037 -0.032834 8.7 0.0064534 -0.0112334 -0.034886 8.8 0.0061387 -0.01414643 -0.036938 8.9 0.0058393 -0.01705947 -0.03899 9 0.0055545 -0.0199725 -0.041042 9.1 0.0052836 -0.02288553 -0.043095 9.2 0.0050259 -0.02579856 -0.045147 9.3 0.0047808 -0.02871159 -0.047199 9.4 0.0045476 -0.03162462 -0.049251 9.5 0.0043258 -0.03453766 -0.051303 9.6 0.0041149 -0.03745069 -0.053355 9.7 0.0039142 -0.04036372 -0.055407 9.8 0.0037233 -0.04327675 -0.057459 9.9 0.0035417 -0.04618978 -0.059512 10 0.003369 -0.04910281 -0.061564 Change the parameters below to see relationship between slope and derivative Z1 Z2 A1 A2 2 8 0.1839397 0.0091578 Secant Slope -0.02913 Secant Intercept 0.2422004 exponential function 0.6 0.5 0.4 A 0.3 0.2 0.1 0 0 -0.1 2 A Column E 0.1839397206 0.0091578194 Column F 0.0410424993 Tangent (i.e. derivative) Z 5 A 0.0410425 Slope -0.020521 Intercept 0.1436487 sinθ/θ -0.165398669 -0.1760258 -0.185656625 -0.194187625 -0.201519028 -0.207555461 -0.212206591 -0.215387743 -0.217020492 -0.217033228 -0.215361685 -0.211949433 -0.206748336 -0.199718957 -0.190830928 -0.180063263 -0.167404623 -0.15285353 -0.136418523 -0.118118257 -0.097981554 -0.076047379 -0.052364777 -0.026992736 3.89817E-017 0.0285351784 0.0585253394 0.0898741754 0.122476942 0.1562209209 0.1909859317 0.2266448899 0.2630644083 0.3001054387 0.3376239501 0.3754716393 0.4134966716 0.4515444447 0.4894583746 0.5270806968 0.5642532788 0.6008184397 0.6366197724 0.671502962 0.7053165985 0.7379129756 0.7691488744 0.7988863252 0.8269933431 0.8533446349 0.8778222702 0.9003163162 0.920725429 0.9389574009 0.9549296586 0.9685697097 0.9798155361 0.9886159295 0.99493077 0.998731244 #DIV/0! 0.998731244 0.99493077 0.9886159295 0.9798155361 0.9685697097 0.9549296586 0.9389574009 0.920725429 0.9003163162 0.8778222702 0.8533446349 0.8269933431 0.7988863252 0.7691488744 0.7379129756 0.7053165985 0.671502962 0.6366197724 0.6008184397 0.5642532788 0.5270806968 0.4894583746 0.4515444447 0.4134966716 0.3754716393 0.3376239501 0.3001054387 0.2630644083 0.2266448899 0.1909859317 0.1562209209 0.122476942 0.0898741754 0.0585253394 0.0285351784 3.89817E-017 -0.026992736 -0.052364777 -0.076047379 -0.097981554 -0.118118257 -0.136418523 -0.15285353 -0.167404623 -0.180063263 -0.190830928 -0.199718957 -0.206748336 -0.211949433 -0.215361685 -0.217033228 -0.217020492 -0.215387743 -0.212206591 -0.207555461 -0.201519028 -0.194187625 -0.185656625 -0.1760258 -0.165398669 What is the limit as q approaches 0? 1.2 1 0.8 0.6 si nx /x θ -300 -295 -290 -285 -280 -275 -270 -265 -260 -255 -250 -245 -240 -235 -230 -225 -220 -215 -210 -205 -200 -195 -190 -185 -180 -175 -170 -165 -160 -155 -150 -145 -140 -135 -130 -125 -120 -115 -110 -105 -100 -95 -90 -85 -80 -75 -70 -65 -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170 175 180 185 190 195 200 205 210 215 220 225 230 235 240 245 250 255 260 265 270 275 280 285 290 295 300 0.4 undefined 0.2 0 -0.2 -0.4 -300 -200 -100 0 de gre e s 100 200 300 Although undefined at θ =0, the limit as θ approaches 0 is 1. ...
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This document was uploaded on 02/29/2012.

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