22_Cal_Solution of Calculus_6e

22_Cal_Solution of Calculus_6e - imum, we Fnd that c 1 = 61...

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1 2 3 4 5 6 2 4 6 8 average of these is 1 . 65 (to two places) and should be a good estimate of the true value of c . Alternatively, you can use a computer algebra program to Fnd c ;in Mathematica, for example, the command ±it will Ft given data points to a sum of constant multiples of functions you specify. We used the commands data = {{ 0 . 25 , 6 . 72 } , { 1 . 0 , 1 . 68 } , { 2 . 5 , 0 . 67 } , { 4 . 0 , 0 . 42 } , { 6 . 0 , 0 . 27 }} ; ±it[data, { 1 / p } ,p ] to Fnd that V ( p )= 1 . 67986 p yields the best least-squares Ft of the given data to a function of the form V ( p )= c/p . We rounded the numerator to 1 . 68 to Fnd the estimates V (0 . 5) 3 . 36 and V (5) 0 . 336 (L). The graph of V ( p ) is shown next. C01S02.082: It seems reasonable to assume that the maximum average temperature occurs on July 15 and the minimum on January 15, so that a multiple of a cosine function should Ft the given data if we take t = 0 on July 15. So we assume a solution of the form T ( t )= c 1 + c 2 cos µ 2 πt 365 ± .
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Unformatted text preview: imum, we Fnd that c 1 = 61 . 25, so we could Fnd c 2 by the averaging method of Problem 81. Alternatively, we could use the ±it command in Mathematica to Fnd both c 1 and c 2 simultaneously as follows: data = {{ , 79 . 1 } , { 62 , 70 . 2 } , { 123 , 52 . 3 } , { 184 , 43 . 4 } , { 224 , 52 . 2 } , { 285 , 70 . 1 }} ; ±it[data, { 1, Cos[2 ∗ Pi ∗ t / 365] } , t] The result is the formula T ( t ) = 62 . 9602 + (17 . 437)cos µ 2 πt 365 ± . The values predicted by this function at the six dates in question are [approximately] 80 . 4, 71 . 4, 53 . 9, 45 . 5, 49 . 8, and 66 . 3. Not bad, considering we are dealing with weather, a most unpredictable phenomenon. The graph of T ( t ) is shown next. Units on the horizontal axis are days, measured from July 15. Units on 11...
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