Question

49. Mean temperature Use Simpson's Rule to approximate the av- erage value of the temperature function f(x) = 37 sin ( 356 (– 101)) + 25 for a 365-day year. This is one way to estimate the annual mean air temperature in Fairbanks, Alaska. The National Weather Service's official figure, a numerical average of the daily normal mean air temperatures for the year, is 25.7°F, which is slightly higher than the average value of f(x).

Answer 1

Given 5 (37)=373in ( 365 (–101) +25 Calculate the average value of the temperature function you using the expression: va-b-abr (2) Substitute O for a, 365 for b and 37 sin ( 37(x-101)] +25 for f(x). va se (59 81 min (–101)+25) (571 in ((101as +237 () (377 din for (x– 101)dx=25(e)*) (37) sim sors (x=101)dx + 25(36–0) est ( sin egen (–101)or +9125 Assume u=x-101 Differentiate the above equation with respect to x. du 1 du=dx Substitute du for dx and u for x-101.

va tos ( 311 air ( 4 ) du +0125) +9125 1 | 21 365 Substitute x-101 for u. ( 13595 cos ( 3 (365–101) - 13505 cos ( 3 (0–101)]+9125 ((-2142.63+2148.398)+9125) Ya 36 (9130.768) = 25.016 = 25°F