Let f(x) = 3e^cos(x). The graph of f and its first four derivatives on [2, 7] ar
ID: 2882004 • Letter: L
Question
Let f(x) = 3e^cos(x). The graph of f and its first four derivatives on [2, 7] are shown on the next page. (Be careful when examining the graphs; look carefully at the vertical scales.) You should assume that the graphs are correct for this problem. Let I = integral^7_2 f(x) dx. Use the graph of f alone to estimate I. Use the information in the graphs to tell how many subdivisions N are needed so that the Trapezoid Rule approximation T_N will approximate I to 5 decimal places. Use the information in the graphs to tell how many subdivisions N are needed so that the Midpoint Rule approximation M_N will approximate I to 5 decimal places. Use the information in the graphs to tell how many subdivisions N are needed so that the Simpson's Rule approximation S_N will approximate I to 5 decimal places.Explanation / Answer
a. Trapezoidal rule for 5 intervals each of 1 unit
= 1/2 [ f(2) +f(3) + f(3) +f(4) -------+f(7) ]
= 1/2 [ ecos2+ 2 ecos3 +2 ecos4+ 2ecos5+2ecos6+ ecos7]
as the angles 2,3,4 ---are very small cos2, cos3 ..=1
= 1/2[ e +2e+2e+2e+2e +e] =10e/2=5e
2. Mid point formula : by taking 5 points 2.5, 3.5, 4.5, 5.5, 6.5 ( width = 1)
area app = 1[ ecos 2.5 +ecos3.5 ---- +ecos6.5] = 5e approximately
3 Simpson rule n=6
= [7-2]/18 { f(2) +4f(3) +2f(4) + 4f(5) +2f(6) +4f(7)}
( 5/18 )[ 17 e] = 4.7e
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