0.5 { (x2+3)/4, if 0.5<x<1.0 S f(x) dx, where f(x) = { 1.5 { (x+4)/(x2+4), if 1.0<x<1.5
minimizing the number of floating point operations and function evaluations. Also, tabulate
i | 0 | 1 | 2 | 3 | 4 | |
xi | ||||||
f i |
i | 0 | 1 | 2 | 3 | 4 | |
xi | 0.5000 | 0.7500 | 1.000 | 1.250 | 1.500 | |
f i | 0.8125 | 0.8906 | 1.000 | 0.9438 | 0.8800 |
FinalAnswer TR5 = h*(0.5*(f0+f4)+f1+f2 +f3) =(r4) 0.9201 in Efficient Form.
b 0.75 S f(x) dx = S (exp(2x)/(x+8)) dx , a 0minimizing the number of floating point operations and function evaluations. Also, tabulate
i | 1 | 2 | 3 | |
ti | ||||
xi | ||||
Fi |
i | 1 | 2 | 3 | |
ti | 0 | 0 | 0 | |
xi | 0.1250 | 0.3750 | 0.6250 | |
Fi | 0.01975 | 0.03159 | 0.05058 |
FinalAnswer 3×G1 = 2*(F1+F2+F3) =(r4) 0.2038 in Efficient Form.
Caution: These old problems may have different answers due to different methods and chopping truncation used, so do not worry if your answers are a few least significant digits different.
1 S f(x) dx, 0where
{ (1+x) on (0,0.5) f(x) = { , { (2-x) on (0.5,1.0)according to the following simple rules:
+1 S ln(1+x2) dx, -1for the following rules:
3 S exp(x)*sqrt(x) dx 0using 7 points and Simpson's composite rule. Minimize the number of function evaluations, multiplications and additions. (Ans.: 27.45 or 27.43 with extra chops)
4 I = S (sqrt(x)/exp(x)) dx 1using a 7 point Simpson's rule. Minimize the number of function evaluations and multiplications in calculating the composite rule.
(Ans.: x = (1.000 1.500 2.000 2.500 3.000 3.500 4.000) f(x) =(4ch) (0.3678 0.2732 0.1913 0.1297 0.8623e-1 0.5645e-1 0.3663e-1), I =(4ch) (0.3678+4*(0.2732+0.1297+0.5649e-1)+2*(0.1913+0.8623e-1)+0.3663e-1)/6. =(4ch) (0.4661,0.4660,0.4663), depending on the chopping sequence.)
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