SharedMTH 505 / sec_3_1_lagrange.sagewsOpen in CoCalc
Author: Jonathan Mitchell
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X = [0.6, 0.7, 0.8, 1] f(x) = sin(e^(x) - 2) Y = [0,0,0,0] for j in(0..3): Y[j] = round(f(X[j]),8) L0(x) = (x-X[1])*(x-X[2])*(x-X[3])/((X[0] - X[1])*(X[0]-X[2])*(X[0] -X[3])) L1(x) = (x-X[0])*(x-X[2])*(x-X[3])/((X[1] - X[0])*(X[1]-X[2])*(X[1] -X[3])) L2(x) = (x-X[0])*(x-X[1])*(x-X[3])/((X[2] - X[0])*(X[2]-X[1])*(X[2] -X[3])) L3(x) = (x-X[0])*(x-X[1])*(x-X[2])/((X[3] - X[0])*(X[3]-X[1])*(X[3] -X[2])) P(x) = Y[0]*L0(x) + Y[1]*L1(x) + Y[2]*L2(x) + Y[3]*L3(x) p = plot(P(x), 0.5,1.1,x,color = 'red') q = plot(f(x),0.5,1.1,x,color = 'blue') show(expand(P(x))) show(p+q)
1.78574124999995x3+4.70928062499987x21.94720472499990x0.318242680000020\displaystyle -1.78574124999995 \, x^{3} + 4.70928062499987 \, x^{2} - 1.94720472499990 \, x - 0.318242680000020
show(P(0.9)) show(f(0.9)) abserror = abs(P(0.9) - f(0.9)) relerror = abserror/abs(f(0.9)) show(abserror) show(relerror)
0.441985002500000\displaystyle 0.441985002500000
0.443592438770281\displaystyle 0.443592438770281
0.00160743627028176\displaystyle 0.00160743627028176
0.00362367824559378\displaystyle 0.00362367824559378
f4(x) = diff(f(x),x,4) show(f4(x))
6cos(ex2)e(3x)+cos(ex2)ex+e(4x)sin(ex2)7e(2x)sin(ex2)\displaystyle -6 \, \cos\left(e^{x} - 2\right) e^{\left(3 \, x\right)} + \cos\left(e^{x} - 2\right) e^{x} + e^{\left(4 \, x\right)} \sin\left(e^{x} - 2\right) - 7 \, e^{\left(2 \, x\right)} \sin\left(e^{x} - 2\right)
P = plot(f4(x),0.5,1.1,x) show(P) round(f4(1),8)
-86.80037916
90*(0.9 - 0.6)*(0.9-0.7)*(0.9-0.8)*(0.9-1)/24
-0.00225000000000000