f(x)=xlnx and h = 1, estimate f^'(4) and f^('')(4)
using Three point Forward difference, Backward difference, Central difference formula numerical differentiation
Also find exact value of f', f'' and error for each estimation
Solution:
Equation is f(x) = xln(x).
:. f^'(x) = ln(x)+1
:. f^('')(x) = 1/x
The value of table for x and y
| x | 2 | 3 | 4 | 5 | 6 |
|---|
| y | 1.3863 | 3.2958 | 5.5452 | 8.0472 | 10.7506 |
|---|
Three-point FDF (Forward difference formula)
f^'(x)=1/(2h)[-3f(x)+4f(x+h)-f(x+2h)]
f^'(4)=1/(2*1)[-3f(4)+4f(4+1)-f(4+2*1)]
f^'(4)=1/2[-3f(4)+4f(5)-f(6)]
f^'(4)=1/2[-3(5.5452)+4(8.0472)-10.7506]
f^'(4)=2.4013
Absolute Error:|"exact value of " f^'(4)-(2.4013)|=|2.3863 -2.4013|=0.015
Three-point BDF (Backward difference formula)
f^'(x)=1/(2h)[f(x-2h)-4f(x-h)+3f(x)]
f^'(4)=1/(2*1)[f(4-2*1)-4f(4-1)+3f(4)]
f^'(4)=1/2[f(2)-4f(3)+3f(4)]
f^'(4)=1/2[1.3863-4(3.2958)+3(5.5452)]
f^'(4)=2.4192
Absolute Error:|"exact value of " f^'(4)-(2.4192)|=|2.3863 -2.4192|=0.0329
Three-point CDF (Central difference formula)
f^'(x)=(f(x+h)-f(x-h))/(2h)
f^'(4)=(f(4+1)-f(4-1))/(2*1)
f^'(4)=(f(5)-f(3))/2
f^'(4)=(8.0472-3.2958)/2
f^'(4)=2.3757
Absolute Error:|"exact value of " f^'(4)-(2.3757)|=|2.3863 -2.3757|=0.0106
Three-point FDF (Forward difference formula) for second derivatives
f^('')(x)=(f(x)-2f(x+h)+f(x+2h))/(h^2)
f^('')(4)=(f(4)-2f(4+1)+f(4+2*1))/((1)^2)
f^('')(4)=(f(4)-2f(5)+f(6))/(1)
f^('')(4)=(5.5452-2(8.0472)+10.7506)/(1)
f^('')(4)=0.2014
Absolute Error:|"exact value of " f^('')(4)-(0.2014)|=|0.25 -0.2014|=0.0486
Three-point BDF (Backward difference formula) for second derivatives
f^('')(x)=(f(x-2h)-2f(x-h)+f(x))/(h^2)
f^('')(4)=(f(4-2*1)-2f(4-1)+f(4))/((1)^2)
f^('')(4)=(f(2)-2f(3)+f(4))/(1)
f^('')(4)=(1.3863-2(3.2958)+5.5452)/(1)
f^('')(4)=0.3398
Absolute Error:|"exact value of " f^('')(4)-(0.3398)|=|0.25 -0.3398|=0.0898
Three-point CDF (Central difference formula) for second derivatives
f^('')(x)=(f(x-h)-2f(x)+f(x+h))/(h^2)
f^('')(4)=(f(4-1)-2f(4)+f(4+1))/(1)^2
f^('')(4)=(f(3)-2f(4)+f(5))/(1)
f^('')(4)=(3.2958-2(5.5452)+8.0472)/(1)
f^('')(4)=0.2527
Absolute Error:|"exact value of " f^('')(4)-(0.2527)|=|0.25 -0.2527|=0.0027
This material is intended as a summary. Use your textbook for detail explanation.
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