f(x)=2x^3+x^2-4 and h = 0.5, estimate f^'(2.5) and f^('')(2.5)
using Five 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) = 2x^3+x^2-4.
:. f^'(x) = 6x^2+2x
:. f^('')(x) = 12x+2
The value of table for x and y
| x | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 | 4.5 |
|---|
| y | -3.5 | -1 | 5 | 16 | 33.5 | 59 | 94 | 140 | 198.5 |
|---|
Five-point FDF (Forward difference formula)
f^'(x)=1/(12h)[-25f(x)+48f(x+h)-36f(x+2h)+16f(x+3h)-3f(x+4h)]
f^'(2.5)=1/(12*0.5)[-25f(2.5)+48f(2.5+0.5)-36f(2.5+2*0.5)+16f(2.5+3*0.5)-3f(2.5+4*0.5)]
f^'(2.5)=1/(6)[-25f(2.5)+48f(3)-36f(3.5)+16f(4)-3f(4.5)]
f^'(2.5)=1/(6)[-25(33.5)+48(59)-36(94)+16(140)-3(198.5)]
f^'(2.5)=42.5
Absolute Error:|"exact value of " f^'(2.5)-(42.5)|=|42.5 -42.5|=0
Five-point CDF (Central difference formula)
f^'(x)=1/(12h)[f(x-2h)-8f(x-h)+8f(x+h)-f(x+2h)]
f^'(2.5)=1/(12*0.5)[f(2.5-2*0.5)-8f(2.5-0.5)+8f(2.5+0.5)-f(2.5+2*0.5)]
f^'(2.5)=1/6[f(1.5)-8f(2)+8f(3)-f(3.5)]
f^'(2.5)=1/6[5-8(16)+8(59)-94]
f^'(2.5)=42.5
Absolute Error:|"exact value of " f^'(2.5)-(42.5)|=|42.5 -42.5|=0
Five-point CDF (Central difference formula) for second derivatives
f^('')(x)=1/(12h^2)[-f(x-2h)+16f(x-h)-30f(x)+16f(x+h)-f(x+2h)]
f^('')(2.5)=1/(12*(0.5)^2)[-f(2.5-2*0.5)+16f(2.5-0.5)-30f(2.5)+16f(2.5+0.5)-f(2.5+2*0.5)]
f^('')(2.5)=1/3[-f(1.5)+16f(2)-30f(2.5)+16f(3)-f(3.5)]
f^('')(2.5)=1/3[-5+16(16)-30(33.5)+16(59)-94]
f^('')(2.5)=32
Absolute Error:|"exact value of " f^('')(2.5)-(32)|=|32 -32|=0
This material is intended as a summary. Use your textbook for detail explanation.
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