Formula
1. Five-point FDF (Forward difference formula)
`f^'(x)=1/(12h)[-25f(x)+48f(x+h)-36f(x+2h)+16f(x+3h)-3f(x+4h)]`
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2. Five-point CDF (Central difference formula)
`f^'(x)=1/(12h)[f(x-2h)-8f(x-h)+8f(x+h)-f(x+2h)]`
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3. 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)]`
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Examples
1. Using Five point Forward difference, Backward difference, Central difference formula numerical differentiation to find solution
| x | 1 | 1.05 | 1.10 | 1.15 | 1.20 | 1.25 | 1.30 |
| f(x) | 1 | 1.02470 | 1.04881 | 1.07238 | 1.09545 | 1.11803 | 1.14018 |
`f^'(1.10) and f^('')(1.10)`
Solution:
The value of table for `x` and `y`
| x | 1 | 1.05 | 1.1 | 1.15 | 1.2 | 1.25 | 1.3 |
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| y | 1 | 1.0247 | 1.0488 | 1.0724 | 1.0954 | 1.118 | 1.1402 |
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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^'(1.10)=1/(12*0.05)[-25f(1.10)+48f(1.10+0.05)-36f(1.10+2*0.05)+16f(1.10+3*0.05)-3f(1.10+4*0.05)]`
`f^'(1.10)=1/(0.6)[-25f(1.10)+48f(1.15)-36f(1.2)+16f(1.25)-3f(1.3)]`
`f^'(1.10)=1/(0.6)[-25(1.0488)+48(1.0724)-36(1.0954)+16(1.118)-3(1.1402)]`
`f^'(1.10)=0.4762`
Five-point CDF (Central difference formula)
`f^'(x)=1/(12h)[f(x-2h)-8f(x-h)+8f(x+h)-f(x+2h)]`
`f^'(1.10)=1/(12*0.05)[f(1.10-2*0.05)-8f(1.10-0.05)+8f(1.10+0.05)-f(1.10+2*0.05)]`
`f^'(1.10)=1/0.6[f(1)-8f(1.05)+8f(1.15)-f(1.2)]`
`f^'(1.10)=1/0.6[1-8(1.0247)+8(1.0724)-1.0954]`
`f^'(1.10)=0.4767`
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^('')(1.10)=1/(12*(0.05)^2)[-f(1.10-2*0.05)+16f(1.10-0.05)-30f(1.10)+16f(1.10+0.05)-f(1.10+2*0.05)]`
`f^('')(1.10)=1/0.03[-f(1)+16f(1.05)-30f(1.10)+16f(1.15)-f(1.2)]`
`f^('')(1.10)=1/0.03[-1+16(1.0247)-30(1.0488)+16(1.0724)-1.0954]`
`f^('')(1.10)=-0.2157`
This material is intended as a summary. Use your textbook for detail explanation.
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