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^'(1.15) and f^('')(1.15)`
Solution:
The value of table for `x` and `y`
| x | 1 | 1.05 | 1.1 | 1.15 | 1.2 | 1.25 | 1.3 |
|---|
| y | 1 | 1.0247 | 1.0488 | 1.0724 | 1.0954 | 1.118 | 1.1402 |
|---|
Five-point CDF (Central difference formula)
`f^'(x)=1/(12h)[f(x-2h)-8f(x-h)+8f(x+h)-f(x+2h)]`
`f^'(1.15)=1/(12*0.05)[f(1.15-2*0.05)-8f(1.15-0.05)+8f(1.15+0.05)-f(1.15+2*0.05)]`
`f^'(1.15)=1/0.6[f(1.05)-8f(1.1)+8f(1.2)-f(1.25)]`
`f^'(1.15)=1/0.6[1.0247-8(1.0488)+8(1.0954)-1.118]`
`f^'(1.15)=0.4663`
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^('')(1.15)=1/(12*(0.05)^2)[-f(1.15-2*0.05)+16f(1.15-0.05)-30f(1.15)+16f(1.15+0.05)-f(1.15+2*0.05)]`
`f^('')(1.15)=1/0.03[-f(1.05)+16f(1.1)-30f(1.15)+16f(1.2)-f(1.25)]`
`f^('')(1.15)=1/0.03[-1.0247+16(1.0488)-30(1.0724)+16(1.0954)-1.118]`
`f^('')(1.15)=-0.199`
This material is intended as a summary. Use your textbook for detail explanation.
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