`f(x)=cosx` and `h = 0.05`, estimate `f^'(1.2) and f^('')(1.2)`
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) = cos(x)`.
`:. f^'(x) = -sin(x)`
`:. f^('')(x) = -cos(x)`
The value of table for `x` and `y`
| x | 1 | 1.05 | 1.1 | 1.15 | 1.2 | 1.25 | 1.3 | 1.35 | 1.4 |
|---|
| y | 0.5403 | 0.4976 | 0.4536 | 0.4085 | 0.3624 | 0.3153 | 0.2675 | 0.219 | 0.17 |
|---|
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^'(1.2)=1/(12*0.05)[-25f(1.2)+48f(1.2+0.05)-36f(1.2+2*0.05)+16f(1.2+3*0.05)-3f(1.2+4*0.05)]`
`f^'(1.2)=1/(0.6)[-25f(1.2)+48f(1.25)-36f(1.3)+16f(1.35)-3f(1.4)]`
`f^'(1.2)=1/(0.6)[-25(0.3624)+48(0.3153)-36(0.2675)+16(0.219)-3(0.17)]`
`f^'(1.2)=-0.932`
Absolute Error:`|"exact value of " f^'(1.2)-(-0.932)|=|-0.932 +0.932|=0`
Five-point CDF (Central difference formula)
`f^'(x)=1/(12h)[f(x-2h)-8f(x-h)+8f(x+h)-f(x+2h)]`
`f^'(1.2)=1/(12*0.05)[f(1.2-2*0.05)-8f(1.2-0.05)+8f(1.2+0.05)-f(1.2+2*0.05)]`
`f^'(1.2)=1/0.6[f(1.1)-8f(1.15)+8f(1.25)-f(1.3)]`
`f^'(1.2)=1/0.6[0.4536-8(0.4085)+8(0.3153)-0.2675]`
`f^'(1.2)=-0.932`
Absolute Error:`|"exact value of " f^'(1.2)-(-0.932)|=|-0.932 +0.932|=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^('')(1.2)=1/(12*(0.05)^2)[-f(1.2-2*0.05)+16f(1.2-0.05)-30f(1.2)+16f(1.2+0.05)-f(1.2+2*0.05)]`
`f^('')(1.2)=1/0.03[-f(1.1)+16f(1.15)-30f(1.2)+16f(1.25)-f(1.3)]`
`f^('')(1.2)=1/0.03[-0.4536+16(0.4085)-30(0.3624)+16(0.3153)-0.2675]`
`f^('')(1.2)=-0.3624`
Absolute Error:`|"exact value of " f^('')(1.2)-(-0.3624)|=|-0.3624 +0.3624|=0`
This material is intended as a summary. Use your textbook for detail explanation.
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