Numerical Methods Important Formulas Business Maths Class 12

Dear Class 12 Samacheer Kalvi students, here are the Business Maths Chapter 5 Numerical Methods important formulas for your reference and study.

Important Formulas to Study

Forward Difference Operator

$\Delta f(x)=f (x + h) − f (x) \\$
$\Delta \left( f(x)+g(x) \right)=\Delta f(x)+ \Delta g(x) \\$
$\Delta c f(x)=c \;\Delta f(x) \\$
$If \; c \; is \; a \; constant\; then\; \Delta\;c=0 \\$
$\Delta ^{k} f(x)=\Delta^{k-1} f(x+h)-\Delta^{k-1} f(x+h)f(x)\\$
$\Delta y_{n}=y_{n+1}-y_{n}\\$

Backward Difference Operator

$\nabla f(x)=f (x) − f (x-h) \\$
$\nabla f(x+nh)=\Delta n f(x)\\$
$\nabla y_{n}=y_{n}-y_{n-1}\\$
$\nabla^{k}y_{n}=\nabla ^{k-1} y_{n}-\nabla ^{k-1}y_{n-1}\; n=1,2,3\cdots \\$

Shifting Operator (E) or Displacement Operator

$E f(x)=f (x+h) \\$
$E^{n}f(x)=f(x+nh) \\$
$E^{-n}f(x)=f(x-nh) \\$

Relations between operators Delta, Nabla, and E

$\Delta =E-1 \\$
$E \Delta =\Delta E \\$
$\nabla \dfrac{E-1}{E} \\$
$( 1+\Delta) ( 1-\nabla)=1 \\$
$(\Delta) (\nabla) =\Delta-\nabla \\$
$\nabla) = E^{-1}\Delta \\$

Newton’s forward interpolation formula:

$y_{(x=x_{0}+ nh)}=y_{0}+\dfrac{n}{1!}\: \Delta (\; y_{0})+\dfrac{n(n-1)}{2!}\: \Delta^{2} (\; y_{0})+\dfrac{n(n-1)(n-2)}{3!}\: \Delta^{3} (\; y_{0})+ \cdots$

Newton’s backward interpolation formula:

$y_{(x=x_{n}+ nh)}=y_{n}+\dfrac{n}{1!} \nabla y_{n}+\dfrac{n(n+1)}{2!} \nabla^{2} y_{n}+\dfrac{n(n+1)(n+2)}{3!} \nabla^{3} y_{n}+\dfrac{n(n+1)(n+2)(n+3)}{4!} \nabla^{4} y_{n}+\cdots\\$

Lagrange’s interpolation formula:

$y=\dfrac{(x-x_{1})(x-x_{2})(x-x_{3})}{(x_{0}-x_{1})(x_{0}-x_{2})(x_{0}-x_{3})} y_{0}+ \\$
$\dfrac{(x-x_{0})(x-x_{2})(x-x_{3})}{(x_{1}-x_{0})(x_{1}-x_{2})(x_{1}-x_{3})} y_{1}+ \\$
$\dfrac{(x-x_{0})(x-x_{1})(x-x_{3})}{(x_{2}-x_{0})(x_{2}-x_{1})(x_{2}-x_{3})} y_{2}+ \\$
$\dfrac{(x-x_{0})(x-x_{1})(x-x_{2})}{(x_{3}-x_{0})(x_{3}-x_{1})(x_{3}-x_{2})} y_{3} \\$

Numerical Methods Important Formulas Business Maths Class 12

Important Formulas to Study

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