Numerical diffrentiation and integration
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Numerical differentiation and integration are techniques for approximating derivatives and integrals using discrete data points rather than continuous functions. Numerical differentiation can be done by deriving a formula that approximates the derivative as a linear combination of function values, or by approximating derivatives of discrete data using interpolating polynomials like Newton's forward difference formula. Numerical integration breaks problems into small pieces that can be approximated using simple formulas like the two-point formula to estimate integrals.
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Numerical diffrentiation and integration
- 2. Numerical Differentiation Basic Steps • Derive a formula that approximates the derivative of a function in terms of linear combination of function values ( Function may be known ) • Approximate the value of a derivative of a function defined by discrete data.
- 3. Numerical Diffrentiation using interpolating polynomial x ……….. f(x) ……… Remark 1: If data is equispaced. We can used Newton’s Forward or Backward formula depends the location of the point. Remark 2: If data is not equispaced then Lagrange interpolating polynomial can be used.
- 4. Using Newton’s forward difference formula
- 5. For a given x Thus, in general
- 6. Practice Question Using Newton forward difference formula ,find the first and the second derivative x=2 for the following tabulated function :
- 8. ,u=1,h=1
- 9. Do yourself
- 11. Where numerical integration method is used ?
- 13. The fundamental idea is: