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Derivatives and Differential Equations. Differentiation. Differential change . Derivative Definition. Taylor Series. Taylor Series Graphically. Numerical Differentiation Based on Taylor Series . Forward finite-divided difference B ackward finite-divided difference

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Presentation Transcript
differentiation
Differentiation
  • Differential change
numerical differentiation based on taylor series
Numerical Differentiation Based on Taylor Series
  • Forward finite-divided difference
  • Backward finite-divided difference
  • Centered finite-divided difference
unequally spaced data
Unequally Spaced Data
  • One way to calculated derivatives of unequally spaced data is to determine a polynomial fit and take its derivative at a point.
  • As an example, using a second-order Lagrange polynomial to fit three points and taking its derivative yields:
derivatives and integrals for data with errors
Derivatives and Integrals for Data with Errors
  • Numerical differentiation amplifies data errors.
  • Solution: fit a smooth, differentiable function to the data and take the derivative of the function.
numerical differentiation with matlab
Numerical Differentiation with MATLAB
  • MATLAB has two built-in functions to help take derivatives, diff and gradient:
  • diff(x)
    • Returns the difference between adjacent elements in x. Not the same size as vector x.
  • diff(y)./diff(x)
    • Returns the difference between adjacent values in y divided by the corresponding difference in adjacent values of x
numerical differentiation function of a single variable
Numerical Differentiation, function of a single variable
  • fx = gradient(f, h)
  • Gradient can also be used to find partial derivatives for matrices: [fx, fy] = gradient(f, h)
numerical differentiation function of a two variables
Numerical Differentiation, function of a two variables

To generate the components of a derivative of a function of two variables x, y use

[fx, fy] = gradient(f, h)

Where h scales the magnitude of vectors displayed