Numerical Differentiation and Integration with a Computer

• If we are working with experimental data that are displayed in a table of [$x,f(x)$] pairs emulation of calculus is impossible.
• We must approximate the function behind the data in some way.
• Differentiation with a Computer:
  • Employs the interpolating polynomials to derive formulas for getting derivatives.
  • These can be applied to functions known explicitly as well as those whose values are found in a table.
• Numerical Integration-The Trapezoidal Rule:
  • Approximates, the integrand function with a linear interpolating polynomial to derive a very simple but important formula for numerically integrating functions between given limits.

• We continue to exploit the useful properties of polynomials to develop methods for a computer to do integrations and to find derivatives.
• When the function is explicitly known, we can emulate the methods of calculus.
• But doing so in getting derivatives requires the subtraction of quantities that are nearly equal and that runs into round-off error.
• However, integration involves only addition, so round-off is not problem.
• We cannot often find the true answer numerically because the analytical value is the limit of the sum of an infinite number of terms.
• We must be satisfied with approximations for both derivatives and integrals but, for most applications, the numerical answer is adequate.