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Part Six Numerical Differentiation and Integration Motivation You encounter differentiation and integration every day !

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part six

Part Six

Numerical Differentiation and Integration

slide2

Motivation

You encounter differentiation and integration every day!

Differentiation:Almost all physical processes/phenomena are best cast in differentiation formExample: Newtons 2nd law: F = (dv/dt)m Heat conduction: Heat flux = -k’ (dT/dx) Our parachutist problem: dv/dt = (mg – cv)/m

Integration:Integration is commonplace in science and engineering

Urban area River cross-section Windblow on rocket

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What are Differentiation and Integration?

Differentiation: rate of change of a dependent variable with respect to an independent variable.

Integration: the integral of the function f(x) with respect to the independent variable x, evaluated between the limits x = a to x = b.

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Why Numerical Methods?

Example: numerical integration

  • Very often, the function f(x) to differentiate or the integrand to integrate is too complex to derive exact analytical solutions.
  • In most cases in engineering, the function f(x) is only available in a tabulated form with values known only at discrete points.

Numerical Solution

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Examples of Numerical Differentiation and Integration

Differentiation

Integration

There exist much more efficient and accurate numerical methods than these two! They are the ones we are to learn!

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Some Often Used Math Derivations

Differentiation

Integration