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# Part Six - PowerPoint PPT Presentation

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

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### Part Six

Numerical Differentiation and Integration

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

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.

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

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!

Some Often Used Math Derivations

Differentiation

Integration