Part Six

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Part Six

• Numerical Differentiation and Integration

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Motivation
You encounter differentiation and integration
every day!
DifferentiationAlmost 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
IntegrationIntegration 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?

• 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.

Example numerical integration
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
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Overall Structure