Differention in Es

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QUANTITATIVE METHODS FORENVIROMENTAL SCIENCES

UNIVERSITAS DIPONEGORO

2012 

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The subject of Calculus

• Derivative

• Integral

• Ordinary (partial )

differential equations

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DERIV TIVE 

Why we must to know derivative.

Most non-mathematicians asked “

what is the derivative at a point x

of the function y=f(x) ”. Usually,

the derivative is denoted by

or

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DERIV TIVE 

The derivative can be viewed

as RATE MEASURER.

Derivative as rate measurerinterpret as follows

1. Whatever be the quantity „y‟

its derivative gives how

fast y is changing with „t‟. 

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2. If is positive, then the rate of

change of y with respect to x ispositive. This means that if x

increase, then y also increases

and if x decreases, then y also

decreases.

3. If is negative, then the rate of

change of y with respect to x is

negative. This means that if xincrease, then y decreases and if

x decreases, then y increase.

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DERIV TIVE 

Generally, The derivative arise in

environmental science in two

ways :

1. Derivative is fundamentally

important (entities)

2. Derivative arise from maximum

and minimum problems.

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NUMERIC L

DIFFERENTI TION There are several reasons why we sometimeswant to differentiate numerically as well:

1. We may need the derivative of a functionthat we know only as a table of values ofthe form [x, f(x)]. For example, thissituation might arise if we had a table ofdaily measurements of the volume ofwater in a reservoir.

2. If a function is very messy, and we needits derivative at only one point, numericaldifferentiation may be the easiest way toobtain it.

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NUMERIC L

DIFFERENTI TION

3. Numerical differentiation can form the

basis for numerical methods for solvingdifferential equations.

4. Numerical derivatives can be very useful

for checking whether an analytic derivativeis correct or not. 

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Approximation to the derivatives can be

Obtained numerically using the following two

approaches

•

Methods based on finite differencesfor

equispaced data.

• Methods based on divided differences or

Lagrange interpolation for non-uniform data

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Methods based on finite differences :

1. Forward difference approximation.

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2. Central difference approximation.

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Consider the data (xi, f(xi)) given at equispaced

points xi = x0 + ih, i = 0, 1, 2, ..., n where h is

the step length.

The Newton’s forward difference formula is

given by

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Dari formula

• Aproksimasi sampai suku ke-1

• Aproksimasi sampai suku ke-2

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Approximation for second derivative

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MAX/MIN PROBLEM : OPTIMIZATION

 Assumed and exist.

Then :

• If then f(x) has relative minimum

• If then f(x) has relative maximum

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Example 1 :

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Example 2 :

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Exer.