differention in es
TRANSCRIPT
8/11/2019 Differention in Es
http://slidepdf.com/reader/full/differention-in-es 1/23
QUANTITATIVE METHODS FORENVIROMENTAL SCIENCES
UNIVERSITAS DIPONEGORO
2012
8/11/2019 Differention in Es
http://slidepdf.com/reader/full/differention-in-es 2/23
The subject of Calculus
• Derivative
• Integral
• Ordinary (partial )
differential equations
8/11/2019 Differention in Es
http://slidepdf.com/reader/full/differention-in-es 3/23
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
8/11/2019 Differention in Es
http://slidepdf.com/reader/full/differention-in-es 4/23
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‟.
8/11/2019 Differention in Es
http://slidepdf.com/reader/full/differention-in-es 5/23
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.
8/11/2019 Differention in Es
http://slidepdf.com/reader/full/differention-in-es 6/23
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.
8/11/2019 Differention in Es
http://slidepdf.com/reader/full/differention-in-es 7/23
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.
8/11/2019 Differention in Es
http://slidepdf.com/reader/full/differention-in-es 8/23
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.
8/11/2019 Differention in Es
http://slidepdf.com/reader/full/differention-in-es 9/23
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
8/11/2019 Differention in Es
http://slidepdf.com/reader/full/differention-in-es 10/23
Methods based on finite differences :
1. Forward difference approximation.
8/11/2019 Differention in Es
http://slidepdf.com/reader/full/differention-in-es 11/23
2. Central difference approximation.
8/11/2019 Differention in Es
http://slidepdf.com/reader/full/differention-in-es 12/23
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
8/11/2019 Differention in Es
http://slidepdf.com/reader/full/differention-in-es 13/23
Dari formula
• Aproksimasi sampai suku ke-1
• Aproksimasi sampai suku ke-2
8/11/2019 Differention in Es
http://slidepdf.com/reader/full/differention-in-es 14/23
Approximation for second derivative
8/11/2019 Differention in Es
http://slidepdf.com/reader/full/differention-in-es 15/23
8/11/2019 Differention in Es
http://slidepdf.com/reader/full/differention-in-es 16/23
8/11/2019 Differention in Es
http://slidepdf.com/reader/full/differention-in-es 17/23
8/11/2019 Differention in Es
http://slidepdf.com/reader/full/differention-in-es 18/23
MAX/MIN PROBLEM : OPTIMIZATION
Assumed and exist.
Then :
• If then f(x) has relative minimum
• If then f(x) has relative maximum
8/11/2019 Differention in Es
http://slidepdf.com/reader/full/differention-in-es 19/23
Example 1 :
8/11/2019 Differention in Es
http://slidepdf.com/reader/full/differention-in-es 20/23
8/11/2019 Differention in Es
http://slidepdf.com/reader/full/differention-in-es 21/23
Example 2 :
8/11/2019 Differention in Es
http://slidepdf.com/reader/full/differention-in-es 22/23
8/11/2019 Differention in Es
http://slidepdf.com/reader/full/differention-in-es 23/23
Exer.