Lecture 5
●
MEEN 357 Homework Solution
1
Lecture05HomeworkSolution2009.doc
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1.
Problem 4.19 of the textbook.
Solution
:
You are given a function
(
)
3
2
4
f
x
x
x
=
−
+
(5.1)
and asked to find a) forward first derivative, b) backward first derivative, c) centered first
derivative, d) forward second derivative, e) backward second derivative and f) centered
second derivative at the points in the interval
2
2
x
−
≤
≤
separated by a step size of
.25
h
=
.
You are also asked to graph these results along with the formulas for the exact
derivatives that you can obtain analytically from (5.1).
The formulas you must use are
given in Chapter 4 of the textbook or in the table
(
)
i
f
x
′
(
)
i
f
x
′′
Forward
(
)
(
)
(
)
1
i
i
f
x
f
x
O h
h
+
−
+
(
)
(
)
(
)
(
)
2
1
2
2
i
i
i
f
x
f
x
f
x
O h
h
+
+
−
+
+
Back
(
)
(
)
(
)
1
i
i
f
x
f
x
O h
h
−
−
+
(
)
(
)
(
)
(
)
1
2
2
2
i
i
i
f
x
f
x
f
x
O h
h
−
−
−
+
+
Centered
(
)
(
)
(
)
1
1
2
2
i
i
f
x
f
x
O h
h
+
−
−
+
(
)
(
)
(
)
(
)
1
1
2
2
2
i
i
i
f
x
f
x
f
x
O h
h
+
−
−
+
+
given in the notes on Lecture 5.
Because the entries in the above table involve the terms
1
1
2
,
,
i
i
i
x
x
x
+
−
+
and
2
i
x
−
, these formulas tell you immediately that you are going to have to
evaluate the given
(
)
f
x
at points in the interval
2.5
2.5
x
−
≤
≤
in order to calculate the
second derivatives at the points
2
x
= −
and
2
x
=
.
This problem can be done by hand, but it is tedious.
MATLAB is ideally suited
for this kind of problem.
The mfile below generates the table of derivatives.
Inserted in
the mfile are comments designed to help you understand the details.
The mfile is long
because it generates both tables and the two plots.
You could break the problem down
into shorter files.
In any case, the file is
%Problem 4.19 of the textbook
clc
clear
all
%Set the given step size
h=.25;
%Call by the symbol x the values in the interval 2<=x<=2 devided by the
%step size h=.25.
This symbol is a matrix of size 1x17.
It does not
%contain the end points in the larger interval 2.5<=x<=2.5.
x=[2:h:2];
%Call by the symbol x_b the values of x required to calculate the backward
%first derivative.
This matrix is also a 1x17 matrix.
It is one step size
%back.
x_b=[2.25:h:1.75];
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Lecture 5
●
MEEN 357 Homework Solution
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Lecture05HomeworkSolution2009.doc
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%Call by the symbol x_f the values of x required to calculate the forward
%first derivative.
This matrix is also a 1x17 matrix.
It is one step size
%forward
x_f=[1.75:h:2.25];
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 Fall '07
 ANAMALAI
 Derivative, Continuous function

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