8. 8
Edit Menu
Copy, cut, paste
Find and replace
phrases
Clear command history,
workspace
Desktop Menu
Change appearance of
desktop
Select windows to
display
MATLAB GUI
9. 9
Getting Help
type one of following commands in the command window:
help lists all the help topic
help command provides help for the specified command
Example:
>> help sqrt
SQRT Square root.
SQRT(X) is the square root of the elements of X. Complex
results are produced if X is not positive.
help help provides information on the help command use
10. 10
Variables
Variables are named memory locations that can be assigned a value by
the user or programmer.
The system can retrieve , or “remember” the value of a variable.
Variable names:
Must start with a letter
May contain only letters, digits, and the underscore “_”
Matlab is case sensitive, i.e. one & OnE are different variables.
Assignment statement:
Variable = number;
Variable = expression;
NOTE: when a semi-colon ”;” is placed at the
end of each command, the result is not
displayed.
11. 11
Variable Examples
>> tutorial = 1234;
>> tutorial = 1234
tutorial = ?
>> a ??? Undefined function or variable 'a'.
>> a=2 a = ?
>> a+3 ans = ?
>> sqrt(a+14) ans = ?
>> a=a+12 a = ?
12. 12
String Variables
>> a = 'This is a string'
a = ?
>> b= 'Another string'
b = ?
>> a + b
??? Error using ==> plus Matrix dimensions must agree.
13. 13
Special Variables
ans : default variable name for the result
pi: π = 3.1415926…………
eps: ԑ = 2.2204e-016, smallest amount by which 2 numbers can differ.
Inf or inf : ∞, infinity
NaN or nan: not-a-number
14. 14
Defining Symbolic Variables & Viewing Variables
Use sym to create a symbolic variable x:
Use syms to create several symbolic variables at one time
Use who to view all variables in the workspace
» x=sym('x');
» syms y a b
» who
Your variables are:
Use whos to view all workspace variables with their associated size,
bytes, and class information
» n=1.0;t=[1.1 2.2 3.3];
» whos
16. 16
MATLAB as a Calculator
MATLAB can be used as an expression evaluator.
For example, to “evaluate” pi type
>> pi
ans = ?
x=pi % the value of pi is assigned to variable x
Try the following examples:
>> r = sqrt(9)
r = ?
>> m = sin(pi/2)
m = ?
17. 17
Arithmetic Operators in MATLAB
The contents of all bracketed expressions are evaluated, starting from the
innermost brackets and working out.
All exponentials are evaluated, working from left to right.
All multiplications and divisions are evaluated, working from left to right.
All additions and subtractions are evaluated, working from left to right.
Operators:
+ for addition
- For subtraction
* for multiplication
/ for division
^ for power
18. 18
Operator Precedence
Precedence is the order in which operations are performed.
(Highest) Parenthesis . anything in parenthesis is done first
Exponentiation
Multiplication, division
(Lowest) Addition, subtraction
Spaces are ignored
Solve the following examples:
>> 2 + 3 * 4
>> 2 * 3 ^ 2
>> (2 + 3) * 4
>> (2^(2 + 3)) * 4
19. 19
Constants and Functions
>> eps
ans = 2.2204e-016
>> sin(pi/2)
ans = 1
>> log(1000)
ans = 6.9078
>> log10(1000)
ans = 3
20. 20
Elementary Math Functions
>> help elfun
Elementary math functions.
Trigonometric.
sin - Sine.
sinh - Hyperbolic sine.
asin - Inverse sine.
asinh - Inverse hyperbolic sine.
cos - Cosine.
cosh - Hyperbolic cosine.
acos - Inverse cosine.
acosh - Inverse hyperbolic cosine.
tan - Tangent.
tanh - Hyperbolic tangent.
21. 21
Elementary Math Functions
exp Exponential (e^x).
log Natural logarithm.
log10 Common (base 10) logarithm.
log2 Base 2 logarithm and dissect floating point number.
sqrt Square root.
Exponential function examples:
>> exp(log(3))
>> log(exp(3))
>> log2(32)
>> log2(pow2(5))
22. 22
Elementary Math Functions
Rounding and remainder
fix Round towards zero.
floor Round towards minus infinity.
ceil Round towards plus infinity.
round Round towards nearest integer.
mod Modulus (signed remainder after division).
rem Remainder after division.
sign Signum.
24. 24
Mathematical Functions in MATLAB
Some of the Matlab built-in functions are:
abs(x) for absolute value
sqrt (x) for square root
sin(x) for sine
cos(x) for cosine
tan(x) for tangent
cot(x) for cotangent
log(x) for natural logarithm
exp(x) for exponentioal
atan(x) for inverse tangent or arctan
sinh(x) for sine hyperbolic
25. 25
disp Function
Used to display numbers and strings.
The values displayed at the screen don’t necessarily include all the
information.
It is possible to change the display format.
Examples
>> disp(3) 3
>> disp('abc') abc
>> format long
>> pi ans = ?
>> format short
>> pi ans = ?
26. 26
fprintf Function
Advanced version of the disp dunction
Displays text and variables simultaneously
C++ principles based
Syntax: fprintf(‘text %var1_place_type text’, var1_name)
Example:
a=65;
b=97;
fprintf('the sum of %i and %i is %i',a,b,a+b)
fprintf('the ASCII code of %s is %i',a,a)
27. 27
fprintf arguments
Type of variables:
%i: integer
%d: double
%f: float
%c: char
%s: string
Possibility to control the number of decimal places
Ex: fprintf(‘a with three decimal places: %.3f',a)
Ex: fprintf('a with 4 numbers: %.4i',a)
All the text in the same line? Use n for a new line
28. 28
Drawing in MATLAB
Possibility to draw figures and curves in MATLAB
The x axis limit need to be specified most of the time
ex: x= -6 : 0.1 : 6
Different plot types:
Plot: draw the curve of an equation that depends on a variable
Ezplot: plot with a predefined range of [-2π, 2π]
Subplot: draw many plot in the same window
Figure: create a separate window for a new plot
Ex: syms x
ezplot(sin(x))
figure
ezplot(cos(x))
29. 29
plot Command Syntax
Plot(x,y,’r’): plot x in function of y with the ‘r’ (red style)
‘r’: red ‘b’:blue ‘black’: black
‘x’: cross ‘o’: circle ‘.’: Points
plot(x,y1,'--',x,y2,'-',x,y3,':') plot multiple plots
xlabel(‘name'): give a name to X-axis
ylabel(‘name'): give a name to Y-axis
legend(‘y1',‘y2',‘y2') display each plot’s legend
title(‘name') display the plot title
axis([0 2*pi -3 3]): define the X & Y axis limits
hold on multiple displays in 1 plot
30. 30
In Matlab, we can plot a symbolic function over one variable by using the
ezplot function.
Plotting Symbolic Function
>> syms a b c x % define symbolic math variables
>> y = sin(x)
>> ezplot(y)
>> f = sin(x);
>> ezsurf(f) % another type of plotting example
>> g = cos(y);
>> ezsurf(f+g);
31. 31
Draw multiple independent plots in the same window
Syntax: subplot(x,y,id)
subplot Command
syms x
subplot(2,2,1)
ezplot(cos(x))
subplot(2,2,2)
ezplot(sin(x))
subplot(2,2,3)
ezplot(tan(x))
subplot(2,2,4)
ezplot(atan(x))
33. 33
Script and Functions: Script Files
A user-created file with a sequence of MATLAB commands in it.
A text file containing script or function or program to run
The file must be saved with a ‘.m’ extension to its name, thereby, making
it an M-file.
Executed by typing its name (without the ‘.m’ extension) at the command
prompt.
34. 34
Create a .m File
From the File menu, select New M-File.
→
A new edit window should appear.
35. 35
Example 1:
in Command window
>> example1
C = ?
Example 1
in m-file:
A= 2;
B=3;
C=A+B
Save as a file name example1.m
36. 36
Find the force if the mass is 10 Kg and the acceleration is 100m/sec2.
Newton’s Law: F = m .γ (Force = mass x acceleration) (γ = 100)
Example 2
Create a new script (example2.m)
m = 10;
gamma = 100;
Force =m * gamma
To run the above file, type in the command window
>>example2
37. 37
Script and Functions: Function Files
A m-file also , like a script file, except that the variables are all local.
A function file begins with a function definition line containing a well-
defined list of inputs and outputs.
Without that line, the file becomes a simple script file.
The syntax of the function definition line is:
function [ output variables ] = function_name ( input variables );
It is recommended for the function_name to be the same as the
filename (without the ‘.m’ extension) in which the function is written.
Example: if the name of the function is projectile it must be written and
saved in a file with the name projectile.m.
38. WARNING!
for the scipt or function name
file, avoid using the name of a
MATLAB command such as plot,
sqrt, abs…
39. 39
Write a MATLAB function called sumy that adds two input numbers and
returns the result minus 2
Example 3
function out = sumy(x,y)
out=x+y-2
Save with filename sumy.m , then type in the command window:
>>sumy(6,5) % 6 and 5 are arbitrary values chosen by the user
out = ?
40. 40
Solution
Function file:
function [area, volume] = cyl(h, r)
% function to compute the area and volume of a cylinder
base = pi * r^2;
volume = base * h
area = 2*pi*r * h + 2*base
Store the function in cyl.m, then, in
command window:
>> cyl(6, 4)
area = ?
vol = ?
Example 3: Measuring a Solid Object
We need to compute the volume and
surface area of this cylinder object:
Formulas: b: base, h: height
Base: b= π r²
Volume = b h
Area=2 π r h + 2 b
42. 42
Create a m-file and enter the following
statements in the m-file:
A = [1 2 3; 3 3 4; 2 3 3];
b = [1; 1; 2];
x = Ab
Save the file, (e.g. example1.m)
In the command line, type:
>> example1
x = ?
Example 1:
Consider the system of equations:
x + 2y + 3z = 1
3x + 3y + 4z = 1
2x + 3y + 3z = 2
Find the solution x to the system of
equations.
43. 43
Create a file, say example2.m, which
contains the following commands:
x = 0:pi/100:2*pi;
y1 = 2*cos(x);
y2 = cos(x);
y3 = 0.5*cos(x);
plot(x,y1,'--',x,y2,'-',x,y3,':')
xlabel('Range')
ylabel('Cosine functions')
legend('2*cos(x)','cos(x)','0.5*cos(x)')
title('Example of multiple plots')
axis([0 2*pi -3 3])
Run the file by typing example2 in the
Command Window.
Example 2:
Plot the following cosine functions, y1
= 2 cos(x)
y2 = cos(x)
y3 = 0.5 * cos(x)
in the interval 0 x 2.
44. 44
game1 = input('Enter the points
scored in the first game ');
game2 = input('Enter the points
scored in the second game ');
game3 = input('Enter the points
scored in the third game ');
average = (game1+game2+game3)/3
The following shows the command
prompt when this script file (saved as
example3) is executed.
>> example3
>> Enter the points scored in the
first game xx
>> Enter the points scored in the
second game yy
>> Enter the points scored in the
third game zz
average = ?
Example 3:
Create a script file that calculates the
average of points scored in three
games.
The point from each game are
assigned to a variable by using the
’input’ command.
45. 45
Create a script in a file computes area of triangle
Example 4
b=5;
h=3;
area=0.5*(b*h)
area = ?
46. 46
Scripts VS Functions
To calculate the area of another triangle using the same script, it is
possible to update the values of b and h in the script and re-run it. the
script stores the result each time in a variable named a that is in the
base workspace.
Instead of manually updating the script each time, it is possible to make
a more flexible program by converting the script to a function.
Replace the statements that assign values to b and h with a function
declaration statement.
The declaration includes
the function keyword
the names of input and output arguments
the name of the function.
47. 47
Redo example 4 using the function method
Example 5
function a = triarea(b,h)
a = 0.5*(b.* h);
After saving the file, call the function with different values
a1 = triarea(1,5)
a1 = ?
a2 = triarea(2,10)
a2= ?
a3 = triarea(3,6)
a3= ?
48. Functions have their own workspace
(local), separate from the base
workspace (global). So, none of the
calls to the function triarea overwrite
the value of a in the base workspace.
Instead, the function assigns the
results to variables a1, a2, and a3.
49. 49
HOMEWORK 1
Page no: 28 problem no:9&11
Page no: 30 problem no:21
Page no: 31 problem no:27
Page no: 164 problem no:6
Submit- next week class
52. 52
The for loop
The general form of for loop is:
for i=beginning :step: final
program statements
end
Where beginning, step, and final are integers or real numbers.
Example 1:
Find the sum of the integers from 5 to 100.
53. 53
Solution
Open an m file (example1.m)
sum=0;
for i=5:100
sum=sum+i;
end
disp(‘sum’)
disp(sum)
To run the above file, type in the command window
>>example1
54. 54
Find the sum of the odd integers between 1 and 100.
Example 2
sum=0;
for i=1:2:100
sum=sum+i;
end
disp(‘sum of odd integers between 1 and 100 is:’);
disp(sum)
55. 55
Find the sum of the first 10 terms as below.
Example 3
sum=0;
for i=1:10
sum=sum+1/i;
end
disp(‘sum of the 10 terms is:’);
disp(sum)
10
1
9
1
8
1
7
1
6
1
5
1
4
1
3
1
2
1
1
56. 56
Functions (reminder)
The general form of a function in MATLAB is
function [rv1 rv2 … rvn]=Function_Name(pv1,pv2,..,pvn)
where
rv1, rv2,.. are the outputs
pv1, pv2,.. are the inputs.
The body of the function will contain the relations between the
outputs and inputs.
57. 57
Write a function “sumN” that finds the sum of the first N term of the series
below.
Example 4
function s=sumN(N)
s=0;
for i=1:N
s=s+1/i;
end
To run this function code, type in the command window:
>>sumN(5);
N
1
....
8
1
7
1
6
1
5
1
4
1
3
1
2
1
1
58. 58
Write a function “expN” that finds the sum of the first N term of the series
below for a given value of x.
Example 5
function s=expN(N,x)
s=0;
for i=0:N
s=s+x^i/factorial(i);
end
In the command window:
>>p=expN(5,1);
N
i
i
i
x
0 !
60. 60
if statement
Conditional statement are used to perform certain task if certain
conditions are met or not
The general structure of if statement is:
if expression
statements
elseif expression
statements
else
statements
end
62. 62
n1=input('enter number1 ');
n2=input('enter number2 ');
if n1>n2
disp('n1>n2')
elseif n1<n2
disp('n1<n2')
else
disp('the numbers are identical')
end
Example 2
Create a program that compares two
numbers inputted by the user and
display the comparison result
63. 63
nrows = 6;
ncols = 6;
A = ones(nrows,ncols)
for c = 1:ncols
for r = 1:nrows
if r == c
A(r,c) = 2;
elseif abs(r-c) == 1
A(r,c) = -1;
else
A(r,c) = 0;
end
end
end
A
Example 3
1-Create a matrix and fill it with 1
2-Loop through the matrix and assign
each element a new value:
2 on the main diagonal
-1 on the adjacent diagonals
0 everywhere else.
64. 64
syms hw pr mt et total
nbr=input('enter the number of student ')
for i=1:nbr
fprintf('student %in',i)
hw(i)=input('enter the homework grade ');
pr(i)=input('enter the project grade ');
mt(i)=input('enter the midterm exam grade ');
et(i)=input('enter the endterm exam grade ');
end
for i=1:nbr
moy(i)=hw(i)+pr(i)+mt(i)+ et(i);
end
for i=1:nbr
if moy(i)<60
fprintf('student %i got %d,
so he has failedn',i,moy(i))
else
fprintf('student %i got %d, so he
has passedn',i,moy(i))
end
end
Example 4
Create a program that collects the n
students’ grades (Homework, Project,
mid-term & End-term) and
determines if the student has passed
or failed the course
65. Homework 2
Redo the previous example
while taking all possible grades
(F, D, D+, C, C+, B, B+, A & a+)
into account
68. 68
Matlab can also compute many integrals and derivatives that you might find
in Calculus or many advanced engineering courses.
The key functions are int for integration and diff for derivation.
Derivation and Integration
>> syms x;
>> f = sin(5*x)
>>diff(f)
ans = ?
69. 69
Derivation and Integration (2nd and Higher Derivative)
>> f = x^3
>> diff(f) % 1st derivative
ans = ?
>> diff(f,2) % 2nd derivative
ans = ?
>> diff(f,3) % 3rd derivative
ans = ?
>> diff(f,4) % 4th derivative
ans = ?
70. 70
Partial Differential Equations
Sometimes, an expression can have multiple variables.
To compute the derivative of the function with respect to one variable,
use a second parameter to the diff function:
>> syms x t;
>> f = sin( x * t )
>> diff(f,t) % derivative of f with respect to t
ans = ?
71. 71
By using the int function, in the same way we use the diff function, we can
ask Matlab to do symbolic integration for us.
Integration
>> syms x t;
>> f = x * x ;
>>int(f)
ans = ?
72. 72
As you (should) know, a definite integral represents the area under a curve from
a given start point to a given end point.
Try to find how much area is under the curve f(x) = x^3; for [0..10], [-10..10], [-
10..0]
Definite Integrals
>> syms x;
f= x^3
>> int(x^3,0,10)
ans = ?
>> int(x^3,-10,10)
ans = ?
>> int(x^3,-10,0)
ans = ?
73. MATLAB can find the derivative of symbolic
expressions. Higher derivatives can also be
found easily. These derivatives can be
evaluated for arbitrary values of the
independent variable.
74. 74
Enter:
syms x
g = x^2*cos(x)
Now enter:
diff(g, x)
Then enter:
diff(g, x, 2)
How would you calculate the third derivative?
Example:1
First we declare x to be a symbolic
variable and define g to be the
function given by g(x) = x2
cos(x).
75. 75
Example 2
To calculate the derivative of an expression that you have not yet
entered, just replace g by the expression. For example, enter:
diff(x^3-x^2+2, x)
Now insert a symbolic constant a into the expression. Enter:
syms a
diff(x^3 - a*x^2 + 2, x)
Then change the final x to an a. That is, enter:
diff(x^3 - a*x^2 + 2, a)
What is the role of the symbol after the comma in the differentiation
expression?
76. 76
Example 3
Evaluate the derivative dg/dx at x = 2.1.
Enter:
gp=diff(g, x)
Then enter:
subs(gp, x, 2.1)
Alternatively, you could enter:
subs( diff(g, x), x , 2.1 )
77. Check your understanding so far by using
MATLAB to calculate the second derivative of
tan(x6
- 3x + 5) at x = 3/2.
78. 78
Calculate the derivative of the function F(x)=x3
- cos(x)
Example 4
>> syms x
>> f=x^3-cos(x);
>> g=diff(f)
g = ?
79. 79
Obtain the derivative of the following function according to y:
F(x,y)=x² +(y+5)3
Example 5
Matlab command entries:
>> syms x y
>> f=x^2+(y+5)^3;
>> diff(f,y)
ans = ?
Note that in this case, the command diff(f,y) is equivalent to
80. 80
Find the derivative of f(x)=sin(ex
).
Example 6
>> syms x
>> f=sin(exp(x));
>> diff(f)
ans = ?
81. 81
To compute the partial derivative of an expression with respect to some
variable we specify that variable as an additional argument ‘diff’
f(x,y)=x3
y4
+ysin(x)
Example 7
>> syms x y
>> f=x^3*y^4+y*sin(x)
>> diff(f,x)
ans = ?
>> diff(f,y)
ans = ?
82. 82
Taylor Series
The Taylor series is a representation of a given function as an infinite
sum of terms calculated from its derivatives values at a single point.
If the series is centered at zero, the Taylor series is Maclaurin series.
It is a common practice to use a finite number of terms to approximate
the function.
Taylor series of order 5 is an approximation of f(x) as a polynomial of
degree 5.
The general expression of Taylor series for an analytic function f(x) about
the base point x=a is:
fm
(a) is the mth
derivative of f(x) at the point a.
0 !
)
(
)
(
)
(
m
m
m
m
a
a
x
x
f
f
83. 83
Use the "taylor" function to create a taylor series
Taylor Command
>> syms x
>> f = taylor(log(1+x))
f = ?
84. 84
Taylor Series
Two methods for MATLAB to find the Taylor series expansion of f(x) at x=a
Method 1: Using the general Taylor series summation and programming
it using function and for loops as explained in the example 5.
Method 2: Using the MATLAB function : taylor, In general:
taylor(f,n,a) returns the Taylor series expansion of f(x) of order n-1 at
the base point x=a.
taylor(f,n) returns the Taylor series expansion of f(x) of order n-1 at
the base point x=0.
taylor(f,a) returns the Taylor series expansion of f(x) of order 5 at the
base point x=a.
Note: f must be a symbolic expression representing f(x)
85. 85
Example 8
Given a function f(x)=sin(x). Write a program that finds the Taylor series
expansion of f(x) at x=0 of degree 3.
Plot f(x) and its Taylor series approximation of degree 3 on the same
graph for x in the range [-pi,pi]
Add Taylor series of degrees 5, 7 to the same plot.
Which Taylor series better approximates f(x).
What is your conclusion?
86. 86
Solution (1/2)
Open a script file example6.m
syms x;
f=sin(x);
t3=taylor(f,4); %Taylor series at x=0 of degree 3
xd=-pi:pi/20:pi;
yd=subs(f,x,xd); %This evaluates f(x) at xd (symbolic).
td3=subs(t3,x,xd); %This evaluates t(x) at xd.
plot(xd,yd,’y’);
hold on;
plot(xd,td3,’m’);
87. 87
t5=taylor(f,6); %5th deg Taylor series at x=0
t7=taylor(f,8); %7th deg Taylor series at x=0
td5=subs(t5,x,xd);
td7=subs(t7,x,xd);
plot(xd,td5,’r’);
hold on;
plot(xd,td7,’b’);
Legend(‘exact’,’Taylor 3’,’Taylor 5’,’Taylor
7’);
xlabel(‘x’);
ylabel(‘function’)
Solution (2/2)
88. 88
syms x;
f=sin(x);
t3=taylor(f,4); %3rd deg Taylor series at x=0
f_inline=inline(char(f)); %f:symbolic function
t3_inline=inline(char(t3));
fplot(f_inline,[-pi ,pi],'b');
hold on;
fplot(t3_inline,[-pi, pi],'r');
Legend('exact','Taylor 3');
xlabel('x');
ylabel('function')
Example 8(Another Method)
89. 89
Find the following integration using Taylor series of order 6:
Example 9
syms x;
f=exp(x^2);
t6=taylor(f,7); %Taylor series at x=0 of degree 6
int(t6,-1,1) %Integration of t6
Ans= ?
dx
ex
1
1
2
90. 90
Example 9 Solution (2nd
Method) (1/3)
The series expansion of ex is known:
The series expansion of ex2
can then be found by replacing x with x²
The integration of ex
2
can then be found
0 !
i
i
x
i
x
e
0
2
0
2
!
!
)
(
2
i
i
i
i
x
i
x
i
x
e
)
(
)
!
)(
1
2
(
!
! 0
1
2
0
2
0
2
2
x
f
i
i
x
dx
i
x
dx
i
x
dx
e
i
i
i
i
i
i
x
)
1
(
)
1
(
1
1
2
f
f
dx
ex
91. 91
So, this method depends on writing the MATLAB function f(x) and finding f(1) -
f(-1)
Example 9 Solution (2nd
Method) (2/3)
function s=f(x)
s=0;
for i=0:100
s=s+ x^(2*i+1)/((2*i+1)*factorial(i));
end
In the command window, type:
>> int_result=f(1)-f(-1)
int_result=?
92. 92
In the previous example, plot f(x) and its Taylor
approximation t6(x) on the same graph for -1 ≤ x ≤ 1
syms x;
f=exp(x^2);
xd=-1:0.1:1;
yd=subs(f,xd);
plot(xd,yd,’r’);
hold on;
t6=taylor(f,7); %Taylor of degree 6
td=subs(t6,xd);
plot(xd,td,’b’);
xlabel(‘x’);
ylabel(‘functions’);
legend(‘f(x)’,’Taylor’)
Example 9 Solution (2nd
Method) (3/3)
93. 93
Find the taylor series expansion of order 3 about the point x=1 of the function
Example 10
syms x;
f = 1/(1+x);
t3 = taylor(f,4,1)
T3 = ?
x
x
f
1
1
)
(
94. 94
Find the taylor series expansion of order 3 about the point x=0 of the function
Example 11
syms x;
f=1/(1+x);
t3=taylor(f,4)
t3 = ?
x
x
f
1
1
)
(
95. Chapter 4: Polar Coordinates in MATLAB &
Integration Using Polar Coordinates
Slim Chtourou
Lecturer
slim.chtourou@ieee.org
www.slim.fr.mu
97. 97
Every point can be expressed in:
Rectangulat coordinates: (x,y)
Polar coordinates: (r,θ)
Polar and Rectangular Coordinates
98. 98
[x,y] = pol2cart(theta,r)
[theta,r] = cart2pol(x,y)
angle comes first
Converting Between Polar And Cartesian in Matlab
>>[x,y] = pol2cart(pi/4,1)
x = ?
y = ?
>>[theta,r] = cart2pol(0,-4)
theta = ?
r = ?
100. 100
Polar Command in Matlab
The Matlab command:
>> polar (theta,r,s);
Plots the points defined in polar coordinates as (r,theta) for all values of r
and theta
s is a string that defines the style of the plot
Ex: ‘y’: yellow | ‘r’: red
Reminder: theta must be in radians
101. 101
Use MATLAB to plot the following points in polar coordinates:
r=5, θ=0° | r=5, θ=60° | r=3, θ=180° | r=3, θ=300° | r=3, θ=-30°
Example 1
Solution:
Create a script file: example1.m
>> r=[5 5 3 3 3];
>> theta=[0 60 180 300 -30]*pi/180; %Convert to radians
>> polar(theta , r, 'x');
105. 105
Solution:
>>r=5;
>>theta=53*pi/180; %convert to radians
>>x = r * cos(theta);
>>x = r * sin(theta);
Answer:
x=3
y=4
Example 3
Find the rectangular coordinates of
the point (5,53°) using Matlab as a
calculator
106. 106
Solution:
Open a script file example1d.m
>>r=[5 5 3 3 3];
>>theta=[0 60 180 300 -30]*pi/180;%to rad
>>polar(theta,r,’x’);
>>hold on;
>>x=r .* cos(theta); % from polar to rectangular
>>y=r .* sin(theta); % from polar to rectangular
>>plot(x,y, ‘o’);
Example 3
Repeat Example 1 and plot the given
points using the polar to rectangular
conversion relation. Add the two
examples on one plot.
108. 108
Using Matlab, draw the curve r = sin(θ)
Example 5
Solution:
Open a script file example5.m
theta=linspace(0,2*pi,200); %generates 200 points between 0 and 2π
r=sin(theta);
polar(theta,r,’r’);
110. 110
Using Matlab, draw the curve r = cos(2θ)
Example 6
Solution:
Open a script file example6.m
>>theta=linspace(0,2*pi,200); %generates 200 points between 0 and 2π
>>r=cos(2*theta);
>>polar(theta,r,’r’);
112. 112
Area in Polar Coordinates (from calculus II)
The area bounded by the curves r1(θ) , r2(θ), θ = θ1 and θ = θ2 is given by:
θ1 and θ2 must be in radians
Example 7:
Find the area bounded by the curves r = 2 sinθ, θ1 = 45°, θ2 = 135° and the
origin
Show all curves in one plot
d
r
r )
(
2
1 2
1
2
2
2
1
Area
114. 114
Solution (2/2)
2-form the integration and put the limits:
3-Go to command window of MATLAB and do the integration using int:
>>syms theta;
>>area=eval(0.5 * int(4 * sin(theta)^2 , pi/4 , 3*pi/4))
Answer:
area=?
d
)
0
sin
4
(
2
1 4
3
4
2
Area
d
r
r )
(
2
1 2
1
2
2
2
1
Area
115. 115
Determine the area of the region that lies inside
r= 3 +2 sinθ and r=2, show the region
Solution
1-plot the two curves:
Open example8.m
>>theta=linspace(0,2*pi)
>>r=2*ones(size(theta));
>>polar(theta,r,’r’);
>>hold on;
>>r=3+2*sin(theta);
>>polar(theta,r);
Example 8
116. 116
Solution (2/2)
2-form the integration and put the limits:
3-Go to command window of MATLAB and do the integration using int:
>>syms theta;
>>area=eval(0.5*int((3+2*sin(theta))^2-4,-pi/6,7*pi/6))
Answer:
area=?
d
Area ]
4
)
sin
2
3
[(
2
1 6
7
6
2
d
Area r
r )
(
2
1 2
1
2
2
2
1
118. Outline
Definite and Indefinite Integrals
Integral Pairs
Closed Form Integrals
Int Command
Closed Form Example
Multiple Independent Variable Example
Numerical Integration
quad Command
Numerical Integration Example
Closed Form and Numerical Integration Example
119. 119
Definite & Indefinite Integrals
A definite integral represents the area under a curve f(x) between the
lower limit ‘a’ and the upper limit ‘b’
An integral is considered to be indefinite if the upper and lower limits
are not specified
b
a
DEFINITE dx
x
f
I )
(
dx
x
f
IINDEFINITE )
(
120. 120
Integral Pairs
Some indefinite integrals can be considered of as the inverse of
differentiation
Common examples:
Note: due to initial conditions, arbitrary integral pairs are not unique
and may differ by a constant of integration, c
c
ax
adx
c
n
ax
dx
ax
n
n
1
1
c
a
e
dx
e
ax
ax
c
b
bx
a
dx
bx
a
)
sin(
)
cos(
c
b
bx
a
dx
bx
a
)
cos(
)
sin(
c
x
a
dx
x
a
)
ln(
121. 121
Closed Form Integrals
Closed form integrals can be expressed as explicit functions.
The integral pairs on the previous slide are examples of closed form
integrals.
Techniques such as partial fraction expansion, integration by parts, and
integration by substitution can be used to express some integrals in
closed form.
It is not always possible to find the closed form for the integral of an
arbitrary function
122. 122
Evaluation of Definite Integral
If a closed form indefinite integral exists, it can be used to evaluate a
definite integral over a region of integration
An integral from an arbitrary lower limit to a fixed upper limit is denoted
as:
Thus definite integral from a lower limit a to an upper limit b can be
evaluated using:
x
du
u
f
x
F )
(
)
(
b
x
x
a
b
a
dx
x
f
dx
x
f
dx
x
f
0
0
)
(
)
(
)
(
b
a
b
a
x
F
a
F
b
F
dx
x
f )
(
)
(
)
(
)
(
123. 123
int command
The int command is used to solve integrals which can be expressed in
closed form
int(s) returns the indefinite integral of the symbolic variable s with
respect to its default symbolic variable
int(s,v) returns the indefinite integral of the symbolic variable s with
respect to the symbolic variable v
int(s,a,b) returns the definite integral of the symbolic variable s with
respect its default symbolic variable from a to b
int(s,v,a,b) returns the definite integral of the symbolic variable s with
respect to the symbolic variable v from a to b
124. 124
Closed form Example
Given the function, f(x):
1- Use the int command to determine the Closed Form
Indefinite Integral, F(x):
2- From the closed form integral, F(x), determine the definite
integral of f(x) from 0 to p/2
3- Use the int command to directly determine the definite
integral of f(x) from 0 to p/2
4- Determine and plot a function A(z) representing the
general area under f from 0 to any arbitrary point z
)
2
sin(
)
( x
x
f
2
/
0
2 )
(
dx
x
f
A
dx
x
f
x
F )
(
)
(
z
dx
x
f
z
A
0
)
(
)
(
125. 125
Indefinite Closed form
Matlab can be used to verify the integral pair shown on a previous slide
The integration variable parameter does not need to be specified since x
is the only variable in the expression, hence it will be the default variable
of integration
» syms x
» f=(sin(2*x));
» F=int(f)
F =
sin(x)^2 - 1/2
)
2
sin(
)
( x
x
f
2
1
sin
)
(
)
( 2
x
x
f
x
F
126. 126
Definite Integral Evaluation
The definite integral can be evaluated from the
indefinite integral using the relationship:
Considering the previous function:
» Fb=subs(F,'x',pi/2)
Fb = ?
» Fa=subs(F,'x',0)
Fa = ?
» A_pidiv2 = Fb - Fa
A_pidiv2 = ?
)
(
)
(
)
( a
F
b
F
dx
x
f
b
a
?
)
2
/
(
F
1
)
2
sin(
2
/
0
2
dx
x
A
2
1
sin
)
( 2
x
x
F
?
)
0
(
F
127. 127
Since f(x) has a closed form, the int command can be used to directly
determine the definite integral
Matlab Direct Evaluation
» f=(sin(2*x));
» F_definite=int(f,0,pi/2)
F_definite =
?
1
)
2
sin(
2
/
0
2
/
dx
x
A
128. 128
A general function, A(z), used to plot the area under the curve f(x) to an
arbitrary point z can be determined as follows:
Area Under Curve
» syms z
» F_z = subs(F,x,z);
» F_0 = subs(F,x,0);
» A_z = F_z - F_0
A_z = ?
» A_z_pidiv2=subs(A_z,z,pi/2)
A_z_pidiv2 = ?
129. 129
Multiple Independent Variables Example
Given the function f(t,x,y):
Use Matlab to determine the closed
form integrals with respect to the
different independent variables:
1-Integrate f with respect to t :
2-Integrate f with respect to x :
3-Integrate f with respect to y :
txy
tx
t
y
x
t
f 3
2
)
,
,
(
dt
y
x
t
f
t
F )
,
,
(
)
(
dx
y
x
t
f
x
F )
,
,
(
)
(
dy
y
x
t
f
y
F )
,
,
(
)
(
130. 130
Integrating with respect to the default independent variable will integrate
with respect to x
Default Integration
» syms t x y
» f=sym('t + 2*t*x + 3*x*y*t');
» Fx = int(f)
Fx = ?
131. 131
Integration of f(t,x,y) with respect to t or y must be explicitly specified as an
input argument to int
Other Independent Variables
» Ft=int(f,t)
Ft = ?
» Fy=int(f,y)
Fy = ?
132. 132
Numerical Integration
Some functions do not have closed form integrals
A definite integral can be numerically approximated provided that the
function is defined over the region of interest
Numerical integration is performed by:
Dividing the integration region into very small regions
Approximating the area in each region
Summing the areas.
If the length of the subintervals tends to 0 and the sum of areas tends to
a unique limit, I, the definite integral over the interval is I
133. 133
quad Command
The quad command is used to numerically evaluate an integral
Q = quad('f',a,b) approximates the integral of f(x) from a to b
'f' is a string containing the name of an m-function to be integrated.
Function f must return a vector of output values when given an input
vector.
Q = Inf is returned if an excessive recursion level is reached, indicating a
possibly singular integral.
Typically, a new m-function needs to be created for f when numerically
evaluating an integral
134. 134
Numerical Integration Example
Given the function, f(x):
and The definite integral, F:
1-Use the symbolic toolbox to verify that the f(x) does not have a closed
form indefinite integral
2-Plot f(x) to ensure that the function is continuously defined over the
integration region
3-Numerically integrate f(x) to determine F
2
)
sin(
)
( x
e
x
x
f
2
0
)
(
dx
x
f
F
135. 135
Solution (1/3): No Closed form
Verify the closed form absence using Matlab’s symbolic toolbox:
» syms x;
>>f_sym = sin(x)*exp(-x^2);
» int(f_sym)
ans = ?
136. 136
The plot confirms that
f(x) is continuous over
the integration region
Numerical integration
is possible
» ezplot(f_sym)
» grid on
Solution (2/3): Continuous Plot
2
0
Continuous
137. 137
Solution (3/3): Numerical Integration
Must numerically evaluate the integral using the quad command which
requires a function as an input
Create a new m-function describing the function:
The function must return a vector so be sure to use .* and .^ notations
where appropriate
function f = f_sin_exp_sqr( x )
f=sin(x).*exp(-x.^2);
2
)
sin(
)
( x
e
x
x
f
138. 138
Use the quad command to perform the numerical integration and evaluate
the definite integral
Solution (3/3): Numerical Integration
» F_num_int = quad('f_sin_exp_sqr',0,pi/2)
F_num_int =
?
139. 139
Closed Form and Numerical Integration Example
Given the function f(x):
The definite integral, F:
1-Plot f(x) to ensure that the function is continuously defined over the
integration region
2-Use the symbolic toolbox to evaluate the definite integral
3-Numerically integrate f(x) to determine the definite integral
)
(
tan
)
( 1
x
e
x
f x
4
0
)
( dx
x
f
F
140. 140
Use ezplot to plot the
function
» syms x
» f_sym=exp(x)+atan(x);
» ezplot(f_sym)
» grid on
Continuity confirmed in
the x = [0,4] interval
Solution (1/3): Continuous Plot
141. 141
Use the symbolic toolbox int command to evaluate definite integral from the
closed form integral
Solution (2/3): Symbolic Evaluation
» defint_sym=int(f_sym,0,4);
defint_sym =
?
>>defint_dbl=double(f_sym)
defint_dbl =
?
142. 142
Solution (3/3): Numerical Evaluation
Create a new m-function to represent f(x):
function f = f_exp_atan( x )
f=exp(x)+ atan(x);
Use quad command to numerically evaluate the definite integral and
verify the previous symbolic result
» defint_numint=quad('f_exp_atan',0,4)
defint_numint = ?
Small difference to numerical approximation
143. Summary
Definite integrals are evaluated over a continuous
interval and result in a number
Closed form indefinite integral functions exist for
some functions independent of the integration
limits
The definite integrals can be evaluated from the
closed form indefinite integral if it exists
If a closed form indefinite integral does not exist,
the definite integral of continuous functions can
still be numerically approximated
144. Chapter 6: Symbolic Math in MATLAB
Slim Chtourou
Lecturer
slim.chtourou@ieee.org
www.slim.fr.mu
146. 146
The factor(f) command factors f into polynomial products
factor Command
» syms x
» f=x^3+4*x^2-11*x-30
» y = factor(f)
y = ?
» y = factor(x^5-1)
y =
(x-1)*(x^4+x^3+x^2+x+1) )
1
)(
1
(
1 2
3
4
5
x
x
x
x
x
x
147. 147
collect Comand
The collect command collects coefficients of a symbolic expression and
rewrites it as powers of a polynomial
collect(s,v)
s is a symbolic expression matrix
v is the independent polynomial variable
If v is omitted, collect uses rules to determine a default variable
148. 148
» syms x t
» f=(1+x)*t+x*t
f_col_x = collect(f,x)
f_col_x = ?
» f_col_t = collect(f,t)
f_col_t = ?
» f_col = collect(f)
f_col = ?
Examples:
-Create symbolic expression
f(x,t)=(1+x)t+xt
-Specify collecting the x terms
-Specify collecting the t terms
-Unspecified independent
variable collects the x variable
149. Part 2: Solving Algebraic Equations
with Multiple Variables
150. Outline
Solve Command for Equations of Multiple
Variables
Single Equation of Two Variables Example
Two Variable Complex Solution Example
System of Equations Example
151. 151
Solve Command for Equations of Multiple Variables
The solve command can be used to solve symbolic expressions with
more than one variable:
solve(f) solves the symbolic expression f using the symbolic variable
closest to x as the independent variable
solve(f, v) solves the symbolic expression f in terms of the independent
variable v
152. 152
Single Equation of Two Variables Example
Given the equation
1- Find the solution of the equation assuming that c is the solution variable
2- Find the solution of the equation assuming that b is the solution variable
0
2
4
2
b
c
b
153. 153
Solve for b as the independent
variable
b must be explicitly specified on the
command line
sol_b=solve(f,b)
sol_b =
[ -1+(1-4*c)^(1/2)]
[ -1-(1-4*c)^(1/2)]
» syms b c
» f=b^2+4*c+2*b;
» sol_c=solve(f)
sol_c =
-1/4*b^2-1/2*b
0
2
4
2
b
c
b
2
4
2
b
b
c
c
b
c
b
4
1
1
4
1
1
2
1
154. 154
Two Variable Complex Solution Example
Given the equation of two variables:
x2
+ 9y4
= 0
Solve for x in terms of y using Matlab:
» syms x y
» xs=solve(x^2+9*y^4)
xs =
[ 3*i*y^2]
[ -3*i*y^2]
Roots are
Complex Conjugates
155. 155
Matlab Command window:
>> syms x
>> f=2*x^2 + 4*x -8;
>> solve(f,x)
ans = ?
Alternately, you may use the following lines in
Matlab to perform the same calculation:
>> f=[2 4 -8];
>> roots(f)
ans = ?
Are the results the same?
Example: finding roots
Solve the following equation with 2
methods:
F(x)=2x² + 4x - 8
Editor's Notes
#5:It’s name is derived from MATrix LABoratory.
Matlab is also a programming language that currently is widely used as a platform for developing tools for Machine Learning
#14:a b x y
Name Size Bytes Class
a 1x1 126 sym object
b 1x1 126 sym object
n 1x1 8 double array
t 1x3 24 double array
x 1x1 126 sym object
y 1x1 126 sym object
Grand total is 12 elements using 536 bytes
![14
Defining Symbolic Variables & Viewing Variables
Use sym to create a symbolic variable x:
Use syms to create several symbolic variables at one time
Use who to view all variables in the workspace
» x=sym('x');
» syms y a b
» who
Your variables are:
Use whos to view all workspace variables with their associated size,
bytes, and class information
» n=1.0;t=[1.1 2.2 3.3];
» whos](https://image.slidesharecdn.com/matlab-250507081126-e4e4ebd8/85/matlab-presentation-fro-engninering-students-14-320.jpg)
![28
Drawing in MATLAB
Possibility to draw figures and curves in MATLAB
The x axis limit need to be specified most of the time
ex: x= -6 : 0.1 : 6
Different plot types:
Plot: draw the curve of an equation that depends on a variable
Ezplot: plot with a predefined range of [-2π, 2π]
Subplot: draw many plot in the same window
Figure: create a separate window for a new plot
Ex: syms x
ezplot(sin(x))
figure
ezplot(cos(x))](https://image.slidesharecdn.com/matlab-250507081126-e4e4ebd8/85/matlab-presentation-fro-engninering-students-28-320.jpg)
![29
plot Command Syntax
Plot(x,y,’r’): plot x in function of y with the ‘r’ (red style)
‘r’: red ‘b’:blue ‘black’: black
‘x’: cross ‘o’: circle ‘.’: Points
plot(x,y1,'--',x,y2,'-',x,y3,':') plot multiple plots
xlabel(‘name'): give a name to X-axis
ylabel(‘name'): give a name to Y-axis
legend(‘y1',‘y2',‘y2') display each plot’s legend
title(‘name') display the plot title
axis([0 2*pi -3 3]): define the X & Y axis limits
hold on multiple displays in 1 plot](https://image.slidesharecdn.com/matlab-250507081126-e4e4ebd8/85/matlab-presentation-fro-engninering-students-29-320.jpg)
![37
Script and Functions: Function Files
A m-file also , like a script file, except that the variables are all local.
A function file begins with a function definition line containing a well-
defined list of inputs and outputs.
Without that line, the file becomes a simple script file.
The syntax of the function definition line is:
function [ output variables ] = function_name ( input variables );
It is recommended for the function_name to be the same as the
filename (without the ‘.m’ extension) in which the function is written.
Example: if the name of the function is projectile it must be written and
saved in a file with the name projectile.m.](https://image.slidesharecdn.com/matlab-250507081126-e4e4ebd8/85/matlab-presentation-fro-engninering-students-37-320.jpg)
![40
Solution
Function file:
function [area, volume] = cyl(h, r)
% function to compute the area and volume of a cylinder
base = pi * r^2;
volume = base * h
area = 2*pi*r * h + 2*base
Store the function in cyl.m, then, in
command window:
>> cyl(6, 4)
area = ?
vol = ?
Example 3: Measuring a Solid Object
We need to compute the volume and
surface area of this cylinder object:
Formulas: b: base, h: height
Base: b= π r²
Volume = b h
Area=2 π r h + 2 b](https://image.slidesharecdn.com/matlab-250507081126-e4e4ebd8/85/matlab-presentation-fro-engninering-students-40-320.jpg)
![42
Create a m-file and enter the following
statements in the m-file:
A = [1 2 3; 3 3 4; 2 3 3];
b = [1; 1; 2];
x = Ab
Save the file, (e.g. example1.m)
In the command line, type:
>> example1
x = ?
Example 1:
Consider the system of equations:
x + 2y + 3z = 1
3x + 3y + 4z = 1
2x + 3y + 3z = 2
Find the solution x to the system of
equations.](https://image.slidesharecdn.com/matlab-250507081126-e4e4ebd8/85/matlab-presentation-fro-engninering-students-42-320.jpg)
![43
Create a file, say example2.m, which
contains the following commands:
x = 0:pi/100:2*pi;
y1 = 2*cos(x);
y2 = cos(x);
y3 = 0.5*cos(x);
plot(x,y1,'--',x,y2,'-',x,y3,':')
xlabel('Range')
ylabel('Cosine functions')
legend('2*cos(x)','cos(x)','0.5*cos(x)')
title('Example of multiple plots')
axis([0 2*pi -3 3])
Run the file by typing example2 in the
Command Window.
Example 2:
Plot the following cosine functions, y1
= 2 cos(x)
y2 = cos(x)
y3 = 0.5 * cos(x)
in the interval 0 x 2.](https://image.slidesharecdn.com/matlab-250507081126-e4e4ebd8/85/matlab-presentation-fro-engninering-students-43-320.jpg)
![56
Functions (reminder)
The general form of a function in MATLAB is
function [rv1 rv2 … rvn]=Function_Name(pv1,pv2,..,pvn)
where
rv1, rv2,.. are the outputs
pv1, pv2,.. are the inputs.
The body of the function will contain the relations between the
outputs and inputs.](https://image.slidesharecdn.com/matlab-250507081126-e4e4ebd8/85/matlab-presentation-fro-engninering-students-56-320.jpg)
![72
As you (should) know, a definite integral represents the area under a curve from
a given start point to a given end point.
Try to find how much area is under the curve f(x) = x^3; for [0..10], [-10..10], [-
10..0]
Definite Integrals
>> syms x;
f= x^3
>> int(x^3,0,10)
ans = ?
>> int(x^3,-10,10)
ans = ?
>> int(x^3,-10,0)
ans = ?](https://image.slidesharecdn.com/matlab-250507081126-e4e4ebd8/85/matlab-presentation-fro-engninering-students-72-320.jpg)
![85
Example 8
Given a function f(x)=sin(x). Write a program that finds the Taylor series
expansion of f(x) at x=0 of degree 3.
Plot f(x) and its Taylor series approximation of degree 3 on the same
graph for x in the range [-pi,pi]
Add Taylor series of degrees 5, 7 to the same plot.
Which Taylor series better approximates f(x).
What is your conclusion?](https://image.slidesharecdn.com/matlab-250507081126-e4e4ebd8/85/matlab-presentation-fro-engninering-students-85-320.jpg)
![88
syms x;
f=sin(x);
t3=taylor(f,4); %3rd deg Taylor series at x=0
f_inline=inline(char(f)); %f:symbolic function
t3_inline=inline(char(t3));
fplot(f_inline,[-pi ,pi],'b');
hold on;
fplot(t3_inline,[-pi, pi],'r');
Legend('exact','Taylor 3');
xlabel('x');
ylabel('function')
Example 8(Another Method)](https://image.slidesharecdn.com/matlab-250507081126-e4e4ebd8/85/matlab-presentation-fro-engninering-students-88-320.jpg)
![98
[x,y] = pol2cart(theta,r)
[theta,r] = cart2pol(x,y)
angle comes first
Converting Between Polar And Cartesian in Matlab
>>[x,y] = pol2cart(pi/4,1)
x = ?
y = ?
>>[theta,r] = cart2pol(0,-4)
theta = ?
r = ?](https://image.slidesharecdn.com/matlab-250507081126-e4e4ebd8/85/matlab-presentation-fro-engninering-students-98-320.jpg)
![101
Use MATLAB to plot the following points in polar coordinates:
r=5, θ=0° | r=5, θ=60° | r=3, θ=180° | r=3, θ=300° | r=3, θ=-30°
Example 1
Solution:
Create a script file: example1.m
>> r=[5 5 3 3 3];
>> theta=[0 60 180 300 -30]*pi/180; %Convert to radians
>> polar(theta , r, 'x');](https://image.slidesharecdn.com/matlab-250507081126-e4e4ebd8/85/matlab-presentation-fro-engninering-students-101-320.jpg)
![106
Solution:
Open a script file example1d.m
>>r=[5 5 3 3 3];
>>theta=[0 60 180 300 -30]*pi/180;%to rad
>>polar(theta,r,’x’);
>>hold on;
>>x=r .* cos(theta); % from polar to rectangular
>>y=r .* sin(theta); % from polar to rectangular
>>plot(x,y, ‘o’);
Example 3
Repeat Example 1 and plot the given
points using the polar to rectangular
conversion relation. Add the two
examples on one plot.](https://image.slidesharecdn.com/matlab-250507081126-e4e4ebd8/85/matlab-presentation-fro-engninering-students-106-320.jpg)
![116
Solution (2/2)
2-form the integration and put the limits:
3-Go to command window of MATLAB and do the integration using int:
>>syms theta;
>>area=eval(0.5*int((3+2*sin(theta))^2-4,-pi/6,7*pi/6))
Answer:
area=?
d
Area ]
4
)
sin
2
3
[(
2
1 6
7
6
2
d
Area r
r )
(
2
1 2
1
2
2
2
1
](https://image.slidesharecdn.com/matlab-250507081126-e4e4ebd8/85/matlab-presentation-fro-engninering-students-116-320.jpg)
![140
Use ezplot to plot the
function
» syms x
» f_sym=exp(x)+atan(x);
» ezplot(f_sym)
» grid on
Continuity confirmed in
the x = [0,4] interval
Solution (1/3): Continuous Plot](https://image.slidesharecdn.com/matlab-250507081126-e4e4ebd8/85/matlab-presentation-fro-engninering-students-140-320.jpg)
![153
Solve for b as the independent
variable
b must be explicitly specified on the
command line
sol_b=solve(f,b)
sol_b =
[ -1+(1-4*c)^(1/2)]
[ -1-(1-4*c)^(1/2)]
» syms b c
» f=b^2+4*c+2*b;
» sol_c=solve(f)
sol_c =
-1/4*b^2-1/2*b
0
2
4
2
b
c
b
2
4
2
b
b
c
c
b
c
b
4
1
1
4
1
1
2
1
](https://image.slidesharecdn.com/matlab-250507081126-e4e4ebd8/85/matlab-presentation-fro-engninering-students-153-320.jpg)
![154
Two Variable Complex Solution Example
Given the equation of two variables:
x2
+ 9y4
= 0
Solve for x in terms of y using Matlab:
» syms x y
» xs=solve(x^2+9*y^4)
xs =
[ 3*i*y^2]
[ -3*i*y^2]
Roots are
Complex Conjugates](https://image.slidesharecdn.com/matlab-250507081126-e4e4ebd8/85/matlab-presentation-fro-engninering-students-154-320.jpg)
![155
Matlab Command window:
>> syms x
>> f=2*x^2 + 4*x -8;
>> solve(f,x)
ans = ?
Alternately, you may use the following lines in
Matlab to perform the same calculation:
>> f=[2 4 -8];
>> roots(f)
ans = ?
Are the results the same?
Example: finding roots
Solve the following equation with 2
methods:
F(x)=2x² + 4x - 8](https://image.slidesharecdn.com/matlab-250507081126-e4e4ebd8/85/matlab-presentation-fro-engninering-students-155-320.jpg)