Matlab Math.pdf
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Matlab Maths Practice 1.Solving AlgebraicEquations and Other Symbolic Tools Solving Basic Algebraic Equations 1.x + 3 = 0 Command window >> x=solve('x+3=0') x = -3 2. x + 8 =0 Command window >> x=solve('x+8') x = -8 3.ax + 5 = 0 >> solve('a*x+5','a') ans = -5/x >> solve('a*x+5','x') ans =
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Transcript of Matlab Math.pdf
Matlab Maths Practice
1.Solving AlgebraicEquations and Other Symbolic Tools Solving Basic Algebraic Equations
1.x + 3 = 0
Command window
>> x=solve('x+3=0')
x =
-3
2. x + 8 =0
Command window
>> x=solve('x+8')
x =
-8
3.ax + 5 = 0
>> solve('a*x+5','a')
ans =
-5/x
>> solve('a*x+5','x')
ans =
-5/a
2.Solving Quadratic Equations
x2 6x 12 = 0
Command window
s=solve('x^2-6*x-12=0')
s =
3+21^(1/2)
3-21^(1/2)
3.Plotting Symbolic Equations
1.x^2-6*x-12
command window
d=('x^2-6*x-12');
ezplot(d)
same equation with a example
d='x^2-6*x-12'
command window
>> d='x^2-6*x-12'
d =
x^2-6*x-12
>> ezplot(d,[2,8])
2.we wanted to plot:
x + 3 = 0
4 < x < 4, 2 < y < 2
Command window
>> ezplot('x+3',[-4 4 -2 2])
>>
Output
3.Find the roots of x3 + 3x2 2x 6 and plot the function for �8 < x < 8, �8 < y < 8. Generate the plot with a grid.
command window
>> f='x^3+3*x^2-2*x-6'
f =
x^3+3*x^2-2*x-6
>> s=solve(f)
s =
-3
2^(1/2)
-2^(1/2)
>> ezplot(f,[-8 8,-8,8]),grid on
>>
Graph
4.Systems of Equations
1. 5x + 4y = 3 x - 6y = 2
Command window
>> s=solve('5*x+4*y=3','x-6*y=2');
>> x=s.x
x =
13/17
>> y=s.y
y =
-7/34
2.
w + x + 4y + 3z = 5
2w + 3x + y 2z = 1
w + 2x 5y + 4z = 3
w-3z = 9
command window
>> eq1=('w+x+4*y+3*z=5')
eq1 =
w+x+4*y+3*z=5
>> eq2=('2*w+3*x+y-2*z=1')
eq2 =
2*w+3*x+y-2*z=1
>> eq3=('w+2*x-5*y+4*z=3')
eq3 =
w+2*x-5*y+4*z=3
>> eq4=('w-3*z=9')
eq4 =
w-3*z=9
>> s=solve(eq1,eq2,eq3,eq4)
s =
w: [1x1 sym]
x: [1x1 sym]
y: [1x1 sym]
z: [1x1 sym]
>> w=s.w
w =
1404/127
>> x=s.x
x =
-818/127
>> y=s.y
y =
-53/127
>> z=s.z
z =
87/127
5. Expanding and Collecting Equations
expand
(x + 2) (x 3) = x2 x 6
command window
>> syms x
>> expand((x-1)*(x+4))
ans =
x^2+3*x-4
>> syms y
>> expand(cos(x+y))
ans =
cos(x)*cos(y)-sin(x)*sin(y)
>> expand(sin(x-y))
ans =
sin(x)*cos(y)-cos(x)*sin(y)
>> expand((y-2)*(y+6))
ans =
y^2+4*y-12
>>
collect
>> collect(x*(x^2-2))
ans =
x^3-2*x
>> syms t
>> collect((t+3)*sin(t))
ans =
sin(t)*t+3*sin(t)
Factors
>> factor(x^2-y^2)
ans =
(x-y)*(x+y)
Solving with Exponential and Log Functions
Find a value of x that satisfi es:
log10 (x) log10 (x 3) = 1
command window
>> eq='log10(x)-log10(x-3)=1';
>> s=solve(eq)
s =
10/3
Series Representations of Functions
>> syms x
>> s=taylor(sin(x))
s =
x-1/6*x^3+1/120*x^5
>> ezplot(s)
Computing Derivatives(Important)
>> syms x t
>> f=x^2;
>> g=sin(10+t);
>> diff(f)
ans =
2*x
>> diff(sin(10+t))
ans =
cos(10+t)
To take higher derivatives of a function f, we use the syntax diff(f,n).
>> syms x
>> f=x^4;
>> diff(f,2)
ans =
12*x^2
Show that f (x) x2 satisfies -(df/dx)+2*x=0
>> syms x
>> f=x^2; g=2*x;
>> h=diff(f);
>> -h+g
ans =
0
Does y (t) 3 sin t 7 cos 5t solve y'' y –5 cos 2t?
>> syms t
>> y=3*sin(t)+7*cos(5*t);
>> z=diff(y,2);
>> x=z+y
x =
-168*cos(5*t)
As from the results we can deduce that y(t) cannot solve the above required equation.
Find the minima and maxima of the function f (x) x3 – 3x2 3x in the
interval
[0, 2].
>> syms x
>> f=x^3-3x^2+3*x;
>> f=x^3-3*x^2+3*x;
>> ezplot(f,[0,2])
>> g=diff(f)
g =
3*x^2-6*x+3
>> pretty(g)
2
3 x - 6 x + 3
>> s=solve(g)
s =
1
1
We see that there is only one critical point, since the derivative has a double root.
We can see from the plot that the maximum occurs at the endpoint, but let’s prove
this by evaluating the function at the critical points x 0, 1, 2.
So let’s check f for x 0, 1, 2. We can check all three on a single line and
have MATLAB report the output by passing a comma-delimited list:
Since f (2) returns the largest value, we conclude that the maximum occurs at
x 0. For fun, let’s evaluate the derivative at these three points and plot it:
>> subs(g,0),subs(g,1),subs(g,2)
ans =
3
ans =
0
ans =
3
Where are the critical points of the derivative? We take the second derivative and
set equal to zero:
>> subs(g,0),subs(g,1),subs(g,2)
ans =
3
ans =
0
ans =
3
>> h=diff(g)
h =
6*x-6
>> solve(h)
ans =
1
>> y=diff(h)
y =
6
>>
Plot the function f (x) x4 –2x3 and show any local minima and maxima.
>> syms x
>> f=x^4-2*x^3;
>> ezplot(f,[-2,3]);
>> g=diff(f)
g =
4*x^3-6*x^2
>> s=solve(g)
s =
3/2
0
0
>> h=diff(g)
h =
12*x^2-12*x
>> subs(h,3/2),subs(h,0),subs(h,0)
ans =
9
ans =
0
ans =
0
>>
INTEGRATION
Find int3y sec(x) dy .
command window
>> syms x y
>> int('3*y^2*sec(x)',y)
ans =
y^3*sec(x)
>> a=int('3*y^2*sec(x)',x)
a =
3*y^2*log(sec(x)+tan(x))
>> pretty(a)
2
3 y log(sec(x) + tan(x))
Definite Integration The int command can be used for definite integration by passing the limits over which you want to calculate the integral.
>> int('x',2,3)
ans =
5/2
>> b=int('x^2*cos(x)',-6,6)
b =
68*sin(6)+24*cos(6)
>> double(b)
ans =
4.0438
Matlab Code
>> syms x
>> a=int(exp(-x^2)*sin(x),o,inf)
??? Undefined function or variable 'o'.
>> a=int(exp(-x^2)*sin(x),0,inf)
a =
1/2*pi^(1/2)*erfi(1/2)*exp(-1/4)
>> pretty(a)
1/2
1/2 pi erfi(1/2) exp(-1/4)
>> double(a)
ans =
0.4244
Matlab code
>> syms x
>> int(pi*exp(-2*x),1,2)
ans =
1/2*pi*exp(-2)-1/2*pi*exp(-4)
>> f=1/2*pi*exp(-2)-1/2*pi*exp(-4)
f =
0.1838
Matlab Code
>> syms x y z
>> int(int(int(x*y^2*z^5,x),y),z)
ans =
1/36*x^2*y^3*z^6
Matlab Code
f =
x^2*y
>> int(int(f,x,2,4),y,1,2)
ans =
28
Matlab Code
>> syms r theta z h a
>> v=int(int(int(r,0,h),theta,0,2*pi),z,0,a)
v =
h^2*pi*a
>> subs(v,{a,h},{3.5,5})
ans =
274.8894
>> syms r theta z h a
>> v=int(int(int(r,0,h),theta,0,2*pi),z,0,a)
v =
h^2*pi*a
>> subs(v,{a,h},{3.5,5})
ans =
274.8894
Transforms
Transforms, such as the Laplace, z, and Fourier transforms, are used throughout science and engineering. Besides simplifying the analysis, transforms allow us to see data in a new light. For example, the Fourier transform allows you to view a signal that was represented as a function of time as one that is a function of frequency. In this chapter we will introduce the reader to the basics of using MATLAB to work with transforms. In this chapter we will introduce the laplace, fourier, and fft commands
command window
>> syms a t
>> laplace(a)
ans =
1/s^2
>> laplace(t^2)
ans =
2/s^3
>> laplace(t^7)
ans =
5040/s^8
>> laplace(t^5)
ans =
120/s^6
>> syms w
>> laplace(cos(w*t))
ans =
s/(s^2+w^2)
>> laplace(sin(w*t))
ans =
w/(s^2+w^2)
>> syms b
>> laplace(cosh(b*t))
ans =
s/(s^2-b^2)
>> syms s
>> ilaplace(1/s^3)
ans =
1/2*t^2
>> ilaplace(s/(s^2+4))
ans =
cos(2*t)
>> F=(5-3*s)/(2+5*s)
F =
(5-3*s)/(2+5*s)
>> z=ilaplace(F)
z =
-3/5*dirac(t)+31/25*exp(-2/5*t)
>> pretty(z)
31
- 3/5 dirac(t) + -- exp(- 2/5 t)
25
Matlab code
>> syms s
>> f=ilaplace(1/(s+7)^2)
f =
t*exp(-7*t)
>> pretty(f)
t exp(-7 t)
>> ezplot(f,[0,5])
>> syms s
>> f=((2*s)+3)/((s+1)^2*(s+3)^2)
f =
(2*s+3)/(s+1)^2/(s+3)^2
>> pretty(f)
2 s + 3
-----------------
2 2
(s + 1) (s + 3)
>> ilaplace(f)
ans =
1/4*exp(-t)*(1+t)-1/4*exp(-3*t)*(1+3*t)
>> ezplot(f,[0,7])
Math Book solve Problems
John and birds
Calculus
Integration
>> syms x
>> int 3*X^4
ans =
3/5*X^5
>> int (2/x^2,x)
ans =
-2/x
>> c=int(sqrt(x))
c =
2/3*x^(3/2)
>> pretty(c)
3/2
2/3 x
>> d=int(3*x+2*x^2-5,x)
d =
3/2*x^2+2/3*x^3-5*x
>> pretty(d)
2 3
3/2 x + 2/3 x - 5 x
>> int(4+(3/7*x)-6*x^2,x)
ans =
4*x+3/14*x^2-2*x^3
>> a=((2*x^3)/(4*x)-(3*x)/(4*x))
a =
1/2*x^2-3/4
>> pretty(a)
2
1/2 x - 3/4
>> int(a)
ans =
1/6*x^3-3/4*x
Matlab code
>> syms t
>> a=int(1-t^2,t)
a =
t-1/3*t^3
>> pretty(a)
3
t - 1/3 t
Matlab code
>> syms t
>> a=(-5/9)*(t^-0.75)
a =
-5/9/t^(3/4)
pretty(a)
111 1
- --- ----
200 3/4
t
>> a=(-5/9)*(t^-0.75)
a =
-5/9/t^(3/4)
>> int(a)
ans =
-20/9*t^(1/4)
>> b=int(a)
b =
-20/9*t^(1/4)
>> pretty(b)
1/4
- 20/9 t
Matlab code
>> syms m
>> a=(2*m^2+1)/m
a =
(2*m^2+1)/m
>> pretty(a)
2
2 m + 1
--------
m
>> b=int(a,m)
b =
m^2+log(m)
>> pretty(b)
2
m + log(m)
>> a=(3*sin(2*x))
a =
3*sin(2*x)
>> pretty(a)
3 sin(2 x)
>> b=int(a,0,pi/2)
b =
3
Matlab code
>> syms t
>> a=4*cos(3*t)
a =
4*cos(3*t)
>> pretty(a)
4 cos(3 t)
>> b=int(a,1,2)
b =
-4/3*sin(3)+4/3*sin(6)
>> pretty(b)
- 4/3 sin(3) + 4/3 sin(6)
>> double(b)
ans =
-0.5607
Matlab code
>> syms x
>> a=3*(sec(2*x))^2
a =
3*sec(2*x)^2
>> pretty (a)
2
3 sec(2 x)
>> int(a,0,pi/6)
ans =
3/2*3^(1/2)
>> b=int(a,0,pi/6)
b =
3/2*3^(1/2)
>> pretty(b)
1/2
3/2 3
>> double(b)
ans =
2.598
Matlab code
>> syms x
>> ezplot(x^3-2*x^2-8*x)
>> %From Graph limits are -2 to 0 and from 0 to 4
>> syms y
>> int(x^3-2*x^2-8*x,-2,0)-int(x^3-2*x^2-8*x,0,4)
ans =
148/3
Matlab Code
Limits are find by plotting both functions
>> syms x
>> int(7-x,-3,2)-int(x^2+1,-3,2)
ans =
125/6
>> f=int(7-x,-3,2)-int(x^2+1,-3,2)
f =
125/6
>> double(f)
ans =
20.8333
Matlab Code
>> x=linspace(-5,5,20);
>> y1=x.^2+3;
>> y2=7-3*x;
>> plot(x,y1,'ko',x,y2,'r*')
>> plot(x,y1,'k',x,y2,'r'
>> plot(x,y1,'k',x,y2,'r')
limits are -4 and 1
>> clear x
>> syms x
>> int(7-3*x,-4,1)-int(x.^2+3,-4,1)
ans =
125/6
Matlab Code
> syms x
>> f=(x^2+4)^2
f =
(x^2+4)^2
>> f1=expand(f)
f1 =
x^4+8*x^2+16
>> f2=(pi*f1)
f2 =
pi*(x^4+8*x^2+16)
>> f3=int(f2,1,4)
f3 =
2103/5*pi
>> double(f3)
ans =
1.3214e+003
Matlab code
>> syms t
>> f=2*(cos(4*t))^2
f =
2*cos(4*t)^2
>> pretty(f)
2
2 cos(4 t)
>> int(f,0,pi/4)
ans =
1/4*pi
Matlab code
>> syms t
> f=(sin(t))^2*(cos(t))^4
f =
sin(t)^2*cos(t)^4
>> pretty(f)
2 4
sin(t) cos(t)
>> int(f)
ans =
-1/6*sin(t)*cos(t)^5+1/24*cos(t)^3*sin(t)+1/16*cos(t)*sin(t)+1/16*t
>> f1=int(f)
f1 =
-1/6*sin(t)*cos(t)^5+1/24*cos(t)^3*sin(t)+1/16*cos(t)*sin(t)+1/16*t
>> pretty(f1)
5 3
- 1/6 sin(t) cos(t) + 1/24 cos(t) sin(t) + 1/16 cos(t) sin(t) + 1/16 t
Matlab code
> syms a x
>> f=1/(a^2-x^2)
f =
1/(a^2-x^2)
>> pretty(f)
1
-------
2 2
a - x
>> f=1/sqrt(a^2-x^2)
f =
1/(a^2-x^2)^(1/2)
>> pretty(f)
1
------------
2 2 1/2
(a - x )
>> f1=int(f)
f1 =
atan(x/(a^2-x^2)^(1/2))
>> pretty(f1)
x
atan(------------)
2 2 1/2
(a - x )
Matlab Code
>> syms x
>> f=1/sqrt(x^2+4)
f =
1/(x^2+4)^(1/2)
> pretty(f)
1
-----------
2 1/2
(x + 4)
>> f1=int(f,0,2)
f1 =
-log(2^(1/2)-1)
>> pretty(f1)
1/2
-log(2 - 1)
>> f2=double(f1)
f2 =
0.8814
Matlab Code
>> syms x
>> f=(3*x^2+16*x+15)/(x+3)^3
f =
(3*x^2+16*x+15)/(x+3)^3
>> pretty(f)
2
3 x + 16 x + 15
----------------
3
(x + 3)
>> f1=int(f,-2,1)
f1 =
-69/16+6*log(2)
>> pretty(f1)
69
- -- + 6 log(2)
16
>> f2=double(f1)
f2 =
-0.1536
Matlab Code
>> syms x
>> f=3*x^2*exp(x/2)
f =
3*x^2*exp(1/2*x)
>> pretty(f)
2
3 x exp(1/2 x)
>> f1=int(f,1,2)
f1 =
-30*exp(1/2)+24*exp(1)
>> pretty(f1)
-30 exp(1/2) + 24 exp(1)
>> f2=double(f1)
f2 =
15.7771
Matlab Code
> syms t
>> f=4*t^3*cos(t)
f =
4*t^3*cos(t)
> pretty(f)
3
4 t cos(t)
>> int(f,1,2)
ans =
20*sin(1)+12*cos(1)-16*sin(2)+24*cos(2)
>> f1=int(f,1,2)
f1 =
20*sin(1)+12*cos(1)-16*sin(2)+24*cos(2)
>> pretty(f1)
20 sin(1) + 12 cos(1) - 16 sin(2) + 24 cos(2)
>> f2=double(f1)
f2 =
-1.2232
Matlab Code
>> syms t
>> f=(sin(t))^2*(cos(t))^6
f =
sin(t)^2*cos(t)^6
>> pretty(f)
2 6
sin(t) cos(t)
>> f1=int(f,0,pi/2)
f1 =
5/256*pi
>> pretty(f1)
5/256 pi
>> f2=double(f1)
f2 =
0.0614
Matlab Code
>> syms x
>> y=12*x^3;
>> y1=diff(y,x)
y1 =
36*x^2
>> pretty(y1)
2
36 x
Matlab code
>> syms x
>> y=3*sin(4*x)
y =
3*sin(4*x)
>> pretty(y)
3 sin(4 x)
>> y1=diff(y)
y1 =
12*cos(4*x)
>> pretty(y1)
12 cos(4 x)
Matlab code
>> syms x
>> y=3*x^2*sin(2*x)
y =
3*x^2*sin(2*x)
>> pretty(y)
2
3 x sin(2 x)
>> y1=diff(y)
y1 =
6*x*sin(2*x)+6*x^2*cos(2*x)
>> pretty(y1)
2
6 x sin(2 x) + 6 x cos(2 x)
Matlab code
>> syms t
>> y=(t*exp(2*t)/2*cos(t))
y =
1/2*t*exp(2*t)*cos(t)
>> pretty(y)
1/2 t exp(2 t) cos(t)
>> y1=diff(y)
y1 =
1/2*exp(2*t)*cos(t)+t*exp(2*t)*cos(t)-1/2*t*exp(2*t)*sin(t)
>> pretty(y1)
1/2 exp(2 t) cos(t) + t exp(2 t) cos(t) - 1/2 t exp(2 t) sin(t)
>>
Matlab Code
>> syms x
>> y=3*(tan(3*x))^4
y =
3*tan(3*x)^4
>> y1=diff(y)
y1 =
12*tan(3*x)^3*(3+3*tan(3*x)^2)
>> pretty(y1)
3 2
12 tan(3 x) (3 + 3 tan(3 x) )
>> 12*tan(3*x)^3*(3+3*tan(3*x)^2)
pretty(y1)
3 2
12 tan(3 x) (3 + 3 tan(3 x) )
Matlab code
>> syms x
>> y=2*x*exp(-3*x)
y =
2*x*exp(-3*x)
>> pretty(y)
2 x exp(-3 x)
>> y1=diff(y)
y1 =
2*exp(-3*x)-6*x*exp(-3*x)
>> pretty(y1)
2 exp(-3 x) - 6 x exp(-3 x)
>> y2=diff(y1)
y2 =
-12*exp(-3*x)+18*x*exp(-3*x)
>> pretty(y2)
-12 exp(-3 x) + 18 x exp(-3 x)
>> y3=y2+6*y1+9*y
y3 =
0
Matlab Code
>> dsolve('Dy=y*t/(t-5)','y(0)=2')
ans =
-2/3125*exp(t)*t^5+2/125*exp(t)*t^4-4/25*exp(t)*t^3+4/5*exp(t)*t^2-2*exp(t)*t+2*exp(t)
Matlab Code
>> s=dsolve('Dy=t+3','y(0)=7')
s =
1/2*t^2+3*t+7
Matlab Code
>> dsolve('D2y-y=0','y(0)=-1','Dy(0)=2')
ans =
-3/2*exp(-t)+1/2*exp(t)
JOHN BIRDS MATHEMATICS PROBLEMS WITH MATLAB
Matlab Code
>> syms a b c
>> y=(a^3*b^2*c^4)/(a*b*c^-2)
y =
a^2*b*c^6
>> pretty(y)
2 6
a b c
>> subs(y,[a,b,c],[3,1/8,2])
ans =
72
Matlab Code
>> syms x y
>> z=((x^2*y^3+x*y^2)/(x*y))
z =
(x^2*y^3+x*y^2)/x/y
>> pretty(z)
2 3 2
x y + x y
------------
x y
>> simplify(z)
ans =
y*(x*y+1)
Matlab Code
> z=((x^2*sqrt(y))*(sqrt(x)*(y^2)^1.5))/((x^5*y^3)^0.5)
z =
x^(5/2)*y^(1/2)*(y^2)^(3/2)/(x^5*y^3)^(1/2)
>> pretty(z)
5/2 1/2 2 3/2
x y (y )
-----------------
5 3 1/2
(x y )
>> simplify(z)
ans =
x^(5/2)*y^(7/2)*csgn(y)/(x^5*y^3)^(1/2)
Matlab Code
>> expand((3*x+2*y)*(x-y))
ans =
3*x^2-x*y-2*y^2
Matlab Code
>> syms a b c
>> expand((2*a-3)/(4*a )+5*6-3*a)
ans =
61/2-3/4/a-3*a
Matlab Code
>> solve('(sqrt(t)+3)/(sqrt(t))=2')
ans =
9
Matlab Code
>> syms x
>> solve('4-3*x=2*x-11')
ans =
3
Matlab Code
>> syms x
>> solve('3/(x-2)=4/(3*x+4)')
ans =
-4
Matlab Code
>> solve('3*x-2-5*x=2*x-4')
ans =
1/2
Matlab Code
>> solve('8+4*(x-1)-5*(x-3)=2*(5-2*x)')
ans =
-3
Matlab Code
>> syms a
>> solve('(1/(3*a-2))+(1/(5*a+3))')
ans =
-1/8
Matlab Code
>> syms t
>> solve('((3*sqrt(t))/(1-sqrt(t))=-6)')
ans =
4
Matlab Code
>> syms x y
s=solve('8*x-3*y=51','3*x+4*y=14');
>> x=s.x
x =
6
>> y=s.y
y =
-1
Matlab Code
>> s=solve('5*a=1-3*b','2*b+a+4=0');
>> a=s.a
a =
2
>> b=s.b
b =
-3
Matlab Code
>> syms x y
>> s=solve('x/5+2*y/3=49/15','3*x/7-y/2+5/7=0');
>> x=s.x
x =
3
>> y=s.y
y =
4
Matlab Code
>> solve('x^2+4*x-32=0')
ans =
4
-8
2nd Code
>> a=[1 4 -32]
a =
1 4 -32
>> roots(a)
ans =
-8
4
Matlab Code
>> solve('8*x^2+2*x-15=0')
ans =
5/4
-3/2
>> a=[8 2 -15]
2nd Code
a =
8 2 -15
>> roots(a)
ans =
-1.5000
1.2500
Matlab Code
>> a=[2 -5]
a =
2 -5
>> a=poly(a)
a =
1 3 -10
RESOLVE THE FRACTION
f (x) =3x3 - 2x2 + 4x - 3
x2 + 3x + 3
Matlab Code
>> b=[3 -2 4 -3]
b =
3 -2 4 -3
>> a=[1 3 3]
a =
1 3 3
>> [q r]=deconv(b,a)
q =
3 -11
r =
0 0 28 30
matlab code means
f (x) = (3x -11) +28x + 30
x2 + 3x + 3
Partial Fraction Expansion
The command RESIDUE can be used to perform partial fraction expansion directly (RESIDUE
itself makes use of both ROOTS and DECONV behind the scenes). You provide two input
vectors, which again represent the numerator and denominator polynomials, just like the
DECONV command. MATLAB returns three outputs: the first is a vector of residues, the
second a vector of poles, and the third a vector of coefficients for any remainder polynomial that
exists.
F(s) =B(s)
A(s)=
3s+1
s3 + 3s2 + 2s
Matlab Code
>> b=[3 1]
b =
3 1
>> a=[1 3 2 0]
a =
1 3 2 0
>> [r p k]=residue(b,a)
r =
-2.5000
2.0000
0.5000
p =
-2
-1
0
k =
[]
answer is
F(s) =-2.5
s+ 2+
2
s+1+
0.5
s
Matlab Code
>> b=[-3 11]
b =
-3 11
>> a=[1 2 -3]
a =
1 2 -3
>> [r p k]=residue(b,a)
r =
-5
2
p =
-3.0000
1.0000
k =
[]
Matlab Code
>> collect((x+1)*(x-2)*(x+3))
ans =
-6+x^3+2*x^2-5*x
>> b=[2 -9 -35]
b =
2 -9 -35
>> a=[1 2 -5 -6]
a =
1 2 -5 -6
>> [r p k]=residue(b,a)
r =
1.0000
-3.0000
4.0000
p =
-3.0000
2.0000
-1.0000
k =
[]
Matlab Code
> b=[1 0 1]
b =
1 0 1
>> a=[1 -3 2]
a =
1 -3 2
>> [r p k]=residue(b,a)
r =
5
-2
p =
2
1
k =
1
Matlab Code
>> b=[1 -2 -4 -4]
b =
1 -2 -4 -4
>> a=[1 1 -2]
a =
1 1 -2
>> [r p k]=residue(b,a)
r =
4
-3
p =
-2
1
k =
1 -3
Matlab Code
>> b=[2 3]
b =
2 3
>> a=[1 -4 4 ]
a =
1 -4 4
>> [r p k]=residue(b,a)
r =
2
7
p =
2
2
k =
[]
Matlab Code
>> syms x
>> q=(x+3)*(x-1)^2
q =
(x+3)*(x-1)^2
>> expand(q)
ans =
x^3+x^2-5*x+3
>> b=[5 -2 -19]
b =
5 -2 -19
>> a=[1 1 -5 3]
a =
1 1 -5 3
>> [r p k]=residue(b,a)
r =
2.0000
3.0000
-4.0000
p =
-3.0000
1.0000
1.0000
k =
[]
Matlab Code
>> syms x
>> q=(x+3)^3
q =
(x+3)^3
>> q=expand(q)
q =
x^3+9*x^2+27*x+27
>> b=[3 16 15]
b =
3 16 15
>> a=[1 9 27 27]
a =
1 9 27 27
>> [r p k]=residue(b,a)
r =
3.0000
-2.0000
-6.0000
p =
-3.0000
-3.0000
-3.0000
k =
[]
Matlab Code
>> syms x
>> b=[12]
b =
12
>> a=[1 0 -9]
a =
1 0 -9
>> [r p k]=residue(b,a)
r =
2
-2
p =
3
-3
k =
[]
Matlab Code
>> syms x
>> b=[4 -4]
b =
4 -4
>> a=[1 -2 -3]
a =
1 -2 -3
>> [r p k]=residue(b,a)
r =
2.0000
2.0000
p =
3.0000
-1.0000
k =
[]
Matlab Code
>> syms x
>> q=x*(x-2)*(x-1)
q =
x*(x-2)*(x-1)
>> expand(q)
ans =
x^3-3*x^2+2*x
>> b=[1 -3 6]
b =
1 -3 6
>> a=[1 3 2 0]
a =
1 3 2 0
>> a=[1 -3 2 0]
a =
1 -3 2 0
>> [r p k]=residue(b,a)
r =
2
-4
3
p =
2
1
0
k =
[]
Matlab Code
>> syms x y z
>> s=solve('x+y+z=4','2*x-3*y+4*z=33','3*x-2*y-2*z=2');
>> x=s.x
x =
2
>> y=s.y
y =
-3
>> z=s.z
z =
5
Matlab Code
>> syms p q r
>> s=solve('p+2*q+3*r=-7.8','2*p+5*q-r=1.4','5*p-q+7*r=3.5');
>> p=s.p
p =
4.1000000000000000000000000000000
>> q=s.q
q =
-1.9000000000000000000000000000000
>> r=s.r
r =
-2.7000000000000000000000000000000
Matlab Code
>> syms x
>> y=x^2
y =
x^2
>> pretty(y)
2
x
>> diff(y)
ans =
2*x
>> subs(ans,2)
ans =
4
Matlab Code
>> syms x
>> y=5*x^4+4*x-1/(2*x^2)+1/sqrt(x)-3
y =
5*x^4+4*x-1/2/x^2+1/x^(1/2)-3
>> pretty(y)
4 1 1
5 x + 4 x - 1/2 ---- + ---- - 3
2 1/2
x x
>> a=diff(y)
a =
20*x^3+4+1/x^3-1/2/x^(3/2)
>> pretty(a)
3 1 1
20 x + 4 + ---- - 1/2 ----
3 3/2
x x
Matlab Code
>> syms x
>> y=3*x^4-2*x^2+5*x-2
y =
3*x^4-2*x^2+5*x-2
>> pretty(y)
4 2
3 x - 2 x + 5 x - 2
>> a=diff(y)
a =
12*x^3-4*x+5
>> subs(a,0)
ans =
5
>> subs(a,1)
ans =
13
Matlab Code
>> syms x
>> y=3*x^2*sin(2*x)
y =
3*x^2*sin(2*x)
>> pretty(y)
2
3 x sin(2 x)
>> a=diff(y)
a =
6*x*sin(2*x)+6*x^2*cos(2*x)
>> pretty(a)
2
6 x sin(2 x) + 6 x cos(2 x)
Matlab Code
>> syms t
>> v=5*t*sin(2*t)
v =
5*t*sin(2*t)
>> pretty(v)
5 t sin(2 t)
>> a=diff(v)
a =
5*sin(2*t)+10*t*cos(2*t)
>> pretty(a)
5 sin(2 t) + 10 t cos(2 t)
>> subs(a,0.20)
ans =
3.7892
Matlab Code
>> syms t
>> y=exp(t)*sin(4*t)
y =
exp(t)*sin(4*t)
>> syms t
>> y=exp(t)*sin(4*t)
y =
exp(t)*sin(4*t)
Matlab Code
>> syms t
>> z=2*exp(3*t)*sin(2*t)
z =
2*exp(3*t)*sin(2*t)
>> pretty(z)
2 exp(3 t) sin(2 t)
>> a=diff(z)
a =
6*exp(3*t)*sin(2*t)+4*exp(3*t)*cos(2*t)
>> pretty(a)
6 exp(3 t) sin(2 t) + 4 exp(3 t) cos(2 t)
>> subs(a,0.5)
ans =
32.3131
Matlab Code
>> syms t
>> y=t*exp(2*t)/2*cos(t)
y =
1/2*t*exp(2*t)*cos(t)
>> pretty(y)
1/2 t exp(2 t) cos(t)
>> a=diff(y)
a =
1/2*exp(2*t)*cos(t)+t*exp(2*t)*cos(t)-1/2*t*exp(2*t)*sin(t)
>> pretty(a)
1/2 exp(2 t) cos(t) + t exp(2 t) cos(t) - 1/2 t exp(2 t) sin(t)
Matlab Code
>> syms x
>> y=sqrt(3*x^2+4*x-1)
y =
(3*x^2+4*x-1)^(1/2)
>> pretty(y)
2 1/2
(3 x + 4 x - 1)
>> a=diff(y)
a =
1/2/(3*x^2+4*x-1)^(1/2)*(6*x+4)
>> pretty(a)
6 x + 4
1/2 -------------------
2 1/2
(3 x + 4 x - 1)
Matlab Code
>> syms t
>> y=2*exp(-0.9*t)*sin(2*pi*5*t)
y =
2*exp(-9/10*t)*sin(10*pi*t)
>> a=diff(y)
a =
-9/5*exp(-9/10*t)*sin(10*pi*t)+20*exp(-9/10*t)*cos(10*pi*t)*pi
>> subs(a,1)
ans =
25.5455
Matlab Code
