math lab

Question 1

when is semicolon used

comment symbol

Answer 1

end of statement and when you want to hide the Matlab output for a expression

x=5; y = x+5 x=5 is not shown

% symbol

Dealing with Matrices and Arrays
2-D and 3-D Plotting and graphics
Linear Algebra
Algebraic Equations
Statistics
Data Analysis
Calculus and Differential Equations
Numerical Calculations
Integration
Transforms
Curve Fitting
Special Functions

Question 2

MATLAB difference compared to other programs

Answer 2

While other programming languages mostly work with numbers one at a time, MATLAB is designed to operate primarily on whole matrices and arrays.

Question 3

addition, subtraction, multiplication left and right division, exponentiation

Answer 3

+, -, *, /(5/3) (3\5) same value ^

Question 4

special variables and constants

Answer 4

ans: most recent answer
eps: floating-point representation
I,j (imaginary unit root of -1)
inf - infinity
NaN (undefined numerical not a number)
pi = pi

Question 5

what is who command and whos command

Answer 5

displays all the variable names we use

whos : gives size, bytes class,

Question 6

clear x
clear
clc

print in matlab

Answer 6

it will delete x
deletes all variables
clears all text from the command window

print in matlab

fprintf

Question 7

Command Window
MATHLAB editor
workspace
command history
command directory

Answer 7

Command Window
the main area where commands can be entered at the command line. It is indicated by the command prompt (»).

MATLAB Editor. This is where you can write, edit, and save your scripts and functions.

commands can be done there and

command window: where all the editing happens variable assignment happens

workspace:
The workspace shows all the variables created and/or imported from files.

command history:
This panel shows or return commands that are entered at the command line.

command directory:
View Folders and m-files

Question 8

format short
format short e
format long
format bank
format rat

Answer 8

displays numbers with four decimal places
format short
7+10/3+5^1.2
x=17.232

format shorte
the command allows displaying in exponential form with 4 decimal places plus the exponent

and = 2.2922e+01

format long
15 digits after the decimal point

format bank
rounds number to 2 decimal places

format rat
gives closest rational expression resulting from a calculation
format rat
4.678*4.9
= 34177/1491

Question 9

two types of vectors

Answer 9

row vectors and column vectors
row vector: X =[7,8,9,10]
X= 7 8 9 10

column vector X = [7; 8; 9; 10]

7
8
9
10

Question 10

matrix

Answer 10

2D array numbers,
A = [1 2 3 4; 4 5 6 7; 7 8 9 10 ]

1 2 3 4
4 5 6 7
7 8 9 10

Question 11

elementary math built

trigonometry function

other functions

Answer 11

sqrt(x), nthroot(x,n) n here is the root value, 2,3….=> real nth root of a real number 25 has 2 roots
exp(x), abs(x), log(x), log10(x), factorial(x)

trigonometry function
sin(x) => radians, sind(x) => degree
cos(x), cosd(x), tan(x), tand(x)
cot(x), cotd(x)
asin(x), asind(x), acos(x), acosd(x)

other function

round(x) (nearest integer), fix(x)=> Round towards 0, other words, chops fraction part

ceil(x) => round towards positive infinity
ceil(27/2) => 14
floor(x) => rounds minus infinity
rem(x,y) remainder after x/y

Question 12

example question y=e^a*sin(x) + 10root(y)
if a=5, x=2, y=8

Answer 12

a=5; x=2; y=8;
y=exp(-a)sin(x)+10sqrt(y)
y=2.829039804

Question 13

special type of arrays

Answer 13

four types of arrays are:
zeros(), +> zero array zeros(5) => 5 by 5
eye() => returns 1 diagonal towards left to right
ones(), => ones(x,y)
rand() => random functions with given range
ex numbers on (0,1)
rand(3,5)
0.8147 0.9134 0.2785 0.9649 0.9572
0.9058 0.6324 0.5469 0.1576 0.4854
0.1270 0.0975 0.9575 0.9706 0.8003

Question 14

~= operator

logical operators symbol

Answer 14

determine inequality, like a not function

&. |, ~(not)

Question 15

what are M-files and what is a script file

Answer 15

.m at the end of their name
ASCII text files
two M-files: Script files: and function files

script files:A script file contains a set of valid commands, just like you would enter directly into the Command Window.

Question 16

how to create a function file matlab

Answer 16

A function is a group of statements that together perform a task.

In MATLAB, functions are defined in separate files.
Functions can accept more than one input arguments and

 may return more than one output arguments. The syntax of the function definition line is as follows: 

Note that the function_name must be the same as the file name
(without the m.extension) in which the function is written.

function [sum,prod]= my name (a,b,c)
sum = a+b+c;
prod=abc
end

on command window
For,&raquo_space; myname(20,21,22), the output is: ans =63
For,&raquo_space; [s,p]=myname(20,21,22), the output is: s = 63, p =9240

Question 17

for loop syntax

while loop syntax

all conditional statements

Answer 17

repeat a statement or a group of statements for a fixed number of times.

for m=1:3
num=1/(m+1)
end
OUTPUT:
num = 0.5000
num = 0.3333
num = 0.2500

while loop:
repeatedly executes statements while condition is true.

> > a = 10;
while(a < 15)
fprintf(‘value of a: %d\n’, a);
a = a + 1;
end
%d is a format specifier, d in this case is integer
\n is next line

all conditional statements

if-end; if-else-end; if-elseif-elseif-else-end; switch case.

Question 18

switch mat lab

Answer 18

is used to test for the equality against a set of known values

 >> grade = 'B'; 
           switch(grade) 
           case 'A' 
                    fprintf('Excellent!\n' ); 
           case 'B' 
                      fprintf('Well done\n' ); 
           case 'C' 
                      fprintf('Well done\n' );
           case 'D' 
                      fprintf('You passed\n' );
            case 'F' 
                          fprintf('Better try again\n' ); 
             otherwise 
                           fprintf('Invalid grade\n' );    end

Question 19

plot y=x^2

Answer 19

The script file is as follows:
&raquo_space; x = [-100:20:100]; 20
is increments, and the others are the range
» y = x.^2; [Here the “dot” is necessary]
&raquo_space; plot(x, y)

Question 20

3D graph examples

Answer 20

Let us create a 3D surface map for the function 𝑧=𝑥𝑒^(−(𝑥^2+𝑦^2 ) )
The script file is as follows:
&raquo_space; [x,y] = meshgrid(-2:0.2:2);
&raquo_space; z = x.* exp(-x.^2 - y.^2);
&raquo_space; surf(x,y,z)

surf to create a surface plot

mesh grid points over the domain of the function(the range of values)

Question 21

another 3d graph examples

Answer 21

Let us consider another example to create a 3D surface map for the function 𝑧=sinx+cosy
The script file is as follows:
&raquo_space; [x,y] = mesh grid(-2:0.2:2); these are coordinates
&raquo_space; z=sin(x)+cos(y);
&raquo_space; surf(x,y,z)

surf to create a surface plot

mesh grid points over the domain of the function(the range of values)

Question 22

xlabel and y label commands for x, y axis for graphs

Answer 22

> > x = [0:0.01:10];
&raquo_space; y = sin(x);
&raquo_space; plot(x, y), xlabel(‘x’), ylabel(‘Sin(x)’), title(‘Sin(x) Graph’)

Question 23

polar coordinates

Answer 23

polarplot(theta,rho). Here thetais the angle in radians andrhois the radius value for each point/height of sin cos wave combo

> > theta = 0:0.01:2pi;
&raquo_space; rho = sin(2theta).cos(2theta);
&raquo_space; polarplot(theta,rho)

Question 24

another example of polar coordinates

Answer 24

Plot the curve 𝑟=1−𝑠𝑖𝑛𝜃
The script file is as follows:
&raquo_space; theta = 0:0.01:2*pi;
&raquo_space; rho = 1 - sin(theta);
&raquo_space; polar(theta, rho)

polarplot(theta,rho). Here thetais the angle in radians andrhois the radius value for each point/height of sin cos wave combo

Question 25

parametric curve

Answer 25

fplot3(xt,yt,zt), whichplots the parametric curve xt=x(t), yt=y(t), and zt=z(t) over the default interval−5<𝑡<5

Let us plot a parametric curve: x=sin(t); y=cos(t); z=t over the default parameter range[-5 5].

> > syms t
xt = sin(t);
yt = cos(t);
zt = t;
fplot3(xt,yt,zt)

Question 26

parametric plots, with different range

Answer 26

Plot the parametric curve 𝑥=𝑒^(−𝑡/10)sin⁡(5𝑡);
𝑦=𝑒^(−𝑡/10)cos⁡(5𝑡);
z=t;
[-10 10]

   >> syms t 
   >> xt = exp(-t/10).*sin(5*t); 
   >> yt = exp(-t/10).*cos(5*t); 
   >> zt = t; 
   >> fplot3(xt,yt,zt,[-10 10])

Question 27

Radius of curvature

Answer 27

RADIUS FORMULA
d=(1+G(x,y)^2)^3/2))/c

rho=d/c

Find the radius of curvature of the curve 𝑥^(2/3)+𝑦^(2/3)=𝑎^(2/3) at any point (x,y) of the curve.
» syms x y a
» F(x,y)=x^(2/3)+y^(2/3)-a^(2/3);
dy_dx = - diff(F,x)/diff(F,y)
Out put: dy_dx(x, y) =-y^(1/3)/x^(1/3)
» G(x,y)=-y^(1/3)/x^(1/3);
» a=diff(G,x);
» b=diff(G,y);
» c=a+bG(x,y)
Out put: c(x, y) =
1/(3x^(2/3)y^(1/3)) +
y^(1/3)/(3x^(4/3))

> > simplify(c)
Out put: (x, y) =(x^(2/3) + y^(2/3))/(3x^(4/3)y^(1/3))
d=(1+G(x,y)^2)^(3/2)
Out put: d =(y^(2/3)/x^(2/3) + 1)^(3/2)
rho=d/c
Out put: rho(x, y) =(y^(2/3)/x^(2/3) + 1)^(3/2)/(1/(3x^(2/3)y^(1/3)) + y^(1/3)/(3x^(4/3)))
simplify(rho(x,y))
Out put: (y^(2/3)/x^(2/3) + 1)^(3/2)/(1/(3x^(2/3)y^(1/3)) + y^(1/3)/(3x^(4/3)))

Question 28

Radius of curvature using theta

Answer 28

theta formula
rho = (r^2+(r^2’’))
^3/2/(r^2+2r^2’‘-r*r^2’’)

syms theta
» r=exp(2theta);
» r1=diff(r,theta);
» r2=diff(diff(r,theta));
» a=(r^2+r1^2)^(3/2)
Out put: a =(5exp(4theta))^(3/2)
» b=r^2+2r1^2-rr2
Out put: b =5exp(4theta)
» rho=a/b
Out put: rho =(exp(-4theta)(5exp(4theta))^(3/2))/5
simplify(rho)
Out put: 5^(1/2)exp(4*theta)^(1/2)

Question 29

why is a dot used in graph plotting and numerical integration

Answer 29

scaler: single value,
vector: 1D-array

MATLAB can only do multiplication on matrix, when ever there is a vector quantity u need to put a dot

Dots are used in MATLAB to specify element-wise operations when working with vectors or matrices.
.*, .^, ./

special case:
trigonometric functions like neither vectors nor matrices themselves. Instead, they are element-wise functions

and exp also

but exp(-x.^2 - y.^2);
inside u should still but dot for these
and sin(-x.^2 - y.^2);

Imagine each will give u a matrix value, if its one
matrix set exp and trig functions can work, but if its multiple u have to put dot

Question 30

partial derivative of a function, syntax for fxy

Answer 30

f=sin(𝑥)+𝑦^3+𝑥^10−𝑦^2+log⁡(𝑥), then find 𝑓𝑥 & 𝑓𝑦.
» syms x y
» f=sin(x)+y^3+x^10-y^2+log(x);
» diff(f,x)
» diff(f,y)
syntax
»diff(f,x,y)
»diff(f,y,x)
Out put: f =log(x) + sin(x) + x^10 - y^2 + y^3
ans =cos(x) + 1/x + 10x^9
ans =3y^2 - 2*y

Question 31

double partial derivation

Answer 31

f=𝑥^2+2∗𝑦^2−22,then find 𝑓𝑥𝑥 & 𝑓𝑦𝑦
&raquo_space; syms x y
&raquo_space; f=x^2+2y^2-22
&raquo_space; diff(f,x,2)
&raquo_space; diff(f,y,2)
Out put: f = x^2 + 2y^2 – 22
ans =2
ans=4

Question 32

Taylor and maclaurin

Answer 32

Expand f(x)=𝑒^𝑥𝑠𝑖𝑛𝑥 about the point x=2 up to third degree terms.
» syms x
» f = exp(xsin(x));
» t= taylor(f, ‘ExpansionPoint’, 2, ‘Order’, 3)
Out put:
t=exp(2sin(2)) + exp(2sin(2))(2cos(2) + sin(2))(x - 2) + exp(2sin(2))(x - 2)^2(cos(2) - sin(2) + (2cos(2) + sin(2))*(cos(2) + sin(2)/2))

Question 33

Taylors Theorem theorem without giving a value only degree

Answer 33

Expand f(x)=log(secx) about the origin up to six degree terms.
» syms x
» f = log(sec(x));
» T= taylor(f, ‘Order’, 7)
Out put:
T = x^6/45 + x^4/12 + x^2/2

Question 34

taylors plot graphs

Answer 34

1. sinx: &raquo_space; t = [0:0.1:2*pi]
   &raquo_space; a = sin(t);
   &raquo_space; plot(t,a)
2. cosx: &raquo_space; t = [0:0.1:2*pi]
   &raquo_space; a = cos(t);
   &raquo_space; plot(t,a)

Question 35

taylors f(x,y)

Answer 35

Expand f(x,y)=e^x𝑐𝑜𝑠𝑦 about the point 𝑥=1,𝑦=𝜋/4 up to three degree
terms.
&raquo_space; syms x y
&raquo_space; f=exp(x)cos(y);
&raquo_space; t = taylor(f, [x, y], [1, pi/4], ‘Order’, 3)
Out put:
T = (2^(1/2)exp(1))/2 - (2^(1/2)exp(1)(y - pi/4)^2)/4 + (2^(1/2)exp(1)(x - 1)^2)/4 - (2^(1/2)exp(1)(y - pi/4))/2 + (2^(1/2)exp(1)(x - 1))/2 - (2^(1/2)exp(1)(y - pi/4)*(x - 1))/2

Question 36

f(x,y)=0

Answer 36

Expand 𝑓(𝑥,𝑦)=𝑒^𝑦 log⁡(1+𝑥) about the origin up to fourth degree
terms.
&raquo_space; syms x y
&raquo_space; f=exp(y)log(1+x);
&raquo_space; T= taylor(f, [x, y], ‘Order’, 4)
Out put:
T= x^3/3 - (x^2y)/2 - x^2/2 + (xy^2)/2 + xy + x

Question 37

why is syms used

and why is ode used

Answer 37

syms command is used to define symbolic variables. These are variables that represent mathematical symbols rather than numeric values. Symbolic variables allow you to perform symbolic computations, such as differentiation, integration, algebraic manipulation,

ODE in MATLAB refers to solving Ordinary Differential Equations numerically or symbolically. MATLAB offers various built-in functions and tools for handling ODEs.

Question 38

differential equations and boundary condition

Answer 38

𝑑𝑦/𝑑𝑡=𝑡𝑦
» syms y(t)
» ode = diff(y,t) == ty;
» ySol(t) = dsolve(ode)
Output: ySol(t) =C1exp(t^2/2)

Solve the differential equation: 𝑑𝑥/𝑑𝑡=𝑥+𝑡, given 𝑥(0)=0.
» syms x(t)
» ode = diff(x,t) == x+t;
» xSol(t) = dsolve(ode)
Output: xSol(t) =C1*exp(t) - t – 1.
To find the value of C1, we use:
» cond = x(0) == 0;
» xSol(t) = dsolve(ode,cond)
Output: ySol(t) =exp(t) - t - 1

Question 39

PDE arbitrary constants:
z=(x-a)^2+(y-b)^2

Answer 39

syms x y a b p q
z=(x-a)^2+(y-b)^2;
eq1=p==diff(z,x)
c1=solve(eq1,a)
eq2=q==diff(z,y)
c2=solve(eq2,b)
pde=subs(z,a,c1)
pde=subs(pde,b,c2)

Question 40

arbitrary another example
2z=x^2/a^2+y^2/b^2

Answer 40

syms x y a b p q
z=x^2/(2a^2)+y^2/(2b^2)
eq1=p==diff(z,x)
c1=solve(eq1,a)
eq2=q==diff(z,y)
c2=solve(eq2,b)
pde=subs(z,a,c1)
pde=subs(pde,b,c2)

Question 41

Plot the graph of solution of first order differential equations

𝑑𝑥/𝑑𝑡=𝑥+𝑡, given 𝑥(0)=0.

Answer 41

𝑑𝑥/𝑑𝑡=𝑥+𝑡, given 𝑥(0)=0.

> > tspan=[0 2]; % specify time span
x0=0; % specify x0
[t,x]=ode45(@(t,x)x+t,tspan,x0);
% now execute ode45
disp([t,x])
𝑡 is a vector with all discrete points ; and 𝑥 contains the values of the variable 𝑥.
plot(t,x) % plot t verses x

Question 42

(𝑥^2−1) 𝑑𝑦/𝑑𝑥+2𝑥𝑦=1, given 𝑦(0)=1.

Answer 42

(𝑥^2−1) 𝑑𝑦/𝑑𝑥+2𝑥𝑦=1, given 𝑦(0)=1.
» y0=1;
» x0=0;
» xend=[1,5,10]; same for all values,
This specifies the end points for x
x where the solution will be evaluated.
» xspan=[x0,xend];
Combines the start (x0) and end points (xend) into an array, defining the intervals over which the ODE is solved.
» [x,y]=ode45(@(x,y)(1-2xy)/(x^2-1),xspan,y0);
» disp([x,y])
» plot(x,y)

Question 43

non-linear differential equation

(dy/dt+y)^2=1, y(0)=0

Answer 43

> > syms y(t)
ode = (diff(y,t)+y)^2 == 1;
ySol(t) = dsolve(ode)
Output: ySol(t) =
C1exp(-t) + 1; C2exp(-t) – 1
To find the values of C1 and C2, we use:
cond = y(0) == 0;
ySol(t) = dsolve(ode,cond)
Output: ySol(t) =exp(-t) – 1; 1 - exp(-t)

Question 44

radius of curvature of the parametric curve

Answer 44

Formula of Parametric Curve
(x1^2+y1^2)^3/2/
(x1y2-y1x2)

=6𝑡^2−3𝑡^4,
𝑦=8𝑡^3.
» syms t
» x=6t^2-3t^4;
» x1=diff(x,t);
» x2=diff(diff(x,t));
» y=8t^3;
» y1=diff(y,t);
» y2=diff(diff(y,t));
» a=(x1^2+y1^2)
Out put: a =(- 12t^3 + 12t)^2 + 576t^4
» b=simplify(a)
Out put: b =144t^2(t^2 + 1)^2
» c=(x1y2)-(y1x2)
Out put: c =48t(- 12t^3 + 12t) + 24t^2(36t^2 - 12)
» d=simplify(c)
Out put: d =288t^2(t^2 + 1)
» e=b^(3/2)
Out put: e =(144t^2(t^2 + 1)^2)^(3/2)
»rho=e/d
Out put: rho =(144t^2(t^2 + 1)^2)^(3/2)/(288t^2(t^2 + 1))
»simplify(rho)
Out put: rho= (6(t^2(t^2 + 1)^2)^(3/2))/(t^2(t^2 + 1))