Notes

1. Realize the given functions sin t, cos t, sinh t and cosh t for the vector t = [0, 10] with
increment 0.01
PROGRAM
clc
close all
clear all
t = 0:0.01:10;
a = sin(t);
b = cos(t);
c = sinh(t);
d = cosh(t);
subplot(2,2,1)
plot(t,a)
title('sin t')
subplot(2,2,2)
plot(t,b)
title('cos t')
subplot(2,2,3)
plot(t,c)
title('sinh t')
subplot(2,2,4)
plot(t,d)
title('cosh t')

ECL 201 Scientific Computing Laboratory 2021-2022

Dept. Of Electronics and Communication Engineering
OUTPUT

2. Compute the first and second derivatives of sin t, cos t, sinh t, cosh t functions using built in
tools such as grad and plot the derivatives over the respective functions for the vector
t = [-5, 5] with increment 0:01

PROGRAM
clc
close all
clear all
t=-5:0.01:5;
a=sin (t);
subplot(3,1,1)
plot(t,a)
title('sin t')
b=cos (t);
c= sinh (t);
d=cosh (t);
da=gradient(a);
subplot(3,1,2)
plot(t,da)
title('first derivative of sin t')

ECL 201 Scientific Computing Laboratory 2021-2022

Dept. Of Electronics and Communication Engineering
d2a=gradient(da);
subplot(3,1,3)
plot(t,d2a)
title('second derivative of sin t')
figure
subplot(3,1,1)
plot(t,b)
title('cos t')
db=gradient(b);
subplot(3,1,2)
plot(t,db)
title('first derivative of cos t')
d2b=gradient(db);
subplot(3,1,3)
plot(t,d2b)
title('second derivative of cos t')
figure
subplot(3,1,1)
plot(t,c)
title('sinh t')
dc=gradient(c);
subplot(3,1,2)
plot(t,dc)
title('first derivative of sinh t')
d2c=gradient(dc);
subplot(3,1,3)
plot(t,d2c)
title('second derivative of sinh t')
figure
subplot(3,1,1)
plot(t,d)
title('cosh t')
dd=gradient(d);
subplot(3,1,2)
plot(t,dd)
title('first derivative of cosh t')
d2d=gradient(dd);
subplot(3,1,3)
plot(t,d2d)
title('second derivative of cosh t')

ECL 201 Scientific Computing Laboratory 2021-2022

Dept. Of Electronics and Communication Engineering
OUTPUT

ECL 201 Scientific Computing Laboratory 2021-2022

Dept. Of Electronics and Communication Engineering

ECL 201 Scientific Computing Laboratory 2021-2022

Dept. Of Electronics and Communication Engineering
3. Find the derivative of following functions.
(a) f = sin (5 ∗ x)
(b) g = exp(x) ∗ cos (x)
PROGRAM
(a)
clc
close all
clear all
syms x
f = sin(5*x);
y = diff(f) %to take the first derivative of f
%can also use gradient(f) instead of diff(f)
vpa(subs(f,x,2)) %find the value of f at x = 2
vpa(subs(y,x,2)) %find the derivative of f at x = 2
(b)
clc
close all
clear all
syms x
g = exp(x)*cos(x);
y = diff(g) %take the first derivative of g
vpa(subs(y,x,4)) %derivative of g for a given value of x=4
z = diff(g,2) %take the second derivative of g
%You can get the same result by taking the derivative twice:
diff(diff(g))
vpa(subs(z,x,1)) %second derivative of g for a value of x=1
OUTPUT
(a)
ans = 5*cos(5*x)
ans = -0.54402111088936981340474766185138
ans = -4.1953576453822622612943197391203
(b)
y = exp(x)*cos(x) - exp(x)*sin(x)

ECL 201 Scientific Computing Laboratory 2021-2022

Dept. Of Electronics and Communication Engineering
ans = 5.6322837041610723980357343265599
z = -2*exp(x)*sin(x)
ans = -4.574710574357684782416343813401
4. Compute d
dt sin(st) ,
d
ds sin(st) and d
2
dt2
sin (st)

PROGRAM
%Derivatives of expressions with several variables
clc
close all
clear all
syms s t
f = sin(s*t);
y = diff(f,t) %calculates the partial derivative ∂f/∂t
diff(f,s) %%calculates the partial derivative ∂f/∂s
diff(f, t, 2) %calculate the partial derivative ∂2f/∂t2
vpa(subs(y,[s,t],[2,1])) %find the ∂f/∂t for s=2 and t=1
OUTPUT
ans = s*cos(s*t)
ans = t*cos(s*t)
ans = -s^2*sin(s*t)
ans = -0.83229367309428477399513645900152
5. Realize the function f(t) = 4t

2 + 3 and plot it for the vector [-5,5] with increment 0.01

PROGRAM
clc
close all
clear all
t=-5:0.01:5;
y=4*(t.^2)+3;
plot(t,y,'k','linewidth',2)
title('Realization of f(t)')
xlabel('t')
ylabel('f(t)')

ECL 201 Scientific Computing Laboratory 2021-2022

Dept. Of Electronics and Communication Engineering
OUTPUT

6. Familiarize the general integration tools such as int and integral.
Compute following integrations.
(a) ∫
−2x
(1+x
2)
2
dx (b) ∫
1
1+u2
du (c) ∫ e
(−x
2) + log(x
2
) dx

(d) ∫
x
1+z
2
dx (e) ∫ sin (aθ + b)

(f) ∫ exp(−x
2
) ∗ log (x
2
)

∞
0

dx (g) ∫
1
x
3−2x−c
2
0
dx given c = 5

(h) ∫ log(x) dx 1
0

(i) ∫ e
−x
2
dx 1
0

(j) ∫ x
n dx given n = 2
1
0

(k) ∫ sin (aθ + b)

π
2
0

da given θ = 90 and b = 5

(l) ∫ sin (aθ + b)
π
0

dθ given a = 1 and b = 2

PROGRAM
%Indefinite Integral of Univariate Expression
(a)

ECL 201 Scientific Computing Laboratory 2021-2022

Dept. Of Electronics and Communication Engineering
clc
close all
clear all
syms x
f = -2*x/(1+x^2)^2;
F = int(f)
vpa(subs(F,x,1)) %to get function value for x=1
(b)
syms u
f = 1/(1+u^2);
int(f)
(c)
syms x
f = exp(-x^2)*log(x)^2;
int(f)
%Indefinite Integrals of Multivariate Function
(d)
clc
close all
clear all
syms x z
f = x/(1+z^2);
Fx = int(f,x) %integral with respect to the variable x
Fz = int(f,z) %integral with respect to the variable z
(e)
syms a b theta
f = sin(a*theta+b);
Fa = int(f,a) %integral with respect to the variable a
Fb = int(f,b) %integral with respect to the variable b
Ft = int(f,theta) %integral with respect to the variable theta
%Definite Integral
(f)
clc
close all
clear all

ECL 201 Scientific Computing Laboratory 2021-2022

Dept. Of Electronics and Communication Engineering
f = @(x) exp(-x.^2).*log(x.^2);
q = integral(f,0,Inf)
(g)
f = @(x,c) 1./(x.^3-2*x-c);
f(4,6) % to get function value for given x=4 & c=6
q = integral(@(x) f(x,5),0,2)
(h)
f = @(x)log(x);
q = integral(f,0,1)
(i)
f = @(x)exp(-x.^2);
q = integral(f,0,1)
(j)
f = @(x,n)x.^n;
q = integral(@(x)f(x,2),0,1)

(k)
f = @(a,theta,b)sin(a*theta+b);
q = integral(@(a) f(a,90,45),0,pi/2)

(l)
f = @(a,theta,b)sin(a*theta+b);
q = integral(@(theta) f(1,theta,2),0,pi)

OUTPUT
(a)
F = 1/(x^2 + 1)
ans = 0.5
(b)

ECL 201 Scientific Computing Laboratory 2021-2022

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ans = atan(u)
(c)
ans = int(exp(-x^2)*log(x)^2, x)

(d)
Fx = x^2/(2*(z^2 + 1))
Fz = x*atan(z)
(e)
Fa = -cos(b + a*theta)/theta
Fb = -cos(b + a*theta)
Ft = -cos(b + a*theta)/a
(f)
q = -1.7401
(g)
ans = 0.0200
q = -0.4605
(h)
q = -1.0000
(i)
q = 0.7468
(j)
q = 0.3333
(k)
q = 0.0117

ECL 201 Scientific Computing Laboratory 2021-2022

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(l)
q = -0.8323
7. Use general integration tool to compute ∫ (4t
2 + 3)
2
−2

dt

Repeat the above steps with trapezoidal and Simpson method and compare the results.
PROGRAM
%General integration tool
clc;
clear all;
close all;
f = @(t)4*t.^2+2;
integral(f,-2,2)

%Numerical integration tool: Trapezoidal Rule
f=@(t)4*t^2+2;
a = input('Enter lower limit a: ');
b = input('Enter upper limit b: ');
n = input('Enter the no. of sub interval: ');
h = (b-a)/n;
sum = 0;
for k = 1:1:n-1
t = a+k*h;
y = f(t);
sum = sum+y;
end
result = h/2*(f(a)+f(b)+2*sum);
fprintf('n The value of integration is %f',result);

%Trapezoidal Rule using function
t = -2:0.4:2;
f = 4*t.^2+2;
trapz(t,f)

%Numerical integration tool: Simpson’s 1/3 Rule
syms x

ECL 201 Scientific Computing Laboratory 2021-2022

Dept. Of Electronics and Communication Engineering
f = @(t)4*t^2+2;
a = input('Enter lower limit a: ');
b = input('Enter upper limit b: ');
n = input('Enter the no. of sub interval: ');
h = (b-a)/n;
odd = 0;
even = 0;
for k = 1:1:n-1
t = a+k*h;
y(k) = f(t);
end
for k = 1:1:n-1
if rem(k,2) == 1
odd = odd + y(k);%sum of odd terms
else
even = even + y(k); %sum of even terms
end
end
result = (h/3)*(f(a)+f(b)+4*odd+2*even);
fprintf('n The value of integration is %f',result);

%Numerical integration tool: Simpson’s 3/8 Rule
syms x
f = @(t)4*t^2+2;
a = input('Enter lower limit a: ');
b = input('Enter upper limit b: ');
n = input('Enter the no. of sub interval: ');
h = (b-a)/n;
sum1 = 0;
sum2 = 0;
for k = 1:1:n-1
t = a+k*h;
y(k) = f(t);
end
for k = 1:1:n-1
if rem(k,3) == 0
sum1 = sum1 + y(k);%sum of f(a+3h), f(a+6h),....
else
sum2 = sum2 + y(k); %sum of f(a+h),f(a+2h),f(a+4h),.....
end
end

ECL 201 Scientific Computing Laboratory 2021-2022

Dept. Of Electronics and Communication Engineering
result = (3*h/8)*(f(a)+f(b)+3*sum2+2*sum1);
fprintf('n The value of integration is %f',result);
OUTPUT
%General integration tool
ans = 29.3333
%Numerical integration tool: Trapezoidal Rule
Enter lower limit a: -2
Enter upper limit b: 2
Enter the no. of sub interval: 10
The value of integration is 29.760000
%Trapezoidal Rule using function
ans = 29.7600
%Numerical integration tool: Simpson’s 1/3 Rule
Enter lower limit a: -2
Enter upper limit b: 2
Enter the no. of sub interval: 10
The value of integration is 29.333333
%Numerical integration tool: Simpson’s 3/8 Rule
Enter lower limit a: -2
Enter upper limit b: 2
Enter the no. of sub interval: 10
The value of integration is 27.864000
8. Use general integration tool to compute 1
√2π
∫ e
−x
2
2 ∞
0
dx

Repeat the above steps with trapezoidal and Simpson method and compare the results.
PROGRAM
%General integration tool
clc;
clear all;
close all;

ECL 201 Scientific Computing Laboratory 2021-2022

Dept. Of Electronics and Communication Engineering
f = @(x)1/sqrt(2*pi)*exp(-x.^2/2)
integral(f,0,inf)

%Numerical integration tool: Trapezoidal Rule
f = @(x)1/sqrt(2*pi)*exp(-x.^2/2)
a = input('Enter lower limit a: ');
b = input('Enter upper limit b: ');
n = input('Enter the no. of sub interval: ');
h = (b-a)/n;
sum = 0;
for k = 1:1:n-1
x = a+k*h;
y = f(x);
sum = sum+y;
end
result = h/2*(f(a)+f(b)+2*sum);
fprintf('n The value of integration is %f',result);

%Trapezoidal Rule using function
x = 0:0.5:9999;
f = 1/sqrt(2*pi)*exp(-x.^2/2);
q = trapz(x,f)

%Numerical integration tool: Simpson’s 1/3 Rule
syms x
f = @(x)1/sqrt(2*pi)*exp(-x.^2/2)
a = input('Enter lower limit a: ');
b = input('Enter upper limit b: ');
n = input('Enter the no. of sub interval: ');
h = (b-a)/n;
odd = 0;
even = 0;
for k = 1:1:n-1
x = a+k*h;
y(k) = f(x);
end
for k = 1:1:n-1
if rem(k,2) == 1

ECL 201 Scientific Computing Laboratory 2021-2022

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odd = odd + y(k);%sum of odd terms
else
even = even + y(k); %sum of even terms
end
end
result = (h/3)*(f(a)+f(b)+4*odd+2*even);
fprintf('n The value of integration is %f',result);

%Numerical integration tool: Simpson’s 3/8 Rule
syms x
f = @(x)1/sqrt(2*pi)*exp(-x.^2/2)
a = input('Enter lower limit a: ');
b = input('Enter upper limit b: ');
n = input('Enter the no. of sub interval: ');
h = (b-a)/n;
sum1 = 0;
sum2 = 0;
for k = 1:1:n-1
x = a+k*h;
y(k) = f(x);
end
for k = 1:1:n-1
if rem(k,3) == 0
sum1 = sum1 + y(k);%sum of f(a+3h), f(a+6h),....
else
sum2 = sum2 + y(k); %sum of f(a+h),f(a+2h),f(a+4h),.....
end
end
result = (3*h/8)*(f(a)+f(b)+3*sum2+2*sum1);
fprintf('n The value of integration is %f',result);
OUTPUT
%General integration tool
ans = 0.5000
%Numerical integration tool: Trapezoidal Rule
Enter lower limit a: 0
Enter upper limit b: 10000
Enter the no. of sub interval: 20000

ECL 201 Scientific Computing Laboratory 2021-2022

Dept. Of Electronics and Communication Engineering
The value of integration is 0.500000

%Trapezoidal Rule using function
ans = 0.5000
%Numerical integration tool: Simpson’s 1/3 Rule
Enter lower limit a: 0
Enter upper limit b: 10000
Enter the no. of sub interval: 20000
The value of integration is 0.500000
%Numerical integration tool: Simpson’s 3/8 Rule
Enter lower limit a: 0
Enter upper limit b: 10000
Enter the no. of sub interval: 20000
The value of integration is 0.499981