Objectives
Realize the functions sin t, cos t, sin ht and cos ht for the vector t = [0; 10] with increment 0:01
Compute the rst and second derivatives of these functions using built in tools such as grad.
Plot the derivatives over the respective functions and appreciate.
Familiarize the numerical integration tools in the language you use.
Realize the function
f(t) = 4t2 + 3
and plot it for the vector t = [-5; 5] with increment 0:01
Use general integration tool to compute
Theory
Trigonometric and angular functions are discussed here
To work the functions of trigonometry we have to import the numpy library and to plot the figures we have to import the matplotlib library
1. sin() :- This function returns the sine of value passed as argument. The value passed in this function should be in radians.
2. cos() :- This function returns the cosine of value passed as argument. The value passed in this function should be in radians.
Realizing the sin, cos,sinh,cosh functions
Program for plotting sine t signal
# program for plotting the sine t signal
#importing libraries
from numpy import *
from matplotlib.pyplot import *
# Get x values of the sine wave
t = arange(0, 10, 0.01);
# calculating the amplitude
y = sin(t)
# Plot a sine wave using time and amplitude obtained for the sine wave
plot(t,y)
title('Sine wave') # Give a title for the sine wave plot
xlabel('Time') # Give x axis label for the sine wave plot
ylabel('Amplitude = sin(time)') # Give y axis label for the sine wave plot
grid(True, which='both') # enabling the grid
show() # showing the figure
Output 1
Program for plotting cos t wave
# program for plotting the Cose t signal
#importing libraries
from numpy import *
from matplotlib.pyplot import *
# Get x values of the wave
t = arange(0, 10, 0.01);
# calculating the amplitude
y = cos(t)
# Plot a wave using time and amplitude obtained for the sine wave
plot(t,y)
title('cose t wave') # Give a title for the wave plot
xlabel('Time') # Give x axis label for the wave plot
ylabel('Amplitude = cos(t)') # Give y axis label for the wave plot
grid(True, which='both') # enabling the grid
show() # showing the figure
Output 2
Program for plotting Sin ht signal
# program for plotting the Sine ht signal
#importing libraries
from numpy import *
from matplotlib.pyplot import *
# Get x values of the wave
t = arange(0, 10, 0.01);
# calculating the amplitude
y = sinh(t)
# Plot a wave using time and amplitude obtained for the sine wave
plot(t,y)
title('Sine ht wave') # Give a title for the wave plot
xlabel('Time') # Give x axis label for the wave plot
ylabel('Amplitude = Sinh(t)') # Give y axis label for the wave plot
grid(True, which='both') # enabling the grid
show() # showing the figure
Output
Program for plotting Cos ht signal
# program for ploting the Cose ht signal
#importing libraries
from numpy import *
from matplotlib.pyplot import *
# Get x values of the wave
t = arange(0, 10, 0.01);
# calculating the amplitude
y = cosh(t)
# Plot a wave using time and amplitude obtained for the sine wave
plot(t,y)
title('Cos ht wave') # Give a title for the wave plot
xlabel('Time') # Give x axis label for the wave plot
ylabel('Amplitude = Cosh(t)') # Give y axis label for the wave plot
grid(True, which='both') # enabling the grid
show() # showing the figure
Output
Program
# program for plotting the Sineht & cosht signal
#importing libraries
from numpy import *
from matplotlib.pyplot import *
# Get x values of the wave
t = arange(-5, 5, 0.01);
# calculating the amplitude
y = sinh(t)
y1= cosh(t)
# Plot a wave using time and amplitude obtained for the sine wave
plot(t,y,label="sinht")
plot(t,y1,label="cosht")
title('Sineht & Cosht wave') # Give a title for the wave plot
xlabel('Time') # Give x axis label for the wave plot
ylabel('Amplitude') # Give y axis label for the wave plot
grid(True, which='both') # enabling the grid
legend()
show() # showing the figure
Output
Computing and plotting first and second derivative of sine and cos functions
Program
# program for plotting the Sin t signal
#importing libraries
from numpy import *
from matplotlib.pyplot import *
rcParams["figure.figsize"] = (15,10) # changing the figure size
# Get x values of the wave
t = arange(0, 10, 0.01);
# calculating the amplitude
y1 = sin(t) # calculating sine wave
sfd1= gradient(y1) # calculating first derivative
sfd2= gradient(sfd1) # calculating second derivative
# calculating the amplitude
y2 = cos(t) # calculating cos wave
cfd1= gradient(y2) # calculating first derivative
cfd2= gradient(cfd1) # calculating second derivative
# plotting sine and cos signals in subplot
fig, axs = subplots(3,2)
axs[0,0].plot(t,y1) # sin signal potting
axs[0,0].set_title("Sin wave")
axs[1,0].plot(t,sfd1)
axs[1,0].set_title("first sin derivative")
axs[2,0].plot(t,sfd2)
axs[2,0].set_title("second sin derivative")
axs[0,1].plot(t,y1) # cose signal potting
axs[0,1].set_title("cose wave")
axs[1,1].plot(t,cfd1)
axs[1,1].set_title("first cos derivative")
axs[2,1].plot(t,cfd2)
axs[2,1].set_title("second cos derivative")
show() # showing the figure
Output
Program 2
# program for plotting the Sinh t signal
#importing libraries
from numpy import *
from matplotlib.pyplot import *
rcParams["figure.figsize"] = (15,10) # changing the figure size
# Get x values of the wave
t = arange(0, 10, 0.01);
# calculating the amplitude
y1 = sinh(t) # calculating sine wave
sfd1= gradient(y1) # calculating first derivative
sfd2= gradient(sfd1) # calculating second derivative
# calculating the amplitude
y2 = cosh(t) # calculating cose wave
cfd1= gradient(y2) # calculating first derivative
cfd2= gradient(cfd1) # calculating second derivative
# ploting sine and cose signals in subplot
fig, axs = subplots(3,2)
axs[0,0].plot(t,y1) # sin signal potting
axs[0,0].set_title("Sinh wave")
axs[1,0].plot(t,sfd1)
axs[1,0].set_title("first sinh derivative")
axs[2,0].plot(t,sfd2)
axs[2,0].set_title("second sinh derivative")
axs[0,1].plot(t,y1) # cose signal potting
axs[0,1].set_title("cosh wave")
axs[1,1].plot(t,cfd1)
axs[1,1].set_title("first cosh derivative")
axs[2,1].plot(t,cfd2)
axs[2,1].set_title("second cosh rivative")
show() # showing the figure
Output
Realize the function f(t)=4t2+3
Progame
#importing libraries
from numpy import *
from matplotlib.pyplot import *
# realizing the function
t = arange(-5, 5, 0.01); #assigning the vector t
ans = (4*t**2)+3
plot(t,ans)
title("function f(t)=4t^2 + 3")
xlabel("vector t")
ylabel("function f(t)")
show()
Output
Familiarization of Numerical integration
SciPy has a number of routines for performing numerical integration. Most of them are found in the same scipy.integrate library. The scipy.integrate sub-package provides several integration techniques including an ordinary differential equation integrator. An overview of the module is provided by the help command: try the below command in colab
integrate?
Problem statement, find the integral of a function f(t) given below
-22f(t) dt
where f(t) = f(t)=4t2+3
Programe
from scipy.integrate import quad
def integrand(t):
return 4*t**2+3
ans, err = quad(integrand, -2, 2)
print("value of integral is : ",ans)
print("estimate of the absolute error is : ",err)
Output
value of integral is : 33.333333333333336
estimate of the absolute error is : 3.7007434154171886e-13
Inference
Realized the functions sin t, cos t, sin ht and cos ht for the vector t = [0; 10] with increment 0:01
Computed the second derivatives of these functions using built in tools such as grad.
Plotted the derivatives over the respective functions.
Familiarized the numerical integration tools in python.
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