Question

MATLAB question：

Task 4 Many real world objects travel in a jagged manner that is best represented by piecewise functions. For example, the velocity of a rocket may change sharply when thrusters are enabled or disabled and due to various external factors. The velocity of a rocket can be modelled by v(t) where t represents the time 112-5t 0sts10 1100-5t10StS20 vt) 502( 20)25 20sIs 30 1520e0-30) 0 t >30 otherwise Write a function that takes a start and end time as inputs and returns the vectors v and t. The vector t should contain values between start_t and end t at 0.01 increments. The vector v should contain v(t) for each value in t. function [t, v VPiecewise(start_t, end _t) Use the function in an m-file to plot the velocities for t--5 to 80. Hint: Use if statements to break up the different profile conditions. Task 5 Recall that the following Taylor series is used to approximate Cosine: cos(x)- (2n) You have been tasked with developing an M-file that allows other engineers to quickly determine the minimum n needed to reduce the truncation error below an error threshold. The truncation error is the absolute error between the approximation and MATLAB's cos) function. Note: You may want to Google the Taylor series to gain a better understanding of it. Your code should do the following: Ask the user to enter an error threshold value (it should work with vectors) Find the number of terms (minimum n) needed so that the truncation error is below the user error threshold for x--2π to 2 Print the minimum n value, the mean truncation error and the maximum truncation error using fprintf. Plot the truncation error against x. 1. 2. 3. 4. Test your code with the following error thresholds: 0.1, 0.01, 1e-10. The answers for the thresholds above should be n-8, 9, 16, respectively. This is a reference to see whether your code is working correctly. Hint: Your code may include both while and for loops and/or a created function to calculate the approximation of Cosine.

Answer 1

TASK 4:

Programming code in MATLAB;

clc;

clear all;

close all;

function [t, v] = VPiecewise(start_t, end_t)
////////////// Creates two vectors v and t
  
//////////// Creating vector t at increments of 0.01
t = start_t:0.01:end_t;
  
///////////// Initializing vector v
v = [];
  
/////////// Using loop to find v values
for i=1:length(t)
  
/////////  Finding profile conditions
if t(i)>=0 && t(i)<=10
//////////// Updaitng vector
v = [v ((11*t(i)*t(i)) - (5*t(i)))];
elseif t(i)>10 && t(i)<=20
///////////// Updaitng vector
v = [v (1100 - (5*t(i)))];
elseif t(i)>20 && t(i)<=30
////////////////// Updaitng vector
v = [v ((50*t(i)) + ( 2 * ( (t(i)-20) ^ (2.5) )))];
elseif t(i)>30
/////////////// Updaitng vector
v = [v ((1520) * ( exp((-0.1)*(t(i)-30)) ))];
else
v = [v 0];
end % If end
  
end % For loop end
  
end % Function end

Script:

//////////// Plotting travel of a real world objects in a jagged manner
[t, v] = VPiecewise(-5, 80);

///////////// Plotting graph
plot(t, v);
title("Travel of a real world objects in a jagged manner");
xlabel("t");
ylabel("v");

MATLAB O/P:

TASK 5):

Thavel of areal voild objedts in a jegged manner 2500 2000 150c 1000 -500 20

MATLAB CODE:

clc;

clear all;

close all;

%%%%%%%%%% the threshold value

Trshold = input('Please enter the error threshold:')

xvl = linspace(-2*pi,2*pi);

cos_trm = ones(size(xvl));

cos_appx = cos_trm;

nvl = 2;

xsqrd = xvl.^2*nvl;

%%%%%%%% Calculation of the approximation

while abs(cos_trm(end)) > Trshold

cos_trm = - cos_trm.*xsqrd./factorial(nvl);

cos_appx = cos_appx + cos_trm;

nvl = nvl + 2;

end

%%%%%%%%%% Calculate truncate error

trun_err = abs(cos_appx - cos(xvl));

fprintf('N value: %d \n',nvl);

fprintf('Mean truncation error: %0.6f \n',mean(trun_err));

fprintf('The error is: %0.6f \n', trun_err);

%%%%%%%%%%%% Perform the plot

plot(xvl,trun_err)

MATLAB O/P:

0 5 1 し0 40