Question

Compute the derivative of f(t) cos(37t) on the interval 1,1) using a centered differences approximation with discretization size N 10,40 and 70. Plot the resulting approximations on the same graph as the exact derivative. Find the maximum of the error for each of the three N values.

Answer 1

Matlab function for finding derivative using central difference clear all close all Function for which derivative have to do E-e(t) cos (3*pi*t) %exact derivative of the function df- (t) (3 pi)*(-sin(3 pi*t)) numerical derivative of the function num_df-e ( t ,h ) (cos ( 3 *pi* (e+h ) )-cos ( 3 *pi*tt-h)))./(2*h); %all time value for which derivative have to find t 1 in space (-1 , 1 , 1000): %a11 discretization size N [10 40 701; %loop for find ind derivative for all N for i=1: length(N) for j-1: length (tt) val(i,j)-num_df (tt (j),h); end hold on plot(tt, val(i,:)) %finding maximum error err-max (df (tt)-squeeze (val(i,:))); tprint f("AtMaximum error for N=%d ǐs %f.\n',n(1),err) end %plot for exact derivative plot (tt,df (tt, 'linewidth',2) title( 'Exact and numerical derivative for various N") xlabel('t') ylabel' derivative of f (t)') legend( For N-10, For N-40,For N-70, 'Exact value) 888886888888888888 End of Code 웅8888888888888888웅 Maximum error for N-10 is 4.669443. Maximum error for N-40 is 0.344964 Maximum error for N=70 is 0.113487.

Exact and numerical derivative for various N 10 -For N. 10 -For N:40 -For N:70 Exact value -4 -6 10 1 08 6.4 -02 06 08 Published with MATLAB R2018a

%%Matlab function for finding derivative using central difference
clear all
close all

%Function for which derivative have to do
f=@(t) cos(3*pi*t);

%exact derivative of the function
df=@(t) (3*pi)*(-sin(3*pi*t));

%numerical derivative of the function
num_df=@(t,h) (cos(3*pi*(t+h))-cos(3*pi*(t-h)))./(2*h);

%all time value for which derivative have to find

tt=linspace(-1,1,1000);

%all discretization size
N=[10 40 70];

%loop for findind derivative for all N
for i=1:length(N)
  
    h=2/N(i);
    for j=1:length(tt)
        val(i,j)=num_df(tt(j),h);
    end
  
    hold on
    plot(tt,val(i,:))
    %finding maximum error
    err=max(df(tt)-squeeze(val(i,:)));
    fprintf('\tMaximum error for N=%d is %f.\n',N(i),err)
end
%plot for exact derivative
plot(tt,df(tt),'linewidth',2)
title('Exact and numerical derivative for various N')
xlabel('t')
ylabel('derivative of f(t)')
legend('For N=10','For N=40','For N=70','Exact value')

%%%%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%%%%