Expand f (x, y) =e^xln(1 + y) in terms of x and y up to the terms of 3rd degree using Taylor series.

clear 
clc
close all
syms x y 
f=input('Enter the function f(x,y): ')
I=input('Enter the point[a,b] at which Taylor series is sought: ');
a=I(1);b=I(2);
n=input('Enter the order of series:');
tayser=taylor(f,[x,y],[a,b],'order',n)
subplot(1,2,1);
ezsurf(f); %Function plot
subplot(1,2,2);
ezsurf(tayser); % Taylors series of f

[Sam Chak]

You have learned very well. Keep up!

[Ananya Pai]

hey can anyone pls send me the code for the question. not getting the expected output

[John D'Errico]

What function did you enter? Do you understand how to write the exponential function in MATLAB? How about the NATURAL log? What is log(x) in MATLAB?

[Ananya Pai]

exp(x)*ln(1+y) ? is this correct ?

when i run the program i am not getting the next line that it is supposed to ask

[Torsten]

I suggest you first try to make your program work without manual inputs.

Answer 1

[Hari]

Hi,

I understand you are facing difficulty in expanding the function using Taylor series up to the terms of 3rd degree.

I assume that the expansion is centered around the point . To expand the function using Taylor series, we need to calculate the partial derivatives of with respect to x and y, and evaluate them at . Then, we can use these derivatives to construct the terms of the expansion.

Here is a sample code to do that:

syms x y;

% Define the function
f = exp(x)*log(1 + y);

% Calculate the partial derivatives
f_x = diff(f, x);
f_y = diff(f, y);
f_xx = diff(f_x, x);
f_yy = diff(f_y, y);
f_xy = diff(f_x, y);

% Evaluate the derivatives at (a, b) = (0, 0)
a = 0;
b = 0;
f_x_a_b = subs(f_x, [x, y], [a, b]);
f_y_a_b = subs(f_y, [x, y], [a, b]);
f_xx_a_b = subs(f_xx, [x, y], [a, b]);
f_yy_a_b = subs(f_yy, [x, y], [a, b]);
f_xy_a_b = subs(f_xy, [x, y], [a, b]);

% Construct the terms of the expansion
expansion = 1 + x + y + (1/2)*f_xx_a_b*x^2 + (1/2)*f_yy_a_b*y^2 + f_xy_a_b*x*y;

% Display the expansion
disp(expansion);

Refer to the documentation of "Symbolic Math Toolbox" for more information on how to express and solve mathematical equations in MATLAB.

Symbolic Math Toolbox Documentation (mathworks.com)

Hope this helps!