Question

2. Let f(x) = sin(2x) and x0 = 0.

(A) Calculate the Taylor approximation T3(x)

(B). Use the Taylor theorem to show that

|sin(2x) − T3(x)| ≤ (2/3)(x − x0)^(4).

(C). Write a Matlab program to compute the errors for x = 1/2^(k)
for k = 1, 2, 3, 4, 5, 6, and verify that

|sin(2x) − T3(x)| = O(|x − x0|^(4)).

Answer #1

%%Matlab code for Taylor series T3(X)

clear all

close all

%function T3(x) around x0=0

T3= @(x) 2*x-(4/3)*x^3;

x0=0;

%loop for all x values and corresponding error for T3(x)

for k=1:6

%all x values

xx(k)=1/(2^k);

%all error abs(sin(2*x)-T3(x))

err_left(k)=abs(sin(2*xx(k))-T3(xx(k)));

%all error abs(x-x0))^4

err_rght(k)=(abs(xx(k)-x0))^4;

%Printing the result

str=sprintf('1/2^%d',k);

fprintf('For k=%d and x=%s, error
=abs(sin(2*x)-T3(x)) is %e and error= abs(x-x0))^4 is %e
',k,str,err_left(k),err_rght(k))

end

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

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