Question

2. The function fi(xo,h) = sin(xo+h)-sin(xo) can be transformed into another form., f2(xo,h), using the trigonometric formula sin()-sin() 2 cos sin Thus, fi and f2 have the same values, in exact arithmetic, for any given argument values xO and h. (a) Derive f2(xo,h). (b) Suggest a formula that avoids cancellation errors for computing the approximation (f(xo +h)- f(xo))/h to the derivative of f(x) sin(x) at x xo. Write a MATLAB pro- gram that implements your formula and computes an approximation of f'(1.2). for h le-20,le-19,

Answer 1

Ond 2 CoA - 2. Herce nothtno 2. 2- 2. 2 (wort)2 Cane So thot

Matlab code for finding derivative clear all close all function for which derivative have to find f-e (x) sin (x) fprintffunction for which derivative have to find f(x)') disp (f) derivative of f ff-e (x,h) (sin (x+h)-sin(x))/h; %x0 value x0-1.2 all h values hh-logspace(0,-20,21); fprintf\n\t h \tval \t ext_val \t error\n\n' ) for i-1:length (hh) h-hh (i val-ff (x0, h); val_ext-cos (x0); error abs (val-val_ext); fprintf(\t% 2 . 2e \t%f \t%f \n',h,val, val_ext,error) \t$e end 웅웅웅움을 음움 움음을 End of Code 융용용용응음음융음음음융융용음융응용음용응용 function for which derivative have to find f(x) @(x) sin (x) h val ext val error 4.859004e-01 1.00e+00 -0.123543 0.362358 1.00e-01 0.315191 0.362358 4.716676e-02 1.00e-02 0.357692 0.362358 4.666196e-03 1.00e-03 0.361892 0.362358 4.660799e-04 1.00e-04 0.362311 0.362358 4.660256e-05 4.660191e-06 1.00e-05 0.362353 0.362358 4.659686e-07 1.00e-06 0.362357 0.362358 1.00e-07 0.362358 0.362358 4.619326e-08 1.00e-08 0.362358 0.362358 4.361052e-10 1.00e-09 0.362358 0.362358 5.594726e-08 0.362358 1.00e-10 0.362358 1.669696e-07 0.362366 0.362488 1.00e-11 0.362358 7.938531e-06 1.00e-12 0.362358 1.300631e-04 0.361933 0.366374 1.00e-13 1.00e-14 0.362358 4.250484e-04 4.015844e-03 0.362358 1.00e-15 1.00e-16 0.444089 0.000000 0.362358 8.173146e-02 3.623578e-01 0.362358 1.00e-17 1.00e-18 0.000000 0.000000 0.362358 3.623578e-01 3.623578e-01 0.362358 0.000000 3.623578e-01 1.00e-19 0.362358 1.00e-20 0.000000 0.362358 3.623578e-01 1

%%Matlab code for finding derivative
clear all
close all

%function for which derivative have to find
f=@(x) sin(x);
fprintf('function for which derivative have to find f(x)=')
disp(f)
%derivative of f
ff=@(x,h) (sin(x+h)-sin(x))/h;
%x0 value
x0=1.2;
%all h values
hh=logspace(0,-20,21);
fprintf('\n\t h \tval \t ext_val \t error\n\n')
for i=1:length(hh)
    h=hh(i);
    val=ff(x0,h);
    val_ext=cos(x0);
    error=abs(val-val_ext);
    fprintf('\t%2.2e \t%f \t%f \t%e \n',h,val,val_ext,error)
end

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