## Transcribed Text

1. The Taylor series for sin x is sin x = x − x3
3! +
x5
5! − x7
7! + ··· . Plot P3(x) = x − x3
3! and
P5(x) = x − x3
3! +
x5
5! on the same graph on [−5, 5] range. Graph should be in different colors and
also include title, x-label, y-label, and legend.(see MATLAB notes on Isidore)
2. Since f′
(x0) ≈ f(x0 + h) − f(x0)
h . Consider f(x) = sin x, x0 = 1.2,fp = cos(1.2),
hence cos(x0) ≈ sin(x0 + h) − f(x0)
h . Use MATLAB to construct a table of values of difference
quotient with decreasing h values. We hope that with decreasing h the error will become smaller
and smaller. After completing table what do you notice? Explain.
h Absolute error
0.1
0.01
0.001
1e − 4
1e − 5
1e − 6
1e − 7
1e − 8
1e − 9
1e − 10
1e − 11
1e − 13
1e − 15
1e − 16
3 Use Matlab to
• Type the following script and run it to get a plot. The plot shows the random nature of
roundoff error.
t=0:0.002:1;
tt=exp(-t).*(sin(2*pi*t)+2);
rt=single(tt)
round_err=(tt-rt)./tt;
plot(t,round_err,’-b’);
title(’error in sampling exp(-t)(sin(2\pi t)+2) single precision’)
xlabel(’t’)
ylabel(’roundoff error’)
%relative error is about eps(single)/2
rel_round_err=max(abs(round_err))/(eps(’single’)/2)

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%Question 1

x=-5:0.1:5;%increments of 0.1

P3=x-x.^3/factorial(3);

P5=x-x.^3/factorial(3)+x.^5/factorial(5);

figure(1);

plot(x,P3,x,P5);

xlabel('x');

ylabel('y');

legend('P3','P5');...