## Transcribed Text

(1) Consider the function
q(x) = r - + 41.x + 32
a. Graph the derivative q (x) over the interval [0, 6].
b. Use the backward difference formula
=
with h = .5
2h
to find the derivative of q at x = 1, 2, 3, 4, 5 and plot the result against the
graph from part a.
c. Use Richardson's extrapolation with h = .5 to find the derivative of q at
x = 1,2,3,4,5 and plot the result on the same graph.
(2) Run the NumList.m file to obtain the data y = f(x) for this problem.
x
y
0.9
49.5491
1.0
49.0000
1.1
47.9511
1.2
46.4096
3.0 -43.0000
3.1 -49.3409
a. Use the center difference formula with step size h = .1 to find f'(x), the
derivative of this list at x = [1 : 0.1 : 3].
b. Use Simpsons method to integrate the derivative function f' '(x) over the
interval [1,3]. (The answer is a numerical estimate of f(3) - f(1) = - -92).
(3) Consider the integral
6
.
da = 2.26173770883048
1 + sin2 (x)
a.
Divide the interval [0, 6] into 3 subintervals of length 2, use Gaussian Quad-
rature method of degree 4 (data provided in the dat.m file) to integrate over
each interval.
b. Find the number of intervals needed to get as accurate an answer as in part
a (not the given answer) if we were to use the trapezoid method.

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x = 1;

h = 0.5;

eps_step = 0.00001;

R(1, 1) = (f(x + h) - f(x - h))/(2*h);

for i=1:5

h = h/2;

R(i + 1, 1) = (f(x + h) - f(x - h))/(2*h);

for j=1:i

R(i + 1, j + 1) = (4^j*R(i + 1, j) - R(i, j))/(4^j - 1);

end

if ( abs( R...