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(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...
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