Question

Problem 2 (1) Approximate If = 1.4" dx using the composite trpezoidal rule with uniform partition Rr(f;P), where h = 1/2. (2) Find the true value of the definte integral using the antiderivative of f(x) = 4". (3) Does your approximation overestimate or underestimate the true value? Use the graph to explain the overesimation or underestimation geometrically.

Answer 1

& VI YODD 27% 25% 12:19 The Simpson's rule states that [ f(x)da ~ (f(x0) + 4f(x1) + 2 f(x2) + 4f(x3) + 2 f(x4) +...+2f(&n-2) + 4f(xn-1) + f(xn)) - where Ax = b - a n We have that a = 0,b= 2, n = 4. 2-0 Therefore, A.x = 6 1 4 21 Divide the interval (0, 2) into n = 4 subintervals of length Ax = 5, with the following endpoints: a = 0, 7, 1, 3, 2 = 6.

Now, we just evaluate the function at these endpoints: f (xo) = f(a) = f (0) = 1 = 1 4f(x1) = 48 () = 8 = 8 2f (x2) = 2f (1) = 8 = 8 4f (83) = 45 (i) = 32 = 32 f (x4) = f(b) = f (2) = 16 = 16 Advertisen Adv 1 Finally, just sum up the above Ax values and multiply by - =(1+8+8+32 + 16) = 10.8333333333333 Answer: 10.8333333333333.

3 In ſur du i henne - my = 10.820 3) Overestimate as if the graph a concave up. The trapezoid approx. a an orevestimate and midpoint is on underestimale. rep Concare Mol)