Here is the problem:

1. (10 points) Consider a periodic mction f(a:) like the one in the Figure below. (Being periodic means that f (51:) = f (5!: | 2%) for all 3:, when the period is 2a.) The values of f(x) at multiples of 1r/2 are the yi's shown in the Figure. Note yo = 3/4. (a) Calculate the estimate 81 for f0?" f (3:) d3: using Simpson's Rule for 4 equal intervals starting at 3: = D, and the estimate 32 for for fig/2\"\" f (:6) dz: = 02" f (:6) dz: using Simpson's rule for 4 equal intervals starting at a: = 7r / 2. Write 81 and 82 in terms of 310,311, ya, ya only. Recall 3/4 = :90- (b) Calculate the average of 81 and 82, (5'1 + 82) / 2. (c) Calculate the estimate T for f02" f (3:) d3: using the Trapezoidal Rule for 4 equal intervals starting at x = 0. How does T compare to your answer in Part (b)? (d) If |f\"(:r:)| g 1 and | f(4)(3:)| g 1, for 3: E [0, 2%], what is your best estimate for the error |T [021T f(:l:) d3:|