1. (10 points) Let f(x) = −cos(x). Compute ˆφ = c0π0 + c1π1 where π0 = 1 and π1 = x. Compute the discrete approximation using

• x0 = π /2 and x1 = (3π)/4 .

• f( π/2 ) = 0 and f( (3π)/4 ) = √ 1 2 .

You may use the XT X|XT Y method or A⃗c = ⃗b method.

2. (10 points) Let f(x) = x 5 . Compute ˆφ = c0π0 + c1π1 for the domain 0 ≤ x ≤ 1. Let π0 = x and π1 = x 2 . • Compute the continuous approximation. • Do not round.

3. (10 points) Suppose you have just finished approximating f(x) = x 3 for 0 ≤ x ≤ α and ended up with φˆ(α, x) = 1 2 α 2x. (a) (8 points) Evaluate E2 (α, x) = ||f − φˆ||2 . (b) (2 points) After you have evaluated ||f − φˆ||2 for α, plug in α = 1 and simplify.

4. (10 points) Reconcile the XT X|XT Y and A| ⃗b methods for the discrete case. More specifically, show that the two are equivalent for n polynomial basis functions (π0, π1, . . . πn−1) and k points.

Topics Covered:

Least square Approximation
Reconcile the XTX|XTY and Ab Methods