Determine the coefficients a, b, c, and d, so that the curve y = ax' + bx² + cx + d have critical point at (0, 4) and (2, 0).

Find c'(t) and ||c'(t) || - c'(6) = (2e',41,2) (2e*, 4t, 2) ||c'(t)|| = V 4e + 4t + 4| 2t 4²? - 4e2t + 16t2 + 4 Also, find f(c(t)). f(c(t)) = f(2e*, 21?, 2t) 2t 8e4 + 32t 8e2t + 32t Step 2 of 2 Evaluate the line integral using u-substitution. f(x, y, z) ds (8e2t + 32t) V 4e2t + 16t2 + 4 dt Vũ du 8,

Two waves on a string are given by the following functions: yı (x, 1) = 4 cos(201 – 30x) (cm) y2(x, 1) = -4 cos(201 + 30x) (cm) where x is in centimeters. The waves are said to interfere constructively when their superposition lys| = ly1 + y2|maximum, and they interfere destructively when lys| is a minimum. (a) What are the directions of propagation of waves y1 (x.f) and y2(x, t1)? (b) At t = (= (T/50) s, at what x location y do the two waves interfere constructively, and what is the corresponding value of ys|? (c) At t = (7/50) s, at what locationx do the two waves interfere destructively, and what is the corresponding value of lys?

Please explain in details