A semi-infinite string is located initially (at time t = 0) along the positive X-axis, 0 < x <∞. The string is looped around a vertical support at the 'far end', corresponding to x → ∞o, which exerts no vertical force on the tape: limu(x, t) = 0. At the near end, corresponding to x = 0, the tape is tethered to a support, which fixes its position. The initial velocity is zero throughout the tape, i.e., at ди t = 0, = 0. at ²u The displacement u(x, t) of the string obeys the wave equation - A t, where c is a at² given positive constant with units m s¹, and A is a given positive constant with units m s ³. Solve the PDE analytically for the displacement of the tape at any location at any time t > 0. = c². ² əx²

Compute the length of the curve r(t) = (-4t, 5t — 3, -2t + 3) over the interval 0 ≤ t ≤ 9 L =

Let's simulate the process of constructing a matrix for a 3D transformation that rotates 13 degrees about the axis n = [0.3, 0.2, 0.5]. First step: is the vector n a unit vector? (on the answers, 0.3^2 means 0.3-squared, and so on) O Yes, since 0.3 +0.2 + 0.5 = 1 O No, since 0.3^2 +0.2^2 + 0.5^2 1 O No, since 0.3^2 +0.2^2 + 0.5^2 = 1

In this problem, p is in dollars and q is the number of units. (a) Find the elasticity of the demand function p + 5q = 150 at (q, p) = (15, 75). (b) How will a price increase affect total revenue? Since the demand is elastic, an increase in price will decrease the total revenue. Since the demand is unitary, there will be no change in the revenue with a price increase. Since the demand is elastic, an increase in price will increase the total revenue. Since the demand is inelastic, an increase in price will increase the total revenue. Since the demand is inelastic, an increase in price will decrease the total revenue.