(a) Find the general solution of the motion of a mass attached to the ceiling by a spring in presence of friction, i.e. solve the ODE mÿj = mg - k(yl) - vý. with m = 1, k3, y = 2, g = 10, l = 5, where y indicates the distance of the mass from the ceiling. (b) What is the limit limt→∞ y(t) for the motion of the mass described in (a)? Describe in words the asymptotic dynamical behaviour of the mass for t→∞. (c) Determine whether the differential equation -—-y² + y cos(x) + (yx + sin(x) — e²) y' = 0 is exact. If it is exact, find its general solution in explicit form.

(i) Sketch (as accurately as possible) below the function y = t², as x varies from -2 to 2. Then calculate analytically the derivative of y with respect to t, i.e., dy/dt, at the following values of t: 0, 1, 2, using the limit formula. (ii) Now calculate the derivative using the 'rise/run' approach, using the Euler approximation dy/dt~ [y(t+At) - y(t)]/At. Calculate at t=0, 1, and 2. Compare with the values calculated above.

Let g(x, y) = 3x + ylny-sin(1-x). Suppose z is a differentiable function of x and y implicitly defined by the equation E: 2 + y² = g(x,y) - 2. a. Find the instantaneous rate of change of g as one moves from the point (1, e) toward the point (e, 1). b. Find the equation of the tangent plane to the graph of E at (1, e. 1). c. If in addition, x and y are functions of r and t given by x = -rt and y = er+²t, use дz multivariate chain rule to evaluate when r = -1 and t = 1.

Suppose you have a conical tank of height 4m and radius 1m filled with a liquid of density p (shown below). R= 1 H = 4 (a) Calculate the work required to pump all of the liquid to the top of the tank. (Your answer will be in terms of p.) (b) Now suppose your tank is inverted. H = 4 R = 1 Calculate the work required to pump all of the liquid to the top of the tank (now the vertex of the cone) in this case. (c) Use your answers to (a) and (b) to determine if the amount of work required to pump the water to the top of the tank depends on the orientation of the tank (i.e. whether or not the tank is inverted). If it doesn't, briefly explain why it makes sense that the work is the same. If it does, briefly explain why it makes sense that work is different.