I have a Multivariate Calculus and Ordinary Different Equations exam starting tomorrow on the 13th of March at 4pm. I will need assistance for this exam.
1. (10 marks total) A plane is given by P = (0, 1,−1) + λ(1, 0, 2) + µ(0, 2,−4), where λ, µ ∈ R. (i) (4 marks) Determine a normal vector to P . (ii) (4 marks) Let X be a point on the line −x = y + 1 3 = 2− z. Write down an expression for the distance from the point X to the plane P . (iii) (2 marks) Using this, determine the point of intersection between the line from (ii) and the plane P . Page 2 of 22 (Working Space Only) Page 3 of 22 2. (10 marks total) Let a conic section be given by 4x2 − y2 + 4x = 6y − 4. (i) (5 marks) Determine the type of conic section. (ii) (5 marks) Sketch the conic section. Be sure to identify the center, foci, or asymptotes. Page 4 of 22 (Working Space Only) Page 5 of 22 3. (10 marks total) Let h(x, y) = 20 + x2 − y2 + xy2 + y3. (i) (5 marks) Find all critical points of h(x, y). (ii) (5 marks) Classify each critical point as a local minimum, local maximum, or saddle point. Page 6 of 22 (Working Space Only) Page 7 of 22 4. (10 marks total) Consider a particle moving through space with velocity v(t) = cos(t)i− sin(t)j + tk. (i) (3 marks) Determine its acceleration vector. (ii) (7 marks) Determine the position vector, supposing the particle starts at position (3,−2, 3) at time t = 0. Page 8 of 22 (Working Space Only) Page 9 of 22 5. (10 marks) Consider the ODE dy dx = x+ y x− y , and show that it becomes separable using the substitution y(x) = xu(x). (You do not need to solve the ODE.) Page 10 of 22 (Working Space Only) Page 11 of 22 6. (10 marks total) Consider a particle moving in the plane along the curve r(t) = ( R cos(ωt), R sin(ωt) ) for some constants R,ω > 0. (i) (5 marks) Determine the distance the particle travels for t ∈ [0, 2π]. (ii) (5 marks) Suppose the plane has a voltage given by V (x, y) = xy + x2 + y3 + 3. Determine the change in voltage the particle experiences at time t. Page 12 of 22 (Working Space Only) Page 13 of 22 7. (10 marks) Find the general solution y(x) to the ODE 4y′′ − 4y′ + 5y = e3x. Page 14 of 22 (Working Space Only) Page 15 of 22 8. (10 marks) Find the maximum and minimum values of the function f(x, y) = x2 + y3 subject to the constraint x4 + y6 − 2 = 0. Page 16 of 22 (Working Space Only) Page 17 of 22 9. (10 marks total) Let a vector field in R3 be defined by F(x, y, z) = (2xy2 + 3x2, 2yx2, 1). Evaluate ∫ C F · dr, where C is the curve (t2 − 1, sin3(πt 2 ), 2t) for 0 ≤ t ≤ 1. Page 18 of 22 (Working Space Only) Page 19 of 22 10. (10 marks total) (i) (5 marks) Determine all equilibrium solutions to the ODE xẏ + x− y = y2xẏ + xy − 1. (ii) (5 marks) Determine if each equilibrium solution is stable or not. Justify your answer. Page 20 of 22 (Working Space Only) END OF EXAM Page 21 of 22 Formula sheet • Linear and quadratic approximations P (x, y) =f(a, b) + fx(a, b)(x− a) + fy(a, b)(y − b) Q(x, y) =f(a, b) + fx(a, b)(x− a) + fy(a, b)(y − b) + 1 2 fxx(a, b)(x− a)2 + fxy(a, b)(x− a)(y − b) + 1 2 fyy(a, b)(y − b)2 • Directional derivative fu = ∇f(a, b) · u ||u|| • Chain rule dz dt = ∂z ∂x dx dt + ∂z ∂y dy dt • Implicit differentiation F (x, y) = 0 =⇒ y′(a) = −Fx(a, b) Fy(a, b) • Hessian D = ∣∣∣∣ fxx(a, b) fxy(a, b) fyx(a, b) fyy(a, b) ∣∣∣∣ • Lagrange Multipliers ∇f = λ∇g • Work W = ∫ b a F(t) · r′(t) dt Page 22 of 22 1. (10 marks) Consider the two planes π1 : 2x+ y + z = 1 and π2 : x+ 2y − z = 1. (a) Find the angle between the two planes. (b) Find the symmetric equation for the line L of intersection of these two planes. Page 2 of 21 (Working Space Only) Page 3 of 21 2. (10 marks) Let the curve C be given by x+ y2 − 4y + 5 = 0. a) Sketch C, clearly indicating any asymptotes, intercepts and axes of symmetry. b) Find y′(x) on C. c) Find the equation for the tangent line to C at the point (−2, 3). Page 4 of 21 (Working Space Only) Page 5 of 21 3. (10 marks) Consider the following function on R2 \ (0, 0) f(x, y) = xy 2y2 + x2 . a) Find limx→0 f(x, αx), for α ∈ R. b) Show that lim(x,y)→(0,0) f(x, y) does not exist. Page 6 of 21 (Working Space Only) Page 7 of 21 4. (10 marks) Compute the length of the curve in R3 given parametrically by the equations: x = t, y = t2 4 , z = √ 2 3 t 3 2 , t ∈ [0, 1]. Page 8 of 21 (Working Space Only) Page 9 of 21 5. (10 marks) CC-Container Company produces steel shipping containers at three different plants in amounts x, y, and z, respectively. Their annual revenue is R(x, y, z) = 8xyz2 − 100000(x + y + z) (in dollars). The company needs to produce 1000 crates annually. How many containers should they produce at each plant in order to maximize their revenue? Page 10 of 21 (Working Space Only) Page 11 of 21 6. (10 marks) Find a function y(t) that satisfies the initial value problem y′′ = −3t, y(0) = 1, y′(0) = 2. Page 12 of 21 (Working Space Only) Page 13 of 21 7. (10 marks) The growth of a bacterial colony is described by the logistic population dynamics model dP dt = rP ( 1− P K ) , where r and K are positive constants. (a) Determine the equilibrium solutions and their stability. (b) Determine the general solution of the model. Page 14 of 21 (Working Space Only) Page 15 of 21 8. (10 marks) Find the general solution y(t) of the differential equation dy dt = y t + t3. Page 16 of 21 (Working Space Only) Page 17 of 21 9. (10 marks) The ODE xy′′− 3y′− 12y/x = 0 has y = x6 as one solution. Find another linearly independent solution. Page 18 of 21 (Working Space Only) Page 19 of 21 10. (10 marks) Solve the second order ODE y′′ − 5y′ + 4y = 0 with initial conditions y(0) = 0 and y′(0) = 1. Page 20 of 21 (Working Space Only) Page 21 of 21