Student name: MATH 350: Multivariate Calculus March 30, 2018 Test 2 - In class part

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1 Student name: MATH 350: Multivariate Calculus March 30, 2018 Test 2 - In class part Instructions: Answer all questions on separate paper (not on this sheet!). The only reference materials allowed on this test are a calculator, and an 8.5x11 inch sheet of formulas (or other information) prepared by you for the purpose of taking the test. Before turning in your test, be sure to attach your reference sheet to it. No other reference sources, including electronic, printed, written or verbal may be used. This test contains questions numbered (1) to (6). It adds up to 24 points: 4 points for each question. (1) Let f(x, y) = 3 x 3 y 3. (a) Does the partial derivative f y (0, 0) exist? (b) Determine whether f is differentiable at (0, 0). (2) Find the equation of the tangent plane to the graph of f(x, y) = x 2 y 2 2y at the point where (x, y) = (2, 1). (3) Show that the velocity vector of a particle moving at constant speed is orthogonal to its acceleration vector. Hint: If you carefully show/justify all needed steps, which is expected, your solution will require no more than about 5 lines. (4) Find the following limit, if it exists, or show that the limit does not exist lim (x,y,z) (0,0,0) xy 2x 2 + 2y 2 + z 2 (5) Let z = f(x, y), with x = s 2 + t 2, y = s 2 t 2. Find an expression for z tt in terms of derivatives with respect to x and/or y. For example, z s = z x x s +z y y s = 2sz x +2sz y. (6) Find z/ x if z is defined implicitly by: sin(xyz) = 2y + zx. End of test

2 Student name: MATH 350 : Multivariate Calculus : Spring 2018 Part I of Test 2 - administered via Science Commons. Test must be taken between March (by 9 AM). The entire test must be completed in one session and it must be taken in the Science Commons. Complete and return test to the service desk within 2 hours after checkout, or before closing time, whichever comes first. Instructions: Answer all questions on separate paper not on this sheet! The only reference materials allowed on this test are a calculator, and an 8.5x11 inch sheet of formulas (or other information) prepared by you for the purpose of taking the test. No other reference sources, including electronic, printed, written or verbal may be used. Before turning in your test, be sure to attach this question sheet to your solutions. For full credit all questions must show clear, detailed steps with justification. This test consists of questions numbered (1) to (3). Each question is worth 8 points. Total points = 24. (1) Suppose the position of a particle is given by the function r(t) = cos(t)i + sin(t)j + tk, where t denotes time. (a) Find the velocity and speed functions of the particle. (b) Find the tangential and normal components of the acceleration function. You may use the following formulas, or any other method, to find the tangential and normal acceleration components a T = r (t) r (t), a r N = r (t) r (t) (t) r (t) (2) Let f(x, y) = 1 x 2 y 2. (a) Find the domain of f, and describe the geometric shape of the surface that is the graph of f. (b) Find the tangent plane to the graph of f at (x, y) = ( 1/2, 1/2). (c) Let v be a vector (in 3D) from the origin to the point on the graph of f where we found the tangent plane in (b). Find the angle between v and the normal vector to the tangent plane. For solving (c), if you are unsure whether you got the right answer to (b), assume the tangent plane is x + 2y + 3z = 4. Note, this is NOT the correct answer to (b). continued on other side

3 (3) Consider the following model for frost penetration, in which the temperature T (in C) at time t (in days) at a depth x (in feet) is given by the function T (x, t) = T 0 + T 1 e λx sin(ωt λx) where ω = 2π/365, and T 0, T 1 and λ are a positive constants. (a) Find T and explain its physical meaning in the context of this application. x (b) Find T and explain its meaning in context. t (c) Show that T satisfies the equation T t = kt xx for a certain constant k. Find that k. End of test

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