Math 273: Some Practice Problems for Exam 1 1. Describe and sketch the following surfaces. Justify your work by discussing traces. (a) 4x − y + 2z = 0 p (b) z = x2 + y 2 (c) z + |y| = 1 (d) x2 + y 2 + z = 4 2. Match the following equations to one of the graphs below. Briefly explain your reasoning by discussing traces. (a) x = y 2 − z 2 (b) x2 + 4z 2 = y 2 (c) 9y 2 + z 2 = 16 3. Find ~r 0 (t) for the following vector functions: 2 (b) ~r(t) = h1, ln(1 + 3t), et i (a) ~r(t) = t cos(t)i + t sin(t)j + tk 4. For the curve ~r(t) = 2t3/2 i + cos(2t)j + sin(2t)k: R (a) Find ~r(t) dt. (b) Find the length of the curve ~r(t) for 0 ≤ t ≤ 1. 2 2t 1 t r(0) = h0, −1, 1i. 5. Find ~r(t) if ~r 0 (t) = h 1+t 2 , (t+1)2 , 2te i and ~ 6. The point (1, 2, d) lies somewhere on the curve ~r(t) = 3ti + 6tj + ( 1t )k. Find d. 7. A particle has a position function ~r(t) = et sin(t)i + et cos(t)j + et k. Find: (a) the velocity, (b) the speed, and (c) the acceleration of the particle. 8. Sketch the space curve ~r(t) = t2 i + t4 j + t6 k. Indicate with an arrow the direction in which t increases. 9. Find the equation of the plane through (2, 1, 0) and parallel to x + 4y − 3z = 1. 10. Given the space curve ~r(t) = ht2 , cos πt, sin πti: (a) If this curve represents the position of a particle in space as a function of time, find the velocity, and acceleration functions for the particle. (b) Find parametric equations for the line that is tangent to this curve at the point (1, −1, 0). (c) Sketch the piece of the curve from t = 0 to t = 2. Label the (x, y, z) coordinates of the initial and final points. 11. Find and sketch the domain of the following functions: (a) H(x, y) = xy + xy √ (b) f (x, y) = x + y ln(x − y) (c) g(x, y) = exy + ln(y − x2 ) + p 1 − x2 − y 2 12. Sketch any four level curves for each of the following functions. (a) f (x, y) = xy (b) h(x, y) = x − y 2 13. Let g(x, y) = x2 y − xy sin(y). (You may review notation for partial derivatives in page 612 in your textbook.) (a) Find the first partial derivatives of the function g(x, y). (b) Find all second-order partial derivatives (including mixed partial derivatives). Is gxy = gyx ? (c) Find gxxy and gxyx . Is gxxy = gxyx ? Could you have known this without resorting to calculation? 14. A student was asked to find the equation of the tangent plane to the surface z = x3 − y 2 at (2, 3). The student’s answer was z = 3x2 (x − 2) − 2y(y − 3) − 1. (a) At a glance, how do you know this answer is wrong? (b) What mistake did the student make? (c) Answer the question correctly. p 15. Let F (x, y) = 4 − x2 − 2y 2 . (a) Find F (1, −1). (b) Find and sketch the domain of f . (c) Find Fx (1, −1) and Fy (1, −1). Interpret these numbers as slopes and as rates of change. (d) Find an equation of the tangent plane to the surface given by z = F (x, y) at the point (1, −1, 1). 16. For the function f (x, y) = ln(x − 3y): (a) Explain why the function f is differentiable at the point (7, 2). (b) Find the linearization L(x, y) of f at (7, 2). (c) Use the linear approximation to approximate f (x, y) at (7.01, 1.98). 17. Find the differential of the following functions. (a) f (x, y) = xy 2 + y sin(x) 18. Use the chain rule to find dz dt (b) g(x, y, z) = x2 ln(y − z) or dw : dt (a) z = sin(x) + cos(x/y), x = t2 , y = 1 (b) w = xy + yz 2 , x = et , y = et sin(t), z = et cos(t). 19. Let w = xe2y + z where x = cos(r − 2t), y = e4r ∂w and ∂w . ∂r ∂t 2 −t2 , and z = 1 . rt Use the chain rule to find 20. Suppose z = f (x, y), where x = g(s, t), y = h(s, t), g(1, 2) = 3, gs (1, 2) = −1, gt (1, 2) = 4, h(1, 2) = 6, hs (1, 2) = −5, ht (1, 2) = 10, fx (3, 6) = 8, and fy (3, 6) = 8. Find ∂z/∂s and ∂z/∂t when s = 1 and t = 2. 21. Find the following: (a) Find dy dx if x3 + 4x2 y − 3xy 3 + 5 = 0. (b) Find ∂z/∂x and ∂z/∂y if xyz = cos(x + y + z). 22. Find the gradient of the given function. (a) f (x, y) = 6 − 3x2 − y 2 23. Let f (x, y) = x2 + y x 2 (b) g(x, y, z) = xey z . and let ~v = h2, 3i. (a) Find the directional derivative of the function f at the point (1, 5) in the direction of ~v . That is, compute D~u f (1, 5), where ~u is a unit vector in the same direction as ~v . (b) If we keep changing ~v and re-doing part (a), what is the largest maximum answer we can possibly get? √ 24. Find the directional derivative of the function f (x, y, z) = x2 y + x 1 + z at the point (1, 2, 3) in the direction of ~v = 2i + j − 2k. 2 −y 2 25. Find the maximum rate of change of the function f (x, y) = e−x (2, −1). In which direction does it occur? (x2 + 2y 2 ) at the point 26. Given the surface sin(xyz) = x + 2y + 3z, (a) Find an equation for the tangent plane to this surface at the point (2, −1, 0). (b) Find parametric equations for the line which is normal (perpendicular) to this surface at the point (2, −1, 0).