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Module MA2E02 (Frolov), Multivariable Calculus Tutorial Sheet 3 Due: at the end of the tutorial session Tuesday/Thursday, 9/11 February 2016 Name and student number: 1. Consider the function f (x, y, z) = p 4y 2 − sin(2x − 3z) , and the point P (0, −1, 0) . (a) Find a unit vector in the direction in which f increases most rapidly at the point P . (b) Sketch the projection of the vector onto the xz-plane (c) Find a unit vector in the direction in which f decreases most rapidly at the point P. (d) Sketch the projection of the vector onto the yz-plane (e) Find the rate of change of f at the point P in these directions. Show the details of your work. 2. Let r = p x2 + y 2 . (a) Show that ∇r = r , r where r = x i + y j . (b) Show that ∇f (r) = f 0 (r)∇r = 3. Consider the surface p 3 z = ln f 0 (r) r. r 2x3 + y 2 + 2 3 (a) Find an equation for the tangent plane to the surface at the point P (2, −3, z0 ) where z0 = f (2, −3). (b) Find points of intersection of the tangent plane with the x-, y- and z-axes. (c) Sketch the tangent plane. (d) Find parametric equations for the normal line to the surface at the point P (2, −3, z0 ). (e) Sketch the normal line to the surface at the point P (2, −3, z0 ). Show the details of your work. 4. Consider the function f (x, y) = 5 − x4 + 2x2 y − 2y 2 + 2y Locate all relative maxima, relative minima, and saddle points, if any. 1