1. [15 marks] Consider the function w(x,y) = y2 exp(-xy). (a) Dete...

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1. [15 marks] Consider the function w(x,y) = y2 exp(-xy). (a) Determine all partial derivatives of the function up to second order, i.e. aw aw ay 2w a²w a²w dy2 axdy (b) Write down the two-dimensional Taylor series of the function about the point (0,0) including all terms up to second order. (c) Write down the two-dimensional Taylor series of the function about the point (0,1) including all terms up to second order. 2. [10 marks] Consider the integral for (a) Sketch the region of integration. (b) Evaluate the integral. 3. [15 marks] Consider the function f(x,y,z) = cos (7xx) + sin (1) and the curve r(t) = ti i+(t-1)j+(t-1)²k - for 05152 (2) where i,j, k are the Cartesian basis vectors of length 1. Denote the coordinate functions by x(t) = t, y(t) = (t - 1) and z(t) = (t-1)². - (a) Which value of t corresponds to the 'lowest' point along the curve, i.e. the point with the smallest value of z(t)? (Remember to also check if the lowest point occurs at the end points at t = 0 or t = 2.) (b) Accurately sketch the curve in a graph. (You may use a computer plotting program) (c) Show by explicitly evaluating df (x(t),y(t),z(t)) /dt or by use of the chain rule that the values # = 0 and t = 1 correspond to the points along the curve where f adopt its extremal values. Determine which of the points corresponds to a minimum and which to a maximum. 4. [10 marks] Consider the if = 0 level surface of the function f(x,y,z) defined in eq. 1, and the points P = (0,1/2,0), q = (0,1/3, - 1/3) and V = (1/2,1/2,1/2). (a) Test if the points p, q and v are on the f = 0 level surface of f(x,y,z). (b) Determine unit normal vectors to the surface for those points amongst p.q and V that are on the level surface f = 0.

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