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Tutorial – 4 Review of Module -2 1 Given w = x.Cos(yz2). If x = Cos t, y = t2 and z = et, find the rate of change of w with respect to t at t = π, (a) using chain rule and (b) using substitutuion, expressing w as a function of t. 2 Let w = f(x, y, z) be differentiable at (1,0,2) with fx = 1, fy = 2 and fz = 3 at that point. If x =t, y = sin (πt) and z = t2 + 1, find dw/dt when t =1. 3 Use total differential to approximate the changes in the function f(x,y,z) = 2xy2z3 as a point moves from P (-1,-2,4) to Q (-1.04, -1.98, 3.97). Compare this with the actual change and see the error in approximating the change in f(x,y,z) using its total differential. 4 Find the linear approximation of the function (4 x ) at the point (1,1,1) ( y +z ) 5 Suppose that a function f(x,y,z) has a local linear approximation L(x, y, z) = x+2y+3z+4 at (0, -1,-2). Then evaluate f, fx, fy and fz at that point. 6 Suppose that the percentage error in measuring two quantities x and y are at the most r% and s% respectively. Use differentials to approximate the maximum possible percetage error in calculating (a) x y and (b) x 3 √y 7 The legs of a right triangle are measured to be 3cm and 4 cm, with a max error of 0.04cm in each measurement. Use differentials to approximate the maximum possible error in calculating (a) hypotenuse and (b) area of the triangle. 8 Find all the critical points and classify them for the function, 2 3 2 2 f (x , y)=3 x y + y −3 x −3 y +2 9 Find the absolute maximum and absolute minimum for the function 2 2 f (x , y )= x −x y+ y −3 x+3 y in the domain, D = {(x, y)/ 0 ≤ x ≤ 3, 0 ≤ y ≤ 3 }