B find 9 9 0 d f dx 0 d f dx solution a 1 1 r 1 r r 2

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(b) Find Solution t 12 + … . 22. (a) Find the first four non-zero terms of the Maclaurin series of the function y= 1x. dxxdxx. Find the first four non-zero terms of the Maclaurin series of the function y= ln (x+ 21x 2 1 (b) Recall that 2 2 1 ln( 1 ) 1 ). Solution 4 6 5 7 23. Find the second partial derivatives zx, zy, zxx, zyy, and zxy, of the function z= xln (2 y x ).
MAT1322D Solution to Review Questions Fall 2017 11 2 1(2)2242(2)(2)(2)yxxyxyxyyxyxyxyx. x . 24. Consider function z= x2y3x2+ y(a) Find the partial derivatives zx, zy, zxx, zyy, and (b) Find the gradient vector of zat x= −1, and y(c) Find the directional derivative of zat the point (−1, 2) in the direction of v= (4, −3(d) Find the equation of the tangent plane of the graph of this function at the point where x= 1, and y= 2. 2 . z xy . = 2. ). Solution = 2 x . = . y = 25. Let z= f(x, y), where x= g(u, v) and y= h(u, v). Then z= f(g(u, v), h(u, v)) = function of uand v. Suppose the following values are known: f(1, 2) = 3, fx(1, 2) = 5, fy(1, 2) = −2;g(3, 5) = 1, gu(3, 5) = 2, gv(3, 5) = −1;h(3, 5) = 2, hu(3, 5) = 4, hv(3, 5) = −3.Find Fuand Fvat u = 3,v F ( u , v ) is a = 5.
MAT1322D Solution to Review Questions Fall 2017 12 26. Consider function z= f(x, y) defined implicitly by theequation x2z+ xyyz(a) Find the gradient vector of the function z= f (x, y) at the point (2, 1, −1).(b) Find the equation of the tangent plane of the graph of the equation at the point (2, 1, −1).(c) Find the directional derivative of this function in the direction of the vector u= (2, −3). (d) Find the maximum value of the directional derivative at (2, 1, −1) among all possible directions. 3 = −1. x 2 − 3 yz 2 .

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