MAT1322D Solution to Final Examination December 2017 6 Solution. (D) f x= 2xy2+ 3y, f y= 2x2y+ 3x+ 3y2. When x= 2 and y= −1, grad f(2, −1) = (1, 1). 12. The directional derivative of the function z= ln (2x2− y2) at the point x= 3 and y= 4 in the direction of the vector u= (3, 4) is Solution. (D) The unit vector in the direction of uis v = 34,55. zx = 2242xxy, zy= −2222yxy. When x= 3 and y= 4, zx(3, 4) = 6, zy(3, 4) = −4. Du(z) = 3426( 4)555 . Part II. Long Answer Questions (26 marks) 1.(5 marks) Use the definition of improper integralsto determine whether improper integral 220(1)xdxxis convergent or divergent. If it is convergent, find its value. Solution. 21222222001000111111limlimlim(1)(1)22112bbbbbxxdxdxduxxub. This improper integral is convergent, and its value is 12. 2.(5 marks) Find function y(t), where y(t) is the solution to the initial-value problem y'= ysin t, y(0) = −1.Solution. 1sindyt dty. ln | y| = −cos t+ C, | y| = K1e−cos t, where K1= eC> 0. Then y= Ke−cost, where K= K10. When t= 0, cos t= 1. y(0) = Ke−1= −1.Then K= −e. Hence, y(t) = −e e−cos t= −e1 − cos t. 3. (6 marks) Use an appropriate test method to determine whether each of the following series is convergent or divergent. 21nnnn; 1( 1)1nnn; 213nnnn.