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### 04Final Exam

Course: MATH 150, Fall 2005
School: McGill
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Word Count: 434

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Exam 12/9/04 GIVE MATH150 Final DETAILED AND COMPLETE SOLUTIONS. FULLY SIMPLIFY YOUR ANSWERS. NO CALCULATORS PERMITTED! 1. (12 marks) Evaluate the following limits or show that they don't exist: (a) lim 1 - sec x x0 tan2 x (b) lim (ln x)1/x x (c) x2 - 4y 2 (x,y)(2,1) x - 2y lim 2. (10 marks) (a) Apply the mean value theorem to the function f (x) = x on the interval [49, 50] 1 to show that 50 = 7 + for...

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Exam 12/9/04 GIVE MATH150 Final DETAILED AND COMPLETE SOLUTIONS. FULLY SIMPLIFY YOUR ANSWERS. NO CALCULATORS PERMITTED! 1. (12 marks) Evaluate the following limits or show that they don't exist: (a) lim 1 - sec x x0 tan2 x (b) lim (ln x)1/x x (c) x2 - 4y 2 (x,y)(2,1) x - 2y lim 2. (10 marks) (a) Apply the mean value theorem to the function f (x) = x on the interval [49, 50] 1 to show that 50 = 7 + for some number c in (49, 50). 2 c (b) Use (a) to show that 7 < 50 < 99 . 14 3. (12 marks) Let f (x) = x2/3 (5 - x), defined for all real numbers x. (a) Find and classify the critical and singular points (if any) and find the intervals of increase and decrease of f . (b) Find all inflection points (if any) and the intervals of concavity of f . (c) Find the global maximum and minimum values of f on the interval [-1, 4]. 4. (10 marks) Consider the implicitly defined curve C : 2x3 - x2 y + 2y 2 = 3. Compute an equation of the tangent line to C at P = (1, 1) by (a) using implicit differentiation. (b) interpreting C as a level curve of a suitable function of 2 variables. 5. (10 marks) Let f (x, y) = x y - y 2 + 2x. (a) Find the equation of the tangent plane to the graph f of at (4, 4, 0). (b) Find the point P on the graph of f at which the tangent plane is parallel to the plane -6x + 5y + 2z - 1 = 0. (c) Is there a unit vector v such that the directional derivative of f at (-2, 1) in the direction v equals 5? Justify! -- Please turn over! -- 6. (12 marks) Let f (x, y) = x3 - 3xy - y 3 . (a) Find and classify all critical points of f . (b) Find the maximum and minimum value of f on the closed triangular region bounded by the lines x = -1, y = 0 and y = -x. 7. (12 marks) Let z = f (x, y) where x = 2st and y = s2 - t2 and f has continuous partial derivatives of all orders. z z z -t = 2(s2 + t2 ) . s t y 2z 2z 2z 2z (b) Show that + 2 = 4(s2 + t2 )( 2 + 2 ). s2 t x y (a) Show that s 8. (10 marks) Let 2 4 2x y - x f (x, y) = x4 + y 2 0 (x,y)(0,0) if (x, y) = (0, 0) if (x, y) = (0, 0) (a) Determine lim f (x, y) or show that the limit does not exist. (b) Is f continuous at (0, 0)? Is f continuous at points (x, y) = (0, 0)? Justify! 9. (12 marks) For each of the following series determine whether it converges or diverges. Fully justify your answers! (a) n=3 1 (-1) n ln n n cos2 n (b) n n2 + 1 n=1 (c) (-1)n n=1 arctan n
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