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Math 150 2006 Exam

# Math 150 2006 Exam - McGILL UNIVERSITY FACULTY OF SCIENCE...

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Unformatted text preview: McGILL UNIVERSITY FACULTY OF SCIENCE FINAL EXAMINATION MATH 150 CALCULUS A Examiner: Axel Hundemer Date: Wednesday April 12, 2006 Associate Examiner: Professor N. Sancho Time: 2:00 PM— 5:00 PM INSTRUCTIONS Please answer the questions in the exam booklets provided. Give detailed and complete solutions and fully simplify your answers. This is a closed book exam No Calculators are permitted. Use of a regular dictionary is not permitted. Use of a translation dictionary is permitted. This exam comprises the cover page, and 8 questions on 2 pages. MATH150 Supplemental and or Deferred Exam May 2, 2006 GIVE DETAILED AND COMPLETE SOLUTIONS. FULLY SIMPLIFY YOUR ANSWERS. NO CALCULATORS PERMITTED! 1. (12 marks) Evaluate each of the following limits or show that it does not exist: (a) hm COS CU — 1 (b) \$138+ ﬁg (0) lim m+v :52 — 4:1: + 1 m—)—OO x—>0+ (B 2. (10 marks) A snowball in the shape of a ball melts so that its surface area decreases at a rate of 2 square centimeters per minute. Find the rate at which the volume of the snowball decreases when its diameter is 20 centimeters. 3. (12 marks) Show that the equation em = m + 2 has exactly two solutions. Carefully explain your reasoning! ln|w| 4. (15 marks) Let f(\$) = a: (a) Determine the (maximal) domain of f and all horizontal and vertical asymptotes (if any). (b) Find and classify the critical and singular points (if any) and ﬁnd the intervals of increase and decrease of f. (c) Find all inﬂection points (if any) and the intervals of concavity of f. (d) Carefully sketch the graph of f using all the information obtained above. (e) Find the global maximum and minimum values of f on the interval [ff—1,62]. 5. (12 marks) Consider the implicitly deﬁned curve C’ : 3:3 — y3 + 2x342 = 2. 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. — Please turn over! — MATH150 Supplemental and or Deferred Exam May 2, 2006 6. (14 marks) Let f(\$,y) = m3 — my — y3. (a) Find and classify all critical points of f. (b) Find the maximum and minimum values of f on the closed triangular region bounded by the lines 31 = —a:, y = :1: and y = 3. 7. (12 marks) Let z = g(m, y) where a: = 7“ cos 0 and y = 7" cos 9 and g has continuous partial 82 derivatives of all orders. Express Q Q and Z 61" ’ 86 (9760 in terms of partial derivatives of z with respect to a: and y. 8. (13 marks) For each of the following series determine whether it is conditionally conver- gent, absolutely convergent or divergent. oo nn 3 oo —7L2 0° n _ (a) 23—1)”1 n) (b) 2 (1+%> (c) Z——”3n:/4 1 7121 77,21 n21 ...
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Math 150 2006 Exam - McGILL UNIVERSITY FACULTY OF SCIENCE...

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