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Unformatted text preview: MCGILI: UNIVERSITY FACULTY OF SCIENCE FINAL EXAMINATION MATH 150
CALCULUS A
Examiner: Professor A. Hundemer Date: Thrusday December 9,2004
Associate Examiner: Professor W. Brown Time: 9:00 AM — 12:00 PM
INSTRUCTIONS 1. Please answer All questions in the Examination Booklet provided.
2. No Calculators are allowed a. .3. No Notes or Text Books are allowed 4. Deﬁnition Dictionaries are allowed provided that permission is given by the
instructor and that the instructor checks the dictionary being used. This exam comprises of the cover page, and 9 questious MATH150 Final Exam 12/9/04 GIVE DETAILED AND COMPLETE SOLUTIONS. FULLY SIMPLIFY YOUR ANSWERS.
NO CALCULATORS PERMITTED! ' . (12 marks) Evaluate the following limits or Show that they don‘t exist: . 1— seea:  , 1/5: $2 _ 4,. 2
a) 11m _ (b) Tlll’ﬂ (111.13) 1. 9'
( w ’0 tangac TOO (r) (males 15 — 29' . (10 marks) (a) Apply the mean value theorem to the function f(:r) = J? on the interval [49, 50] to show that l
V 5 r 7 + 2—\/; for some number (3 in (49, 50). (b) Use (a) to Show that 7 < m < gg. . (12 marks) Let f(rr) = 372/363 w 3;), deﬁned for all real numbers 3:.
(a) Find and clams“)r 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]. . (10 marks) Censider the implicitly deﬁned curve C : 2:133 — 2323; + ‘2ng2 = 3. Compute an equation of
the tangent line to C at P : (1,1) by
(a) using implicit (liﬂ'erentiation. (b) interpreting C as a level curve of a suitable function of 2 variables. . (10 marks) Let f(:r,y) = .L‘\/_ — y2 + 235. (a) Find the equation of the tangent plane to the graph of f at (4, 4, 0). (1)) Find the point P on the graph of f at which the tangent plane is parallel to the plane
—6:L'+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? J ustifyl — Please turn over! — 6. (12 marks) Let f(:c,y) = m3 w 33:3; — 313. (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
m: ﬁ1,y=Oandy= —33. 7. (12 marks) Let 2 : f(:c, y) Where :2: = 2.916 and y = 32 — t2 and f has continuous partial derivatives
of all orders. Oz 82 . 82
S} tl ——— .2 Hm.
(a) now iatsas tdt= 4(9 + )3y
02: +822 2 2 823 022
(l 7) Show thata —— +0—2 2 4(3 +t )(W + w).
8. (10 marks) Let
22:21) * 3:4 _
f(a:.,y) : W 1f (Lay) 7‘9 (0,0)
0 if (may) = (010)
(a) Determine lim f(:c, y) or show that the limit does not exist.
(IslWW) (b) Is f continuous at (U, 0)? Is f continuous at points (my) # (0,0)? Justify! 9. (12 marks) For each of the following series determine Whether it converges or diverges. Fully justify
your answers! 00 (2») 2H)“ 1 (ME—11m W: n 1n n
n:3 na—rctann
n21 ' 71:1 ...
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