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**Unformatted text preview: **University of Ottawa
Dept. of Mathematics and Statistics
Calculus III for Engineers
MAT 2322 3x - Spring-Summer 2015
Midterm I - V.2
Professor: Abdelkrim El basraoui
Duration: 80 minutes. Name: ACQA—‘(m ID Number: Instructions: (Please read carefully.) o This exam has 8 pages and you have 80 minutes to complete it.
o This is a closed book exam. 0 The only calculators which are allowed are those approved by the faculty
of science such as Texas Instruments TI—30, TI—34, Casio fx-260 and fx—300, scientiﬁc and non programmable.
o Read each question carefully before answering. 0 Questions 1 to 3 are multiple choice questions. These questions are worth 2 points
each and no partial marks are possible. Please record your answers in the cor—
responding boxes in the grid below. 0 Questions 4 to 6 are long answer questions. Questions 4 and 6 are worth 6 marks
each, and question 5 is worth 7 marks, so organize your time accordingly. A correct
answer requires a full, clearly-written and detailed solution. 0 Answer each question in the space provided, using backs of pages or the extra pages
at the end if necessary. 0 Do not unstaple the test. 0 Good luck!
Question Q.1 Q2 Q3 Q4 Q5 Q6 I Total ]
Maximum 2 2 2 6 7 | 6 25 Answer 3 E 6 X X X X Score 3’60 W964i {M 4,4353%. MAT 2322 3X - Midterm I 2
1. If f (11:, y) = 2x2 +3xy— 6y2, which of the following expressions corresponds to the tangent
plane to the graph of f at the point (1, 1, «1)? A. z = —1 B. z=7(a:—1)—9(y—1)—1 C. 2 = 7:1: — 9y — 1 D. z = (4x + 3y)(:1: — 1) + (3x —12y)(y — 1) — 1 E. z = 72"— 9} F. This function is not differentiable at the indicated point, so the tangent plane does not
exist. 2. If f (:17, y) = 62’32‘33’2, and 12' is the unit vector along {+31 which of the following corresponds
to the directional derivative D1; f (1, 2)? ~ . 2\/26“10;— riﬂe-103'
_3\/§e-10 . 0 . —8e‘1° —4\/26‘1° —26_10 swwow> MAT 2322 3X - Midterm I 3 3. If f (:13, y) = $2313 and R is the rectangular region deﬁned by 0 5 :1: S 1, 0 g y g 2, what
is the value of the double integral // f (M?
R A. 5/3
B. 4/3
C. 1
D. 2/3
E. 1/3
F. 0 MAT 2322 3X — Midterm I 4 4. Find and classify the critical points of the function f (3:, y) = 6“ (2:1:2 — y2). 6 CY\M $3" “56 \xawe VS: Xex(°z>¢1+um«‘bl)] ~10 6»
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Db 9.2.67 x3 4ka MAMA .’ MAT 2322 3X — Midterm I 5 5. Find the absolute maxima and minima of the function f (3:, y) = a: + 3/ on the region A= {(azy) 61R<2|$2+y2 S9}- 3 C»th (k MVM A £W1\ +0 )‘ £ﬂ;\ 4:0 -% q&_© 84 MW mm \m UGBVCAQ ‘31 x o Euktwo. WQAMS m “WW \NVNMAM‘Y‘»: TN \OOMKNW I: QQ‘T)\ .11~.ub7'.,0(\ H1\\\°A\ . M \Nméowy \S uv'vﬁeé EXM N ““45 3 3914/4 5
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“\IQ‘E’ & \xb\' \maci‘e— 3V0 C/WWJS ‘Ahsk $9 , MAT 2322 3X - Midterm I 5 6. Let R be the bounded region enclosed between the parabola y = :52 + 1, the parabola
y = —$2, and the lines a: = ~—1 and x = 2 in the my~plane. Sketch the region R then compute the double integral // my dA.
R .MJQ: éhquagaléﬁyeIﬂﬁ‘wghaw
VQ:lC~Cx\0\_\§oc\<V x -302» “my We 3w: Bil“ 7/
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- Summer '14
- Calculus, Statistics, A Closed Book, 3X — Midterm