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Summer Final 2015 Version 2.pdf

# Summer Final 2015 Version 2.pdf - University of Ottawa Dept...

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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» W V‘%: [Z 1 Q1") €\$C1w1+VX*‘b7/) CO { ~17“? :0 .2 64 ”W 3" =0 2—2.? =0 “Ease ‘LUMM; mtixccq} or} 3‘30 \$476 9%“ “ca“ iuu+L¥X3° ’9 1Q... (a) &%\K+1\ :ﬁm g) {w w=~Z Wwvrtﬁw~%w,§‘:-a2°(;mwm) . ‘QESE‘FEW “Hwy: mm :4? <0 b) {mo} {3 a MAAXL "99}, ~ “Mam: Mm) :Ye‘“>o b« h 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 rm 32-4%,)? ; bug's "C M 2w)=3€b¢w 4}):va g“ .erCI-XL Tm 3m:\+5——~ \{W—XLL ,_/ ,2 7, %QX\’O c» 33:; >~‘ (—2 aﬁﬂ’q’ﬁ %3 9° “”4 Mac} <25 vzir— 3 i :3? Q7“. Aka (30.3 is mo¥w Ox? 96: ”Ci? 1’33 ywr’isffx. 9C3, 2 m Nuﬁcﬂ \$3 58* c3 Cw m «kw. whmwm WED WV; 3J3) ﬁnk-is 94 30;):0 )4 a? z)“ b‘wu ”*5KGMOM b4 ’A 3E {;VMMS\%mEg‘3‘- MAT 2322 3X - Midterm I f V4-1? m ﬁsba 7-0 (/3 \$31" 3f“ - {M cxéﬁwx ‘0‘} i ‘ / L Cl) —— i’S 9,2 «31%. :‘Y": M“ 2’ wwﬂ’ﬂﬁ RCiL}:i’5 ) 1/ 7/ R W‘QRMR/‘Ak *\a\' {2'3 \Smwxm \$4 113%: ken/Moor Sit 6m 2531‘ ms «Rafts (BOWVQNi “6’3: *(ﬁMVOSmsm We, \mm §K1\O3ZE}£(3%I-ZEL)3EVZS fQ'b‘e) )"'3 y :9“? ‘3f\:3 J2, MAT 2322 3X - Midterm I 5 ” Mg m Rmmmm We; ﬁx :q 2 ‘cér k ‘Qu Wﬁkwu’ wwwﬁl’i ‘ \Qq, \wwo: ERIE] ; A <13 :k[::] («:5 09‘ :‘ZMR LTWVLJVMX' A\$b\‘ m2; s; MW 5“ «we, J\\Q,\MRN—>§\£ Gk WM mert MC W 9 “\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/ :: B ”(ye/(17410 A?“ \$- «\ ...
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