Math 1AA3/1ZB3 Sample Test 3, Version #1

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1 Math 1AA3/1ZB3 Sample Test 3, Version 1 Name: (Last Name) (First Name) Student Number: Tutorial Number: This test consists of 16 multiple choice questions worth 1 mark each (no part marks), and 1 question worth 1 mark (no part marks) on proper computer card filling. All questions must be answered on the COMPUTER CARD with an HB PENCIL. Marks will not be deducted for wrong answers (i.e., there is no penalty for guessing). You are responsible for ensuring that your copy of the test is complete. Bring any discrepancy to the attention of the invigilator. Calculators are NOT allowed. 1. Find the foci of the following ellipse. Ð $ß &Ñ (b) Ð$ß &Ñ (c) Ð!ß &Ñ (d) Ð &ß $Ñ (e) Ð & $ß Ñ 2. Find the vertices of the following conic section. &B %C &!B "'C œ *"Þ Ð$ß Ñß Ð"ß Ñ (b) Ð!ß &Ñ (c) Ð"ß (Ñß Ð"ß $Ñ (d) Ð"ß Ñß Ð$ß Ñ (e) Ð"ß $Ñß Ð"ß (Ñ 3. Evaluate the following limit, lim ÐBßCÑÄÐ!ß!Ñ C BC/ B %C % " "! (b) d.n.e. (c) " (d) (e) %

2 4. Let AÐ=ß >Ñ œ J Ð?Ð=ß >ÑÑ, where J,?, are differentiable,?ð"ß!ñ œ ß? Ð"ß!Ñ œ ß? Ð"ß!Ñ œ œ Ð"ß!Ñ œ Ð"ß!Ñ œ %ß J Ðß $Ñ œ "ß J@ Ðß $Ñ œ "!. Find A> Ð"ß!Ñ. %" (b) & (c) "' (d) ) (e) $% = > = >? and 5. Which of the following differential equations is satisfied by the function?œln B C?? BC? BB œ! (b)? BC? BB œ! (c)? BB? CC œ! (d)?? œ! (e)?? œ! BB CC BB CC 6. Find the linearization of 0ÐBß CÑ œ C ln B at Ðß "Ñ. " " " BÐ ln ÑC" (b) BÐ ln ÑC" (c) BÐ ln ÑC" (d) B " (e) B Ð ln ÑC " 7. Find the equation of the tangent plane to the surface D œ&b C at the point Ðß"ßÑ. D œ %B C ) (b) D œ %B C ) (c) D œ %B C ) (d) D œ %B C ) (e) D œ %B C ' 8. Estimate the volume of the solid that lies below the surface DœBCand above the rectangle VœÖÐBßCÑl!ŸBŸ'ß!ŸCŸ% Use a Riemann sum with 7 œ $ß 8 œ, and take the sample point to be the lower left corner of each square. 32 (b) 24 (c) 16 (d) 48 (e) Set up a double integral which represents the volume under the function D œ BC and above the triangular region with vertices Ð!ß!Ñß Ð"ß Ñß and Ð!ß $Ñ. " B$ " $ " B$ BC.C.B (b) BC.C.B (c) BC.C.B! B! B!! " B " B$ (d) BC.C.B ( e)! B! B BC.C.B 10. Evaluate "!! BC C/.C.B " " " % $ Ð/ $Ñ (b) Ð/ Ñ (c) / " (d) / $ (e) Ð/ "Ñ

3 11. Sketch the graph of the function 0ÐBßCÑ œ B C (b) (c) (d) (e) Which of the regions in the figure to the right represents the domain of 0ÐBß CÑ œ lnðbcñ B C? Regions II, III, and IV (b) Regions II and IV (c) Reions II and V (d) Regions I and IV (e) Regions I, VI, and V 3 2 VI 1 II x 1 V III 2 IV 3 I 4

4 13. Draw a contour plot of 0ÐBßCÑ œ lnðc " sin BÑ. (b) (c) (d) (e) 14. If D œ B BC $C and ÐBß CÑ changes from Ð$ß "Ñ to ÐÞ*'ß!Þ*&Ñ, find the value of the differential.d. ($ (b) "( (c) ($ (d) "( (e) "( "!! "!! "!! "!! "! 15. Find the directional derivative of 0ÐBß CÑ œ BsinÐBCÑ at the point Ðß!Ñ in the direction indicated by the angle 1Î$. $ " " $ " (b) $ (c) $ (d) (e)

5 16. Which of the below integrals is equal to ln B "! 0ÐBß CÑ.C.B / C ln ln B 0ÐBß CÑ.B.C (b) 0ÐBß CÑ.B.C (c) 0ÐBß CÑ.B.C / "! / C! " " " " ln C (d) 0ÐBß CÑ.B.C (e)! / C!! 0ÐBß CÑ.B.C

6 Math 1AA3/1ZB3 Sample Test 3, Version 1 Name: (Last Name) (First Name) Student Number: Tutorial Number: This test consists of 16 multiple choice questions worth 1 mark each (no part marks), and 1 question worth 1 mark (no part marks) on proper computer card filling. All questions must be answered on the COMPUTER CARD with an HB PENCIL. Marks will not be deducted for wrong answers (i.e., there is no penalty for guessing). You are responsible for ensuring that your copy of the test is complete. Bring any discrepancy to the attention of the invigilator. Calculators are NOT allowed. 1. Find the vertices of the following ellipse. &B!B $C 'C œ ) Ð!ß &Ñ (b) Ð &ß!Ñ (c) Ð $ ß!Ñ (d) Ð $ß!Ñ (e) Ðß " &Ñ 2. Find an equation of the hyperbola with vertices Ðß "Ñß Ðß (Ñ, and foci Ðß Ñ and Ðß )Ñ. ÐC $Ñ ÐB Ñ ÐB Ñ ÐC $Ñ œ " (b) œ " "' & & * ÐC $Ñ ÐB Ñ ÐC Ñ ÐB "Ñ ÐB "Ñ ÐC $Ñ (c) œ " (d) œ " (e) œ " "' * & % "' * 3. Which of the regions in the figure below represent the domain of 0ÐBß CÑ œ lnð" BCÑ? Regions I and II not including the curve (b) Regions II, IV, and V not including the curve (c) Regions III, IV, V, and VI not including the curve (d) Regions I, III, and VI not including the curve (e) Regions II, III, IV, V, and VI not including the curve

7 4. Let BC BC if ÐBß CÑ Á Ð!ß!Ñ 0ÐBßCÑ œ,! if ÐBß CÑ œ Ð!ß!Ñ and let $BC B C if ÐBß CÑ Á Ð!ß!Ñ 1ÐBß CÑ œ.! if ÐBß CÑ œ Ð!ß!Ñ Which of the above functions are continuous at Ð!ß!Ñ? 0 only (b) 1 only (c) 0 and 1 (d) neither 5. Draw a contour plot of 0ÐBß CÑ œ lnðb CÑ. (b) (c) (d) (e)

8 6. Matilda wants to use a Riemman sum to approximate the integral 0ÐBßCÑ.E H where HœÖÐBßCÑl!ŸBŸ%ß!ŸCŸ%. To do so, she divided H into four equal squares and chose a sample point from the middle of each square. The sum of the four pieces was '! " " ) œ "". This approximation is appropriate if her function 0ÐBß CÑ was $B * sinð1bîñsinð1cîñ C B %/ C (b) ' (c) "B C " BC (d) lnðb C *Ñ% (e) ' "B$ % ÐC $Ñ 7. Sketch the graph of 0ÐBßCÑ œ C /. B (b) (c) (e)

9 B > => 8. Let D œ C, B œ <=/ ß C œ </. Find `< when < œ "ß = œ "ß > œ!.! (b) (c) " (d) (e) " `D BC 9. Find the direction in which 0ÐBß Cß DÑ œ D/ increases most rapidly at the point Ð!ß "ß Ñ. Ð"ß "ß Ñ (b) Ðß "ß!Ñ (c) Ð"ß ß "Ñ (d) Ð"ß!ß Ñ (e) Ðß!ß "Ñ 10. Find the linear approximation of QÐBß CÑ œ Ð %B Ñ C at Ðß %Ñ, and use it to approximate QÐÞ"ß $Þ)Ñ. ( $ ln * (b) % ln $" (c) % ln * (d) $ ln ( (e) ln % ln 11. Evaluate "!! ln $ (b) ln ' (c) ln % (d) ln * (e) ln *B C "B $.C.B 12. A contour map is given for a function 0ÐBßCÑ. Based on the contour map, what are the signs of 0 and 0 at the point Ð"ß $Ñ? 0B!à 0C! (b) 0B!à 0C! (c) 0B!à 0C! (d) 0!à0! (e) 0 œ!à0! B C B C B C

10 13. Find the equation of the tangent plane to " $ at the point Ð"ß ß Ñ. Dœ B "BC $BC*Dœ" (b) B$CDœ! (c) BC*Dœ! (d) œ" (e) BC*Dœ% 14. If / œ `D then `B is equal to " CD/ " CD/ "C (b) (c) "/ "BC/ "BC/ BC/ ÎD "C (d) CD/ " C (e) "/ 15. Let 1Ð>Ñ œ 0Ð> sin >ß cos >Ñ. Suppose that 0BÐ!ß "Ñ œ and 0CÐ!ß "Ñ œ $. Find the value of w 1Ð!Ñ. 0 (b) 1 (c) 2 (d) 3 (e) The two legs of a right triangle are measured as 5 m and 12 m with a possible error in measurement of at most 0.1 cm in each. Use differentials to estimate the maximum error in the calculated value of the length of the hypotenuse. & & "( "$ $ (b) (c) (d) (e) "(!!! $!!! "$!!! &!!! "&!!

11 Answers for 1 st Version of Sample Test 3 1. b 2. a 3. b 4. e 5. c 6. a 7. a 8. d 9. e 10. a 11. d 12. b 13. e 14. a 15. c 16. b Answers for 2 nd Version of Sample Test 3 1. e 2. c 3. c 4. b 5. d 6. b 7. a 8. e 9. e 10. b 11. d 12. a 13. e 14. b 15. a 16. c