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Unformatted text preview: STUDENTS NAME (underline Family name):
STUDENT NUMBER:
SECTION NUMBER: Examiner: Professor M. Makkai
Associate Examiner: Professor P. Russell 10. . All problems are of equal weight. McGILL UNIVERSITY FACULTY OF SCIENCE
FINAL EXAMINATION MATH 222 CALCULUS 3 Date: Thursday December 21,, 2006
Time: 2:00PM  5:00PM INSTRUCTIONS . This is a closed book exam. No notes or books are permitted. . Calculators are not permitted. Numerical answers may be given in the forms
1 + v’i \/6 + 7r, cos(23°), arccos 3 ,etc. . Show all calculations. Give sufﬁcient reasons for your answers.
. Use of a regular dictionary is permitted. . No translation dictionaries are allowed. Do not write anything on the separate pink sheet summarizing the questions. . This exam has eight questions. All questions carry the same weight. . Do all your work on the sheets provided. Do not tear or separate sheets that have been stapled together. Make sure that you write your name, student number and section number at the top
of this page. (If your instructor is Michael Makkai , you are in section 1; if your
instructor is Peter Russell, you are in section 2.) This examination is comprised of a cover pages , 7 pages of questions , and
14 extra (blank) pages. There is pink sheet summarizing the problems of this ﬁnal
exam that you may keep. McGILL UNIVERSITY
FINAL EXAMINATION
MATH 222 CALCULUS 3 Thursday December 21. 2006 MATH 222, Calculus 3, Fall 2006, Final Ezramlnatlon 1
1. Consider the function f = (a) Determine the Taylor (“Maclaurin”) polynomials T1($),T2 (ac),T3 of f(ac) centered at
a = 0.
(b) Determine a constant C such that l f — T2(w) S C  x3 in the range % g a: 3 (Hints: Show and use the fact that 1 — a: + m2 2 i always. When calculating repeated derivatives, use temporary abbreviations such as the single letter s for 1 — a: + .732.) E
4. MATH 222, Calculus 3, Fall 2006, Final Examination This page is for the continuation of problem 1; it may also be used for rough work. MATH 222, Calculus 3, Fall 2006, Final Examination This page is for the continuation of problem 1; it may also be used for rough work. MATH 222, Calculus 3, Fall 2006, Final Emaminatlon 2. Consider the space curve P(t) given by m(t) 22:2
r(t) = W) = 4t a
z(t) ln(t2) the parameter t varying in the range t Z 1. (a) Compute the following as functions of t: (i) the arc length s(t) of the curve from to = 1 to t,
(ii)the unit tangent vector T(t),
(iii) the curvature 1405). (b) Compute the principal unit normal vector N(1) at t = 1. (Note: the results for (ii) and (iii) should come out as rational functions.) MATH 222, Calculus 3, Fall 2006, Final Examination This page is for the continuation of problem 2; it may also be used for rough work. MATH 222, Calculus 3, Fall 2006, Final Examination This page is for the continuation of problem 2; it may also be used for rough work. MATH 222, Calculus 3, Fall 2006, Final Examination
3. Let f(x, y) = $3 + 22723; + 4y3 (a) Calculate the gradient Vf(2, 1) of the function f at the point P = (2,1). (b) Determine the equation of the tangent line of the level curve f (10,24) = 20 of f at the
point P = (2,1). (0) Suppose that, for the unit vector u = (a,b), we have that the directional derivative 1
Duf(2, 1) equals §Vf(2, Determine the angle between the vectors Vf(2, 1) and u.
((1) Determine all unit vectors u = (a, b) for which Duf(2, 1) = %Vf(2, MATH 222, Calculus 3, Fall 2006, Final Examination This page is for the continuation of problem 3; it may also be used for rough work. MATH 222, Calculus 3, Fall 2006, Final Examination This page is for the continuation of problem 3; it may also be used for rough work. MATH 222, Calculus 5’, Fall 2006, Final Examination
4. Let f(:r, y) = 03231 + 22:14 + y3. (a) Find all the critical points of the function f (x,y), and classify them. For the local
maximum (maxima) and minimum (minima), determine the local maximum/ minimum value of f(a:, y). (b) Consider the closed square D on the (m,y) plane deﬁned by —3 S :1: S 1, —1 S y S 1.
As we know, f (ac, g) has an absolute maximum value M, and an absolute minimum value monD. (i) Decide whether f (:3, 3;) takes the value M in the interior or on the boundary of D.
(ii) Do the same for m. Give the reasons for your answers. (Hint: Use the values obtained in (a), but do not try to
determine the values M and m; it would take too long.) I0 MATH 222, Calculus 3, Fall 2006, Final Examination This page is for the continuation of problem 4; it may also be used for rough work. 'l MATH 222, Calculus 3, Fall 2006, Final Examination This page is for the continuation of problem 4; it may also be used for rough work. MATH 222, Calculus 3, Fall 2006, Final Examination 5. A solid in the (x, y, z)—coordinate system is bounded from below by the disk inside the circle
£172+y2 2 6y in the (a, y)plane, and from above by a portion of the paraboloid z = $2+2y2+4.
It is deﬁned by the inequalities, ﬁ+ﬁ£®
nggm2+2y2+4 (a) Find the inequalities that deﬁne the solid in cylindrical coordinates. (b) The solid has a mass density whose value at (00, y, z) is p(:c, y, z) = y+z. Using cylindrical
coordinates, set up, but do not calculate, an iterated integral equal to the total mass M of the solid. l5 MATH 222, Calculus 3, Fall 2006, Final Examination This page is for the continuation of problem 5; it may also be used for rough work. MATH 222, Calculus 3, Fall 2006, Final Examination This page is for the continuation of problem 5; it may also be used for rough work. 05 MATH 222, Calculus 3, Fall 2006, Final Examination 2
6. Calculate the surface area of the part of the surface z(:1:,y) = 3302 + 3—1— 2
2
portion of the elliptical disk 69!:2 + 3% g 1 that is, over the region D deﬁned by 2
6w2+%§1, 3020,1420. over the ﬁrstquadrant Hint: to compute the double integral expressing the surface area, use the new variables
7“ and t with the coordinate transformation a: = 'r cos(t), y = 67' sin(t). You will have to determine the (r, t) region P that corresponds to the given region D under
the coordinate transformation. Kn MATH 222, Calculus 3, Fall 2006, Final Examination This page is for the continuation of problem 6; it may also be used for rough work. I?— MATH 222, Calculus 3, Fall 2006, Final Examination This page is for the continuation of problem 6; it may also be used for rough work. MATH 222, Calculus 3, Fall 2006, Final Examination . A particle of mass 1 moves in the (m,y)plane on the curve :54 + y2 = 1. Another particle
of mass 1 is located at the origin. By Newton’s law, the gravitational attraction between the particles is proportional to 213, where d is the distance between the particles. Using the
method of Lagrange multipliers ﬁnd all the points on the curve where the attraction between the particles is largest and all points where it is the smallest. ‘9 MATH 222, Calculus 3, Fall 2006, Final Examination This page is for the continuation of problem 7; it may also be used for rough work. 10. MATH 222, Calculus 3, Fall 2006, Final Examination This page is for the continuation of problem 7; it may also be used for rough work. SUMMARY OF FINAL EXAM FOR MATH 222 McGill University MATH 222  Calculus 3 Final exam Fall 2006, Final Examination Ques
tions. For your convenience on the two sides of this sheet, all problems are listed on this exam.
You may keep this sheet, but do not write anything on it during the exam. All the problems are
of equal weight. 1 Numerical answers may be given in the forms x/e + 7r, cos(23°), arccos 1+\/27
3 , etc. Show all calculations. Give sufﬁcient reasons for your answers. 1
1. Consider the function f(m) = — 9: an (a) Determine the Taylor (“Maclaurin”) polynomials T1 (:17),T2(:L'),T3(:B) of f centered at a = 0.
1 3
(b) Determine a constant C such that I f(ac) — T2(m) IS 0  m3 in the range 5 S m g 3
Hints: Show and use the fact that 1 — :1: + m2 2 — always. When calculatin repeated
4 , g derivatives, use temporary abbreviations such as the single letter 3 for 1 9m, + $2.) 2. Consider the space curve given by a:(t) 27:2
rm = ya) = 4t ,
2(t) ln(t2) the parameter t varying in the range If 2 1. (a) Compute the following as functions of t:
(i) the arc length 3(t) of the curve from to = 1 to t,
(ii) the unit tangent vector T(t),
(iii) the curvature H(t). (b) Compute the principal unit normal vector N (1) at t = 1. (Note: the results for (ii) and (iii) should come out as rational functions.)
3. Let f(:1:, y) = m3 + 21223; + 43/3 (a) Calculate the gradient Vf(2, 1) of the function f at the point P = (2,1). (b) Determine the equation of the tangent line of the level curve f (51:,y) = 20 of f at the
point P = (2,1). (c) Suppose that, for the unit vector u = (a,b), we have that the directional derivative
1
Duf(2, 1) equals §Vf(2, Determine the angle between the vectors Vf(2, 1) and u. ((1) Determine all unit vectors u = (a, b) for which Duf(2, 1) = %Vf(2, 4. Let f(:c,y) = 51:21; + 2mg + 1/3. (a) Find all the critical points of the function f (m,y), and classify them. For the local
maximum (maxima) and minimum (minima), determine the local maximum/ minimum
value of f(:v,y). (b) Consider the closed square D on the (:10, y) plane deﬁned by —3 g x S 1, —1 g y g 1. As we know, f (ac, g) has an absolute maximum value M, and an absolute minimum value
m on D. (i) Decide whether f(:c, y) takes the value M in the interior or on the boundary of D.
(ii) Do the same for m. Give the reasons for your answers. (Hint: Use the values obtained in (a), but do not try to
determine the values M and m; it would take too long.) 5. A solid in the (m, y, z)coordinate system is bounded from below by the disk inside the circle
m2+y2 = 6y in the (m, y)plane, and from above by a portion of the paraboloid z = m2+2y2+4.
It is deﬁned by the inequalities, w2+y236y
ng§$2+2y2+4 (a)Find the inequalities that deﬁne the solid in cylindrical coordinates. (b)The solid has a mass density whose value at (x, y, z) is p(a:, y, z) = y + 2. Using cylindrical
coordinates, set up, but do not calculate, an iterated integral equal to the total mass M
of the solid. 2 6. Calculate the surface area of the part of the surface z(m, y) = 3x2 + y? over the ﬁrst—quadrant
2
portion of the elliptical disk 6:1;2 + S 1 that is, over the region D deﬁned by U2 , '
(3.1.  51,..6 20, y 2 0. Hint: to compute the double integral expressing the surface area, use the new variables
7" and t with the coordinate transformation a: = rc0s(t), y = 6r sin(t). You will have to determine the (7313) region P that corresponds to the given region D under
the coordinate transformation. 7. A particle of mass 1 moves in the (:13, y)plane on the curve x4 + y2 = 1. Another particle of
mass 1 is located at the origin. By Newton’s law, the gravitational attraction between the
particles is proportional to dig, where d is the distance between the particles. Using the method of Lagrange multipliers ﬁnd all the points on theicurve where the attraction
between the particles is largest and all points where it is smallest. ...
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This test prep was uploaded on 04/19/2008 for the course MATH 222 taught by Professor Karlpeterrussell during the Spring '08 term at McGill.
 Spring '08
 KarlPeterRussell
 Math, Calculus

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