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Mathematics & Statistics (Sci)

MATH 222

Christa Scholtz

Fall

Description

STUDENT NAME:
STUDENT NUMBER:
FACULTY OF SCIENCE
FINAL EXAMINATION
MATHEMATICS 189-222A
CALCULUS III
Examiner: W. Jonsson Date: ????????, December ???, 2010
Associate Examiner: N. Sancho Time: 9:00 AM - 12:00 PM
Instructions
1. Total number of points: 100.
2. No books, calculators or notes are allowed for the exam. Do not rip pages from the examination book.
3. There are 4 versions of this examination. This version belongs to Group 1.
4. Answers to Part I are to be entered on the machine readable sheet with a soft lead pencil.
5. Answers to part II are to be written in the space provided on the examination paper.
6. Your answers may contain … or other expressions that cannot be computed without a calculator, e.g.
1=2 ¡3=2
ln2, 300 + 13 ¢ 150 .
7. All material (question papers, machine readable sheets) must be turned in.
8. Name, Student number and group number of your examination MUST be entered on the question
paper and on the machine readable sheet.
GOOD LUCK!
Score Table
Part I
Multiple Choice
Part II
Problems Points
1.
2.
3.
4.
5.
Total:
This exam comprises 7 pages, including this cover. Final Examination December ???, 2010 Mathematics 189-222A 2
PART I. Multiple choice questions. Group 1
Each question is worth 3 points.
2 2
1. The equation of the plane tangent to the surface z = 2x + y at the point where x = 3, y = 2 is
(a) z = ¡12x ¡ 4y + 66, (b) z = 12x ¡ 4y ¡ 6, (c) z = ¡12x + 4y + 50,
(d) z = ¡12x + 4y ¡ 30, (e) z = 12x + 4y ¡ 22.
2 x
2. The fourth degree Taylor polynomial of f(x) = x e centered at a = 0 is
x2 x3 x4 x2 x3 x 3 x4
(a) 1 + x + + + , (b) 1 + x + + , (c) x + x + + ,
2 6 24 2 6 2 6
x4 x 2 x3 x4
(d) x + x + , (e) + + .
2 2 6 24
3. The vector i + 2j + 3k is perpendicular to the vector 9i ¡ 3j + c ¢ k when
(a) c = ¡1, (b) c = 1, (c) c = 2, (d) c = 3, (e) c = 0 .
4. For any two vectors a;b the cross product (3a + 2b) £ a is the same vector as
(a) 2a £ b, (b) 2b £ a, (c) 3a £ b, (d) 3b £ a, (e) 5b £ a.
µ ¶
X1 ¡1 n
5. The series
2
n=0
(a) has sum 1=2; (b) has sum 1=3; (c) has sum 2=3;
(d) diverges to 1; (e) diverges to ¡1.
X 1
6. The p-series
np
n=1
(a) converges for p ‚ 1 and diverges for p < 1,
(b) converges for p > 1, diverges for p < 1 and we cannot say whether is converges or diverges for
p = 1,
(c) diverges for p ‚ 1 and converges for p < 1,
(d) diverges for p > 1 and converges for p • 1,
(e) converges for p > 1 and diverges for p • 1.
X1 2n+1 3 5
7.The power series (¡1)n x = x ¡ x + x ¡ ¢¢¢ represents the function
(2n + 1)! 3! 5!
n=0
(a) e ; (b) sinx; (c) cosx; (d) arctanx; (e) tanx: Final Examination December ???, 2010 Mathematics 189-222A 3
Group 1
X n 2n+1
8. The power series (¡1) x
n!(n + 1)!2n+1
n=0
(a) converges only for x = 0, (b) has radius of convergence 2,
(c) converges for all real numbers x, (d) has interval of convergence ¡4 < x < 4,
(e) has radius of convergence 1.
9. A particle is moving along the trajectory r(t) = 2costi + 3sintj + tk. At time t = …=2 the velocity
vector v(…=2) and the acceleration vector a(…=2) are
(a) v(…=2) = ¡2i + k; and a(…=2) = ¡3j,
(b) v(…=2) = ¡2j + k; and a(…=2) = ¡3i,
(c) v(…=2) = 2i + k;and a(…=2) = 3j,
(d) v(…=2) = 2j + k; and a(…=2) = 3i,
(e) v(…=2) = 3i + k;and a(…=2) = 2j.
10. The directional derivative of the function f(x;y) = x + 2y in the direction of the unit vector u =
p
(i ¡ j)= 2 and at the point (2;3) is
p p p p
(a) 0, (b) 2= 2, (c) ¡4= 2, (d) 6= 2, (e) ¡8= 2.
Z
x2
11. The s

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