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MATH 261 Final Exam - Fall 2000, Exams of Calculus

Dr. B.R. Ambedkar Open UniversityCalculus

This is the final exam for math 261 from fall 2000. It covers various topics in vector calculus, including equations for planes, parametric equations for lines, particle motion, level surfaces, curl of vector fields, directional derivatives, partial derivatives, tangent planes, and line integrals.

Typology: Exams

2012/2013

Uploaded on 02/14/2013

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MATH 261 โ€“ FALL 2000 โ€“ FINAL EXAM
STUDENT NAME โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”-
STUDENT ID โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”-
RECITATION HOUR โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”-
RECITATION INSTRUCTOR โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€“
INSTRUCTOR โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”-
INSTRUCTIONS
1. This test booklet has 14 pages including this one. There are 25
questions, each worth 8 points.
2. Fill in your name, your ID number, your recitation hour, the name
of your recitation instructor and the name of your instructor above.
3. Use a number 2 pencil on the mark-sense sheet (answer sheet) to do
the following:
3.1. On the top left side, write your name (last name, first name)
and fill in the little circles.
3.2 On the bottom left side, under SECTION, write in your division
and section number and fill in the little circles.
3.3. On the bottom, under STUDENT IDENTIFICATION NUMBER,
write in your student ID number, and fill in the little circles.
3.4 Blacken your choice of the correct answer in the spaces provided
for questions 1-25.
4. After you have finished the exam, turn in BOTH the answer sheet
and the question sheets to your instructor.
5. No books or notes or calculators may be used.
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MATH 261 โ€“ FALL 2000 โ€“ FINAL EXAM

STUDENT NAME โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”-

STUDENT ID โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”-

RECITATION HOUR โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”-

RECITATION INSTRUCTOR โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€“

INSTRUCTOR โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”-

INSTRUCTIONS

  1. This test booklet has 14 pages including this one. There are 25 questions, each worth 8 points.
  2. Fill in your name, your ID number, your recitation hour, the name of your recitation instructor and the name of your instructor above.
  3. Use a number 2 pencil on the mark-sense sheet (answer sheet) to do the following: 3.1. On the top left side, write your name (last name, first name) and fill in the little circles. 3.2 On the bottom left side, under SECTION, write in your division and section number and fill in the little circles. 3.3. On the bottom, under STUDENT IDENTIFICATION NUMBER, write in your student ID number, and fill in the little circles. 3.4 Blacken your choice of the correct answer in the spaces provided for questions 1-25.
  4. After you have finished the exam, turn in BOTH the answer sheet and the question sheets to your instructor.
  5. No books or notes or calculators may be used. 1

  1. Which of the following is an equation for the plane that contains the point (3, 2 , 1) and is perpendicular to the line with vector equation ~r(t) = (1 โˆ’ 2 t) ~i + 3t ~j + (4 + t) ~k? A) 2x โˆ’ 3 y โˆ’ z = โˆ’ 1 B) 2x + 3y + z = 14 C) x + 2y โˆ’ z = 8 D) 2x โˆ’ y + z = 0 E) x + y + z = 4

  2. Which of the following are parametric equations for the line that passes throug (1, 1 , 2) and is parallel to the line xโˆ’ 2 3 = yโˆ’ 3 1 = z โˆ’ 4. A) x(t) = 1 + 2t, y(t) = 1 + 2t, z(t) = 2 + t B) x(t) = 1 + 2t, y(t) = 1 + 3t, z(t) = 2 + t C) x(t) = 1 + t, y(t) = 1 + 2t, z(t) = 2 + 3t D) x(t) = 1 + 4t, y(t) = 1 + 3t, z(t) = 2 + 3t E) x(t) = 1 + 2t, y(t) = 1 + 3t, z(t) = 2 + 2t

  1. The curl of the vector field F~ (x, y, z) = x ~i + y ~j + xyz ~k is A) x ~i + y ~j + xyz ~k B) xz ~i โˆ’ yz ~j C) โˆ’xz ~i + yz ~j D) x^2 ~i + y^2 ~j + z^2 ~k E) xyz ~i + 2xy ~j + 3z ~k

  2. Let f (x, y, z) = x^3 + โˆš 6 y^2 + z^4. The directional derivative of f at the point (1, 1 , 1) in the direction in which f increases most rapidly is A) 2 B) 3 C) 5 D) 7 E) 4

  1. Let f (x, y) = sin(xy). Find (^) โˆ‚xโˆ‚yโˆ‚^2 f. A) cos(xy) B) cos(xy) + xy sin(xy) C) โˆ’xy sin(xy) D) sin(xy) โˆ’ cos(xy) E) cos(xy) โˆ’ xy sin(xy)

  2. Find an equation of the tangent plane to the surface z = โˆšx^2 + y^2 at the point (3, โˆ’ 4 , 5). A) 2x + y โˆ’ 5 z = โˆ’ 23 B) 4x โˆ’ y + z = 0 C) 3x โˆ’ 4 y + 5z = 0 D) 3x + y + z = 10 E) 3x โˆ’ 4 y โˆ’ 5 z = 0

  1. The function f (x, y) = x^3 โˆ’ x โˆ’ y^2 + 2y has A) Two relative minima B) A relative maximum and a relative minimum C) Two saddle points D) A relative maximum and a saddle point E) A relative minimum and a saddle point

  2. Find the point (x, y), which satisfies y โˆ’ x^2 = 0, at which f (x, y) = (x โˆ’ 16)^2 + (y โˆ’ 12 )^2 is minimum. A) (1, 1) B) (2, 4) C) (3, 9) D) (^12 , 14 ) E) (6, 36)

  1. Evaluate โˆซ โˆซ D sin (^ ฯ€ 2 x^2 )^ dA, where D is the region of the plane bounded by x = y, y = 0 and x = 1. A) ฯ€ B) (^1) ฯ€ C) (^2) ฯ€ D) 2ฯ€ E) 3ฯ€

  2. The area of the region inside the circle r = 2 cos ฮธ and above the line y = โˆš 3 x is given by A) โˆซ^0 ฯ€^2 โˆซ^0 2 cos ฮธr dr dฮธ B) โˆซ^ ฯ€ 6 ฯ€^2 โˆซ^0 2 cos ฮธr dr dฮธ C) โˆซ^ ฯ€ 3 ฯ€^2 โˆซ^0 2 cos ฮธr dr dฮธ D) โˆซ^0 ฯ€^3 โˆซ^0 2 cos ฮธr dr dฮธ E) โˆซ^0 ฯ€^6 โˆซ^0 2 cos ฮธr dr dฮธ

  1. The surface area of the portion of the hemisphere z = โˆš 2 โˆ’ x^2 โˆ’ y^2 inside the paraboloid z = x^2 + y^2 is A) โˆซ^02 ฯ€^ โˆซ^01 โˆš1 + r^2 r dr dฮธ B) โˆซ^02 ฯ€^ โˆซ^01 โˆš^2 โˆ’^2 r 2 r dr dฮธ

C) โˆซ^02 ฯ€^ โˆซ^01

2+r^2 r dr dฮธ D) โˆซ^02 ฯ€^ โˆซ^01 โˆš 22 โˆ’r 2 r dr dฮธ E) โˆซ^02 ฯ€^ โˆซ^01 (2 + r^2 ) r dr dฮธ

  1. Compute the line integral โˆซ C y sin x dx + y^2 dy along the curve composed of the line segment from (0, 0) to (2, 0) and the line segment from (2, 0) to (2, 1). A) (^13) B) 1 C) 2 sin 2 D) 2 sin 2 + sin 1 + 1. E) sin 2 + sin 1

  1. Evaluate โˆซ C xy dx + xy dy, where C be the circle x^2 + y^2 = 4 oriented counterclockwise. A) 2ฯ€ B) 3ฯ€ C) 0 D) 32 ฯ€ E) 2

  2. ~ Let f (x, y, z) = 8x^3 y^4 + 9yzex^ + y sin(z^3 + x^4 ). Let F (x, y, z) = โˆ‡f (x, y, z) and let C be parametrized by ~r(t) = cos t ~i + sin t ~j + t ~k, 0 โ‰ค t โ‰ค ฯ€. Then โˆซ C F~ ยท d~r is equal to A) 3ฯ€ B) 2ฯ€ C) โˆ’ฯ€ D) 1 E) 0.

  1. Let ฮฃ be the part of the paraboloid z = x^2 + y^2 inside the cylinder x ~^2 + y^2 = 4, and let the unit normal ~n on ฮฃ be directed upwards. If F (x, y, z) = x ~i โˆ’ ~j + 2x^2 ~k, then โˆซ โˆซ ฮฃ F~ ยท ~n dS is equal to. A) 0 B) 23 ฯ€ C) ฯ€ D) 323 ฯ€ E) 34 ฯ€

  2. If F~ (x, y, z) = y^2 z^3 ~i โˆ’ xz ~j + z ~k, ฮฃ is the sphere x^2 + y^2 + z^2 = 4, and ~n is the outward unit normal of ฮฃ, then โˆซ โˆซ ฮฃ F~ ยท ~n dS equals A) 323 ฯ€ B) 16ฯ€ C) 8ฯ€ D) 203 ฯ€ E) 163 ฯ€

  1. Let F~ (x, y, z) = cos(y^2 z^3 ) ~i โˆ’ xz^8 y^3 ~j + z^16 ~k, and let ฮฃ be the ellipsoid โˆซ โˆซ x 42 + y 92 + z 162 = 1. Let ~n be the outward unit normal of ฮฃ, then ฮฃ(curl^ F~^ )^ ยท^ ~n dS^ equals A) 1 B) 5 C) 3 D) 2 E) 0