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MA 227: Fall 2010 Final Exam in Multivariable Calculus, Exams of Advanced Calculus

Aliah UniversityAdvanced Calculus

The final exam questions for ma 227: multivariable calculus, held on december 10, 2010. The exam covers various topics such as finding equations of planes, directional derivatives, partial derivatives, tangent planes, local maxima, minima, and saddle points, as well as integrals and transformations of coordinates.

Typology: Exams

2012/2013

Uploaded on 03/16/2013

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FALL 2010 — MA 227 — FINAL EXAM
FRIDAY DECEMBER 10, 2010
Name:
There are 11 questions, each worth 10 points; 100 (or more) points is equiv-
alent to 100% for the exam. Partial credit is awarded where appropriate.
Show all working; your solution must include enough detail to justify any
conclusions you reach in answering the question.
1. (a) Find the equation of the plane containing the points (1,1,2), (0,1,−1) and
(−1,2,1).
(b) Let r(t) = (2t2,sin(t3−1),2). Find the unit tangent vector at the point on the
curve corresponding to t= 1.
1
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FALL 2010 — MA 227 — FINAL EXAM

FRIDAY DECEMBER 10, 2010

Name:

There are 11 questions, each worth 10 points; 100 (or more) points is equiv- alent to 100% for the exam. Partial credit is awarded where appropriate. Show all working; your solution must include enough detail to justify any conclusions you reach in answering the question.

  1. (a) Find the equation of the plane containing the points (1, 1 , 2), (0, 1 , −1) and (− 1 , 2 , 1). (b) Let r(t) = (2t^2 , sin(t^3 − 1), 2). Find the unit tangent vector at the point on the curve corresponding to t = 1.

1

  1. (a) Let f (x, y, z) = xz cos(y) − xyz^2. Find the third partial derivative f (^) xyz′′′. (b) Let f = xy^2 z and F = (xz, y, x^2 z). Find ∇f (the gradient of f ), div F (the divergence of F), and curl F (the curl of F).
  1. (a) Let z = x^2 y − x^3. Find the equation of the tangent plane at the point (1, 2). (b) Find equation of the tangent plane to the surface x + 2y^2 − z^3 = 3 at the point (2, − 1 , 1).
  1. Find the local maximum, minimum and saddle points (if any) of the function

f (x, y) = x^2 − xy + y^2 + 9x − 6 y + 10.

  1. Find the absolute maximum and absolute minimum points of the function

f (x, y) = x^2 + y^2 + x on the region − 1 ≤ x ≤ 1 , − 1 ≤ y ≤ 1. Be sure to provide the coordinates of the points and the values of absolute maximum and minimum.

  1. Evaluate, by making an appropriate change of variables, the integral ∫ ∫

D

(x + y)^2 ex−^2 y^ dA

where D is the parallelogram enclosed by the lines x− 2 y = 0, x− 2 y = 2, x+y = −1, and x + y = 1.

  1. (a) Change (1,
  1. from rectangular into spherical coordinates. (b) Using spherical coordinates evaluate ∫ ∫ ∫

E

(x^2 + y^2 + z^2 ) dV,

where E is the half-ball x^2 + y^2 + z^2 ≤ 4, z ≥ 0.

  1. Use polar coordinates to find the mass of the lamina that lies within the annual region 1 ≤ x^2 + y^2 ≤ 16, if the material in the lamina has density (mass per unit volume) given by ρ(x, y) = x^2 + y^2.