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MA 227 Final Exam Spring 2010, Exams of Advanced Calculus

Aliah UniversityAdvanced Calculus

The final exam questions for ma 227, a university-level mathematics course, from spring 2010. The exam covers various topics in multivariable calculus, including finding equations of planes and tangent planes, directional derivatives, partial derivatives, and integrals. Students are required to find solutions and justify their conclusions.

Typology: Exams

2012/2013

Uploaded on 03/16/2013

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SPRING 2010 — MA 227 — FINAL EXAM
FRIDAY APRIL 30, 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 (1,2,2), (1,1,−1) and
(−1,2,1).
(b) Let r(t) = (4t1/4, et2
−1,2t). Find the unit tangent vector at the point on the
curve corresponding to t= 1.
1
pf3
pf4
pf5
pf8
pf9
pfa

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SPRING 2010 — MA 227 — FINAL EXAM

FRIDAY APRIL 30, 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, 2 , 2), (1, 1 , −1) and (− 1 , 2 , 1). (b) Let r(t) = (4t^1 /^4 , et (^2) − 1 , 2 t). Find the unit tangent vector at the point on the curve corresponding to t = 1.

1

  1. (a) Let f (x, y) = x cos(y) − x^2 y. Find the second partial derivative fxy. (b) Let f = x^2 z and F = (xz, y, z^2 y). Find ∇f (the gradient of f ), div F (the divergence of F), and curl F (the curl of F).
  1. (a) Let z = x^3 y^2 − x. Find the equation of the tangent plane at the point (2, 1). (b) Find equation of the tangent plane to the surface x^2 + 2y^2 − 2 z^2 = 4 at the point (2, − 1 , 1).
  1. Find the local maximum, minimum and saddle points (if any) of the function

f (x, y) = 2x^2 + 4xy − y^2 + 6x − 3.

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

f (x, y) = 2x^2 + 3y^2 − 4 x − 3 on the region 0 ≤ x ≤ 2 , − 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) sin(x − y) dA

where D is the rectangle enclosed by the lines x − y = 0, x − y = 4, x + y = 1, and x + y = 2.

  1. (a) Switch the order of integration in the iterated integral

∫^2

0

∫^ x^3

0

f (x, y) dy

 (^) dx.

(b) Using a double integral, find the area of the triangle with vertices (0, 0), (1, 1), (1, 2).

  1. Use cylindrical coordinates to find the mass of the solid that lies within both the cylinder x^2 + y^2 = 16 and the sphere x^2 + y^2 + z^2 = 36 and above the plane z = 0, if the material in the solid has density (mass per unit volume) given by ρ(x, y, z) = 2.