Download MATH 234 Final Exam - December 11, 2001 and more Exams Calculus in PDF only on Docsity!
MATH 234 FINAL EXAM December 11, 2001
Name(print)__________________________ Student Number_________ Section Number_________
Page 1 2 3 4 5 6 Total
Points
Instructions: 1.Since grading will be based on method you must show all work.
2.Boldfaced or lined letters indicate vectors such as F or
r k. 3.Check that your exam has the 12 problems.
- No calculators or formula sheets are allowed.
- (16 pts.) Let A = –2 i + 2 j + k and B = 2 i + 6 j – 2 k
a) – A + 2 B = _______________________
b) A = __________
c) A B = ___________
d) A B = _________________
- (18 pts.) A particle moves along a curve with velocity vector r v =
r i + 2 t
r j + t^2
r k ,1 ≤ t ≤ 2
a) At t = 1 the particle is at the point (0,
). Find
r r ( t ) and
r r ( 2 ).
b) How far did the particle travel in going between these two points (that is, from t = 1 to t = 2)?
c) Find the acceleration vector of the particle.
- (16 points) Let f ( x , y , z ) = z + x^2 + 2 y^2
a) Sketch the part of the level surface f(x,y,z) = 4 that lies above the plane z = 0.
b) Write the equation of the tangent plane to this level surface at the point (–1,1,1)
- (18 pts.) Let I = ydydx 0
ln x
1
e
a) Sketch the region of integration (e is approximately 2.7).
b) Write I with the order of integration reversed.
c) Evaluate one of the integrals.
- (16 pts.) Write an integral which gives the surface area of the surface cut from the
hemisphere x^2 + y^2 + z^2 = 6, z ≥ 0 by the cylinder ( x −1)^2 + y^2 = 1. Your final answer should be written in cylindrical coordinates. DO NOT EVALUATE the integral
- (18 pts.) Let the curve C be given by r r ( t ) = (cos t + t sin t )
r i + (sin t − t cos t )
r j , 0 ≤ t ≤ 2
a) Evaluate ( x^2 C
∫ +^ y
(^2) ) ds
b) Evaluate
r F • d
r r where
r F = x
r i + y
r j C
- (16 pts.) Use a double integral to evaluate (2 x − 6 y ) dx +( y − 4 x ) dy C
∫ where C is the
cardoid r = 1 + cos θ with counter-clockwise rotation.
- (16 pts.) a) Find a potential function for
r F = ( y^3 + z^3 )
r i + (3 xy^2 + 1)
r j + 3 xz^2
r k
