Lebanese American University
Summer I
Byblos
2007
Calculus IV
Test #2
Date: 23/07/2007
Duration: 2h
1. We consider the surface (S) : z = ln(x2 + y 2 ) and the point P0 (0, 1, 0).
(a) Verify that P0 is on the surface (S).
(b) Find the equation of the tange
Lebanese American University
Summer I
Byblos
2007
Calculus IV
Test #2
Solution Keys
1. (a) Easy.
(b) The gradient at P0 is 2 j + k then the equation of the tangent plane is
2y + z + 2 = 0.
(c) x = 0, y = 2t + 1 and z = t.
2. The solutions of fx = fy = 0 a
Lebanese American University
Fall 2007
Byblos
Calculus IV
Test #2
Date: 05/01/2008
Duration: 2h
1. We consider the surface (S) : z = e(x
2 +y 2 )
and the point P 0(0, 1, e).
(a) Verify that P0 is on the surface (S).
(b) Find the equation of the tangent pl
Lebanese American University
Summer I
Byblos
2007
Calculus IV
Test #1
Date: 12/07/2007
Duration: 1h 30
1. The position vector of a particle in the plane at time t is given by
r(t) = et cos t i + et sin t j
(a)
(b)
(c)
(d)
Prove that r(t) is a smooth curve
Lebanese American University
Fall 2007
Byblos
Calculus IV
Test #2
Date: 05/01/2008
Duration: 2h
1. We consider the surface (S) : z = e(x
2 +y 2 )
and the point P 0(0, 1, e).
(a) Verify that P0 is on the surface (S).
(b) Find the equation of the tangent pl
Lebanese American University
Summer I
Byblos
2007
Calculus IV
Test #2
Date: 23/07/2007
Duration: 2h
1. We consider the surface (S) : z = ln(x2 + y 2 ) and the point P0 (0, 1, 0).
(a) Verify that P0 is on the surface (S).
(b) Find the equation of the tange
Lebanese American University
Summer I
Byblos
2007
Calculus IV
Test #1
Date: 12/07/2007
Duration: 1h 30
1. The position vector of a particle in the plane at time t is given by
r(t) = et cos t i + et sin t j
(a)
(b)
(c)
(d)
Prove that r(t) is a smooth curve
Lebanese American University
Summer I
Byblos
2007
Calculus IV
Final exam
Date: 30/07/2007
Duration: 2h
1. We consider the velocity field F = (zex + ey ) i + (xey + ez ) j + (yez + ex ) k.
(a) Prove the F is conservative.
(b) Find a potential function for
Lebanese American University
Summer I
Byblos
2008
Calculus IV
Test #1
Date: 15/07/2008
Duration: 1h 30
1. We consider the plane curve
(C ) : y = ln(sin x),
0 < x < .
(a) Find the curvature of (C ).
(b) Find the osculating circle of (C ) at x0 = /4.
2. Let
Lebanese American University
Summer I
Byblos
2008
Calculus IV
Test #2
Date: 26/07/2008
Duration: 2h
1. We consider the surface (S) : z = exy x y 1 and the point P0 (1, 1, 2).
(a) Verify that P0 is on the surface (S).
(b) Find the equation of the tangent p
Lebanese American University
Fall 2008
Byblos
Calculus IV
Test #2
Date: 14/01/2009
Duration: 1h 30
1. We consider the surface (S) : z = x cos y + y 2 sin x and the point P0 (1, 0, 1).
(a) Verify that P0 is on the surface (S).
(b) Find the equation of the
Lebanese American University
Fall 2008
Byblos
Calculus IV
Final Exam
Date: 02/02/2009
Duration: 2h
1. We consider the vector field:
F = yz 2 i + (xz 2 + 2) j + (2xyz 1) k
(a) Prove the F is conservative.
(b) Find a potential function for the field F.
(c)
Lebanese American University
Spring 2009
Byblos
Calculus IV
Final Exam
Date: 23/06/2009
Duration: 2h
1. Find the osculating circle of the plane curve (C) : y = x4 2x2 at x0 = 1.
2. We consider the vector field:
F = (sin(yz) + 2x) i + (xz cos(yz) + ez ) j
Lebanese American University
Summer I
Byblos
2009
Calculus IV
Final Exam
Date: 08/08/2009
Duration: 2h
1. We consider the velocity field:
F = (eyz + yzexz ) i + (xzeyz + exz ) j + (xyeyz + xyexz + 2z) k
(a) Prove the F is conservative.
(b) Find a potentia
Lebanese American University
Spring 2009
Byblos
Calculus IV
Test #2
Date: 21/05/2009
Duration: 1h 30
1. Find the equation of the tangent plane to the surface z = ex + y sin x at the
point P0 (0, 1, 1).
2. We consider the function f (x, y) = sin xy.
(a) Fi
Lebanese American University
Summer I
Byblos
2009
Calculus IV
Test #2
Date: 30/07/2009
Duration: 1h 30
1. Find the equation of the tangent plane to the surface xy + xez + zey = 1 at
the point P0 (1, 0, 0).
2. We consider the function f (x, y) = xexy .
(a)
Lebanese American University
Fall 2009
Byblos
Calculus IV
Test #2
Date: 07/01/2010
Duration: 2h
1. Find the equation of the tangent plane to the surface x2 y + 2xz 2 zexy = 1
at the point P0 (1, 0, 1).
2. We consider the function f(x, y) = x3 y2 x2 y + 1.
Lebanese American University
Summer I
Byblos
2010
Calculus IV
Final Exam
Date: 27/07/2010
Duration: 2h
1. We consider the velocity field: F = (3x2 y + z 2 ) i + (x3 2yz) j + (2xz y2 ) k. We
denote by (C1 ) and (C2 ) are the following two curves:
(C1 ) :
Lebanese American University
Spring 2010
Byblos
Calculus IV
Test #2
Date: 05/05/2010
Duration: 1h 30
1. We consider the function f(x, y, z) = x2 + 2y2 + 3z 2 and the point P0 (2, 1, 1).
(a) Find the equations of the tangent plane and normal line to the el