TEXAS A&M UNIVERSITY
DEPARTMENT OF MATHEMATICS
LAST NAME
FIRST NAME
ID
MATH 251
Section 504
EXAM 1
FALL 2008
O.Shatalov
Part credit possible. Show all work.
1. Consider two planes:
x+y−z =2
(1)
3x − 4y + 5z = 6.
(2)
and
(a) Find the normal vectors of these planes.
(4 points)
(b) Find cosine of the angle between the planes.
(8 points)
(c) Find equations for the line of intersection of these planes.
(13 points)
2. Determine whether each of the following limits exists and
explain why or why not. If the limit exists, find it.
(a)
lim
(x + y) cos
(x,y)→(0,0)
1
x2 + y 2
(10 points)
x3 y
(b)
lim
(x,y)→(0,0) x4 + y 4
(10 points)
3. Consider the function
z = ln(x − 3y).
(3)
(a) Find zx (7, 2) and zy (7, 2).
(5 points)
(b) Show that the function is differentiable at the point
(7,2).
(5 points)
(c) Find the differential of the function at the point (7,2).
(5 points)
(d) Using linear approximation approximate the value of
the function at the point (6.9,2.06).
(10 points)
4. Show that
y
∂z
∂z
−x
=0
∂x
∂y
if z = f (x2 + y 2 ).
(10 points)
5. Consider the surface
z = x2 + y 2 − 2x + 6y + 10.
(4)
(a) Find an equation of the tangent plane to the given
surface at the point (2, 0, 10).
(10 points)
(b) Classify and sketch the graph of given surface.
(10 points)
6. Bonus question +20%
Suppose that the equation
F (x, y, z) = 0
(5)
implicitly defines each of the three variables x, y, z as functions of the other two:
z = f (x, y),
y = g(x, z),
x = h(y, z).
If F is differentiable and Fx , Fy , Fz are all nonzero, show
that
∂z ∂x ∂y
·
·
= −1.
(6)
∂x ∂y ∂z