MATH 201 Exam 2, Fall 2014
American University of Beirut · Calculus and Analytical Geometry III · 5 pages · Mr. Hassan Yamani
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Math 201, Quiz II Long Sample 1
K. Aziziheris (2013)
1) Let ( ) √
b) From definitions, show that (0, 0) = 0 = (0, 0)
c) Prove that f is differentiable at (0, 0) d) sketch the graph of z = √ (upper sphere)
a) Find the domain of f and describe the level curves of f .
2) (i) Show that
.0
)
cos(
8
2
9
( ,0 0)
,
( lim
x
y
y
x
xy
x y
( 0,0 )
y ) for ( ,x y)
x
y
7cos( x
(ii) Let f (x, y)
8
2
4
/ 3
4
Prove or disprove that
f ( 0,0 )
can be defined so that
( , )
f x y
is continous at (0, 0).
3) Suppose
100
( , , , )
F x y z w
and all components of F are never zero.
Find
x
y
y
x
x
z
.
.
given that
e3x 10y 7 .z
z
x
Justify your answer .
4) The function
( , , )
f x y z
at a point P increases most rapidly in the direction of the vector v=(3,4,5)
with directional derivative
10 2
. (i) Find
f (P)
(ii) Find the directional derivative of
( , , )
f x y z
at P in the direction of the vector w=(4, 0, 3).
(iii) Is it possible to find a vector v such that ( ) ( ) Explain.
5) The derivative of of f(x; y) at P(1; 2) in the direction of i + j is
and in direction of –2j is –3. Find the derivative of f in the direction of –i–2j.
(big Hint: Suppose grad(f)(P)= <a, b>. So you have 2 equations in 2 unknowns
0)
cos5z
e
)z,y
Find a,b if f( ,x
axby
satisfies Laplace equation
0
zz
yy
xx
f
f
f
.
Baby 6) Given the surface containing the point P(1; –1; –2)
a) Find an equation of the tangent plane to the surface at P.
b) Find an equation of the normal line to the surface at P.
7a) Investigate the critical points of
17
2
6
2
( , )
3
3
y
xy
x
f x y
for local maxima, local minima, or saddle points.
7b) Locate all local extrema and saddle points of ( )
7c) Locate all local extrema and saddle points of ( ) .
8) Find the parametric equations for the line tangent to the curve of intersection of the surfaces
xyz = 1 and at the point P(1; 1; 1). (big Hint: cross products).
9) By about how much will f(x; y; z) = ln √ change if the point p(x; y; z) moves from P0
(3; 4; 12) a distance of 0.1 units in the direction of 3i + 6j 2k ?
10) Find the set of points on the surface
xyz
y
x
8
36
2
2
where the tangent plane is
(i) perpendicular to the x-y plane. (ii) parallel to the x-y plane.
11) (14.8) Use Lagrange multipliers to find the (absolute) maximum and minimum of the function
30
2
2
17 on the surface 2
2
5
( , , )
z
y
x
z
y
x
f x y z
12) (Chain Rule) Suppose
k
j
5i
4
3
)1,1,1(
f
and f(1,1,1)=5.
Let
)
,
,
(
2
2
4
tx
t
P f t
where
( , , )
f u v w
is a differentiable function. Then at t=1, x=1,
t
P
i
/
( )
……
t
x P
ii
/)
( 3
( )
=….
t
t P
iii
/)
( 3
)
(
=…….
12*) Suppose
( , )
)
,
(
5
t f x y
f tx ty
for all values of x, y, t (where
( , )
f u v
is a differentiable
function). Show that
f
yf
xf
i
y
x
5
( )
( Hint: Partial w.r.t t both sides, then set t=1).
f
y f
xyf
x f
ii
yy
xy
xx
20
2
)
(
2
2
(Hint: Double partial w.r.t t both sides, then set t=1).
13) Find the absolute minimum and maximum of the function ( )
a) Over the region *( ) }
b) Over the region *( ) }
Good Luck