Math 2511 – Calc III Practice Exam 2
This is a practice exam. The actual exam consists of questions of the type found in this practice exam, but
will be shorter. If you have questions do not hesitate to send me email. Answers will be posted Tuesday.
1.
a)
b)
c)
d)
Definitions: Please state in your own words the following definitions:
partial derivative of a function f(x,y)
directional derivative of a function f(x, y) in the direction of a unit vector u
definition of a function being differentiable
gradient
e)
The double integral of f over the region R
f)
g)
h)
i)
j)
k)
l)
Surface Area
Fubini’s theorem
Vector field
Conservative vector field
Potential function of a vector field
curl or a vector field F
divergence of a vector field F
 f ( x, y)dA
R
2.
a)
b)
c)
d)
e)
f)
g)
Theorems: Describe, in your own words, the following:
a theorem relating differentiability with continuity
a theorem relating differentiability with partial derivatives
a theorem stating criteria for a function to have relative extrema
a result that classifies critical points into relative max., min., or saddle points
the procedure to find relative extrema of a function f(x, y)
the procedure to find absolute extrema of a function f(x, y)
a theorem that allows you to evaluate a double integral easily
a)
b)
c)
d)
e)
f)
g)
h)
i)
True/False questions:
If f(x, y) is continuous, it must be differentiable
If f(x, y) is differentiable, it must be continuous
If f(x, y) has partial derivatives fx and fy, then f must be differentiable
If f(x, y) has partial derivatives fx and fy and both are continuous then f must be differentiable
If f(x, y) is a function such that fxx, fyy, fxy, and fyx exist then fxy = fxy
If f(x, y) is a function such that all second order partials exist and are continuous then f xy = fyx
If f(x, y) is a function such that all second order partials exist and are continuous then fxx = fyy
If the divergence of a vector is zero, the vector field is conservative.
If F ( x, y, z ) is a conservative vector field then curl ( F )  0 How about div(F) ?
3.
bd
j)

a c
4.
a)
b
 d

f ( x) g ( y )dydx   f ( x)dx    g ( y )dy 

 

a
 c



Differentiation: Find the indicated derivatives for the given function:
Suppose f ( x, y)  2 x 3 y 2  2 y  4x , find fx, fy, fxx, fxy, fyy, and fyx


c)
Find the rate of change with respect to y of x 2  y 2  z 2  1 at P 2 , 1 , 2 .
3 3 3
2 x
Let f ( x, y)  y e  y . Find fxyy, fyxy, and fyyx
d)
Suppose f (  ,  , )   2 cos( ) sin( ) , find f , f , f
e)
For more examples, please see homework assignments
b)
5.
Total differential:
a)
Suppose f ( x, y )  x 2  y 2 . Use the total differential to approximate the change of f as (x, y) varies
b)
from (3, 4) to (3.04, 3.98).
Suppose the radius of a right cylinder is measured with a 2% error, while the height is measured with
an error of 4%. What is the maximum relative error in V, the volume of that cylinder.
R1 R2
.
R1  R2
Suppose R1 is measured at 200 Ohm, with an error of 2%, and R2 is measured at 400 Ohm, with an
error of 2% as well. What is the error in computing R from this data?
The total resistance R of two resistances R1, R2 that are connected in parallel is R 
c)
6.
Direction Derivatives:
Find the directional derivative of f(x, y) = xy exy at (-2, 0) in the direction of a vector u, where u
makes an angle of Pi/4 with the x-axis.
b)
Suppose f ( x, y)  x 2 e y . Find the maximum value of the directional derivative at (-2, 0) and compute
a unit vector in that direction.
c)
For more examples, please see homework assignments
a)
7.
Max/Min Problems: Compute the extrema as indicated
2
f ( x, y )  3 x 2  2 xy  y  8 y . Find relative extreme and saddle point(s), if any.
a)
f ( x, y)  4 xy  x 4  y 4 . Find relative extrema and saddle point(s), if any
Let f ( x, y)  3xy  6 x  3 y  7 . Find absolute maximum and minimum inside the triangular region
spanned by the points (0,0), (3, 0), and (0, 5).
For more examples, please see homework assignments
b)
c)
b)
8.
Evaluate the following integrals:
1 2
a)
  xy
2
dxdy
2
 2 y dydx
0 0
2 x
b)
  x

0 x2
3 9 x 2
c)
 
x 2  y 2 dydx
3 0
ab c
d)
x
2
 y 2  z 2 dxdydz
000
 1
e)
  cos(x
2
)dxdy
0 y
f)

x 2  y 2 dA , where R is the part of the circle in the 1st quadrant
R
  /2
g)
  sin(x) cos(y)dydx
0 0
h) For more examples, please see homework assignments
9.
The pictures below show to different ways that a region R in the plane can be covered. Which picture
corresponds to the integral
 f ( x, y )dxdy
R
10.
Suppose you want to evaluate
 f ( x, y )dA where R is the region in the xy plane bounded by
y 0,
R
y  2  x 2 , and y  x . According to Fubini’s theorem you could use either the iterated integral
 f ( x, y )dxdy or
 f ( x, y )dydx to evaluate the double integral. Which version do you prefer? Explain.
9.
a)
Use a multiple integral and a convenient coordinate system to find the volume of the solid:
bounded by z  x 2  y  4 , z  0 , y  0 , x  0 , and x  4
b) bounded by z  e  x and the planes y  0 , y  x , and x  1
2
c)
bounded above by z  16  x 2  y 2 and bounded below by the circle x 2  y 2  4
d) evaluate
y
 x 2  y 2
where R is a triangle bounded by y  x , y  2x , x  2
R
e)
bounded by the paraboloid z  4  x 2  2 y 2 and the xy plane
a)
Find the following surface areas:
of the plane z  2  x  y above the rectangle 0  x  2 and 0  y  3
10.
b) of the cylinder z  9  y 2 above the triangle bounded by y  x , y   x , and y  3
c)
of the surface z  16  x 2  y 2 above the circle x 2  y 2  9
a)
Find the following centers of mass
center of mass of a lamina corresponding to the parabolic region 0  y  4  x2 where the density at a point
( x, y) is proportional to the distance between ( x, y) and the x-axis (i.e.  ( x, y)  ky )
11.
b) center of mass of a lamina with density  ( x, y)  x 2  y 2 and region inside x 2  y 2  9 , x  0 , y  0
12.
Below are four algebraic vector fields and four sketches of vector fields. Match them.
[A]
[B]
[C]
[D]
(1) F ( x, y)  x, y  , (2) F ( x, y)   y, x  , (3) F ( x, y)  x,1  , (4) F ( x, y)  1, y 
13.
Suppose that F ( x, y, z)  x 3 y 2 z, x 2 z, x 2 y  is some vector field.
a) Find div(F)
b) Find curl(F)
c) Find curl(curl(F))
d) Find div(curl(F))
14.
Decide which of the following vector fields are conservative. If a vector is conservative, find its potential
function
a) F ( x, y)  2xy, x 2 
c)
F ( x, y)  e x cos(y), e x sin( y) 
F ( x, y, z )  sin( y),  x cos y,1 
d)
F ( x, y, z)  2xy, x 2  z 2 ,2zy 
b)
15.
Prove the following facts:
a)
A function f is said to satisfy the Laplace equation if
2 f
x 2

2 f
y 2
 0 . Show that the function
f ( x, y)  ln( x 2  y 2 ) satisfies the Laplace equation.
b)
A function (not a vector field) f ( x, y, z ) is called harmonic if
any function f ( x, y, z ) the function
c)
d)
e)
f)
2 f
x
2

2 f
y
2

 2f
z 2
 0 . Show that for
1
is harmonic.
f ( x, y, z )
Two function u(x, y) and v(x, y) are said to satisfy the Cauchy-Riemann equations if
u v
and

x y
u
v
  . Show that the functions u( x, y)  e x cos(y) and v( x, y)  e x sin( y) satisfy the Cauchyy
x
Riemann equations.
Show that if u(x, y) and v(x, y) are functions such that all second-order partials are continuous and u
and v satisfy the Cauchy-Riemann equation. Then both u and v also satisfy the Laplace equation.
Use the definition of differentiability to show that f ( x, y)  xy is differentiable.
 xy
, for ( x, y )  (0,0)

Let f ( x, y )   x 2  y 2
Then show that f has partial derivatives at (0, 0) but f is not
0,
for ( x, y )  (0,0)
differentiable at (0, 0).
g)
h)
i)
11.
a)
If F ( x, y, z )  M ( x, y, z ), N ( x, y, z), P( x, y, z )  is any vector field where M , N , P are twice
continuously differentiable then div(curl( F ))  0
Prove that the volume of a Sphere with radius R is 4/3 * Pi * r^3
Prove that the surface area of a Sphere with radius R is 4 * Pi * r^3
Story problem (motion)
Either max/min problem, or a problem with error estimation, or a Lagrange multiplier problem.