MVC
Final Exam Practice
Instructions: You must show enough work so that I can recreate your results!
1.
Given the points A(2, 3, 5), B(–2, 3, –1), and C(1, 0, 2), find each of the following.
a.
b.
2.
3.
an equation of the plane containing points A, B, and C.
equations (parametric or non-parametric) for the line through points A and B.
 xy 4 
Show that the following limit does not exist.
lim


 x, y    0,0   x 2  2 y8 
Find an equation of the tangent plane to the surface 2x2 – y2 + ln z3 = 1 at (3, 2, 1).
4.
5.
Change the order of integration:

4
6 x
0
x

e x cos y dy dx . DO NOT integrate.

Let T  u, v   uv, u 2  v2 . This transformation maps the region 0  u  1, 0  v  1 to the
region 0  x  1, x2 – 1  y  1 – x2.
a.
What curve in the xy–plane is the image of the top edge v = 1, 0  u  1 from the
region in the uv–plane? b. Use change-of-variables to evaluate
1 1 x 2

2
0 x 1
6.
y 2  4 x 2 dy dx . Show ALL steps (No Calculator).
If C be the curve that starts at the origin, goes to (1,1) along the graph of x  y 2 , and
returns to the origin along the graph of y = x3.
 e x2  3 y  dx  e y  5x 2 dy to a double integral using
a.
Transform the integral



C 
Green’s Theorem, providing appropriate limits of integration.
 e x2  3 y  dx  e y  5x 2 dy either directly or by using
b. Evaluate the integral



C 
your integral in part a.







x2
7. Let F ( x, y)   x sin y  i   y  cos y  j . Evaluate
2


2
(0,1) to (5,4) along y  x  6 x  1 .

 F ( x, y) ds ,
where C is the curve from
C
8. Prove that for any two non-zero vectors a and b , a  b  a  b if and only if a and b are
orthogonal.
IMSA
pg. 1
Fall 14 MVC Practice
9. The temperature at (x,y,z) of a ball centered at the origin is T ( x, y, z )  100e( x  y  z ) .
a. By inspection, decide where the ball the hottest?
b. Find a vector pointing in the direction of greatest increase of temperature at (1,1,1).
c. Find the rate of change of the temperature in the direction of the vector found in b.
2
2
2
10. Let C be the curve that starts at the point (0,1) and goes to (1,e) along the graph of y  e x and
returns to the point (0,1) along a straight line.
a. Transform the integral  2 y dx  y 2e y dy to a double integral using Green’s theorem,
C
providing appropriate limits of integration.
b. Calculate  2 y dx  y 2e y dy either directly or by using your integral in part a.
C
11. Use an appropriate change of variables to evaluate
first quadrant bounded by the hyperbolas
 y
D
2
2
xy
dA where D is the region in the
 x2
x  y  1 and x 2  y 2  4 , and the ellipses
2
x2
x2 y 2
 y 2  1 and

 4.
4
16 4
12. Find the volume of the solid bounded by the graph of y  x 2  2 and the planes y = 4,
z = 0, and 3y – 4z = 0.
13
Let C be an arbitrary simple, closed oriented curve and F be a vector field. Write an
integral that represents the flux of F across C.
14. Let S be the funnel-shaped surface defined by x 2  y 2  z 2 for 1  z  3 and x 2  y 2  1 for 0  z  1 .
a. Sketch the surface S.
b. Set up the triple integral (or integrals) that give the volume of the solid inside S (bounded
between z = 0 and z = 3).
15. Find the absolute maximum and minimum values
f ( x, y)  x 2  6 x  y 2  8 y  7 on the set {( x, y) : x 2  y 2  1}.
of
the
function
16. Find the equation of the tangent plane to f ( x, y)  x 2  6 x3  y 2 at the point (2,1).
17.

integral


If x  t   cos t ,sin t , t 2 with 0  t   is a path and f ( x, y, z )  x  e yz , write the line
f ds as a single integral using the variable t.
x
IMSA
pg. 2
Fall 14 MVC Practice
18. Evaluate the vector line integral


 F  ds where F ( x, y, z)  xi  yj  zk and the path is
x
given by x  t   2t  1, t ,2t  1 with 0  t  1.
2
19. Parameterize the portion of the surface r   , 0    6 , in cylindrical coordinates between
the planes z  2 and z  2 . Give the parameterization in terms of x,y, and z.
20. a.
Determine which of the following line integrals is independent of path in going from
(0,0) to (1,2).
i.
 ( x  3x
2
y 2 ) dx  (2 x3 y  3y 2 ) dy
C
ii.  sin x cos y dx  sin x sin y dy
C
iii.
b.
21.
(2 xy 5  2xy 2 )
(5 y 4  2y )
dx

dy
C ( x 2  1)2
x2  1
Evaluate one of the above integrals by finding the scalar potential function and the using
the fundamental theorem of line integrals.
Let C be a curve in the plane starting at (1, 0) moving to (1,0) along the curve 1  x 2 ,
and then returning to (1,0) along the x-axis.
a. Parameterize the path to express the line integral
 ( x  y )dx  x
2
y dy as an integral or
C
integrals in the single variable t. DON’T integrate.
b. Apply Green’s Theorem to the integral in part a. to obtain a double integral, making
sure to provide appropriate limits of integration. DON’T integrate.
c. Given the vector field F( x, y)  ( x  y) i  x 2 y j , write the integral(s) in the single
variable t you would need to evaluate to find the outward flux of F across the curve C.
DON’T integrate.
d.
Apply the divergence form of Green’s theorem to obtain a double integral that would
calculate this flux. DON’T integrate.
22. Find the flow line x (t ) for each vector field through the given point.
a.
F ( x, y, z )   x,  y  with x(0)  (1,5)
 1

b. F ( x, y, z )   2 ,1, z  3  with x(0)  (1,5, 2) .
x

IMSA
pg. 3
Fall 14 MVC Practice
23. Determine
whether
Green’s
theorem
can
be
used
to
evaluate
1
2
2
C ( x  3)2  ( y  3)2 dx  ( x  3)  ( y  3) dy where C is the curve pictured. Explain carefully


a. Circle of radius 3 centered at (3,3)
b. Unit circle centered at origin
c. C is the union of the two circles.
d. Parallelogram with vertices (1,0),(1,0),(3,3), and (1,3)
IMSA
pg. 4
Fall 14 MVC Practice