Math203, Demo Exam #1 (Chapters 12-14)
Directions: Answer all of the questions. The problems will have their weights assigned next
to them. Show all work for maximal credit. No notes, cellular phones, pagers, calculators,
books or friends. Good luck.
1) Definitions.
a. What is a Vector? What does it represent? Provide an example.
b. What does the Dot Product signify? Provide an example.
c. For a single variable function f(x), what does it mean when we state that )(
lim
3
xf
xโ
exists?
d. For the multivariable function f(x,y), what does it mean when we state that
),(
lim
2,3
yxf
yx โโ
exists?
e. What do contour lines represent?
f. Define the directional derivative and provide an example.
2) For the function,
( ) ( )
22 35
4
1++โ= yxz , perform the following. Be sure to explain
your reasoning.
a. Draw the function on the x-y plane as a set of contours representing static values of
z.
b. Draw the function on the y-z plan as a set of contours representing static values of x.
c. Identify any points which are discontinuous for the above function and describe why.
d. Describe the shape of the function.
e. Graph the function in 3-dimensional space.
3) For the vectors, kjiA
๎
๎
๎
๎
23 +โ= and kiB
๎
๎
๎
3โ= , perform the following. Be sure to
explain your reasoning.
a. Graph both vectors in 3 dimensional space and determine them angle between them.
b. Find 45 +โ
BA
๎
๎
. What does this signify?
c. Find
(
)
ABA
๎
๎
๎
โ
โร 22 . What does this signify?
d. Find
(
)
BBA
๎
๎
๎
โ
. What does this signify?
4) For the function,
(
)
(
)
2
22 435),,( zyxzyxf โ++โ= , perform the following. Be sure to
explain your reasoning.
a. Find x
f,
z
f
โ
โ
, f
โ
. What do each of these represent?
b. Find
)0,1,1(
y
f
โ
โ and the directional derivative at (1,1,0) going in the direction kj
๎
๎
+.
c. Find yx
f
โโ
โ2
.
d. Is this function differentiable for the entire region of 3-space?