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The instructions and problems for a demo exam in math203, covering topics such as vectors, dot product, contour lines, directional derivatives, and multivariable functions. Students are required to answer all questions, show their work, and perform various calculations and graphical representations.
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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.
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
( )
3
f x
x โ
exists?
d. For the multivariable function f(x,y), what does it mean when we state that
(, )
3 , 2
f x y
x โ y โ
exists?
e. What do contour lines represent?
f. Define the directional derivative and provide an example.
2 2
z = x โ + y + , 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 โ + 2 and B i k
= โ 3 , 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 5 A โ B + 4
. What does this signify?
2 ร โ 2 โ . What does this signify?
โ . What does this signify?
2 2 2
f ( x , y , z )= x โ 5 + y + 3 โ 4 z , perform the following. Be sure to
explain your reasoning.
a. Find
x
f ,
z
f
, โ f. What do each of these represent?
b. Find
( 1 , 1 , 0 )
y
f
and the directional derivative at (1,1,0) going in the direction j k
c. Find
x y
f
2
.
d. Is this function differentiable for the entire region of 3-space?