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Math 3012 Mathematics, Lecture notes of Engineering Mathematics

Math 3012 MathematicsMath 3012 MathematicsMath 3012 Mathematics

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Yilin Wang MTH212 Fall2016
Assignment 01 Intro 12.1 12.2 due 09/19/2016 at 11:59pm EDT
1. (1 point) Find the derivative of h(w) = 2w3+14w.
h0(w) =
Solution:
SOLUTION
Recall that w=w1/2, so that
h0(w) = (2)·(3)w31+14
2w1/2=6w4+7w1/2.
Correct Answers:
2*3*wˆ[-1*(3+1)]+14/2*wˆ(-1/2)
2. (1 point) A cube is located such that its top four cor-
ners have the coordinates (5,2,2),(5,3,2),(0,2,2)and
(0,3,2). Give the coordinates of the center of the cube.
center =
Solution:
SOLUTION
By drawing the top four corners, we find that the length of
the edge of the cube is 5. See the figure below.
We also notice that the edges of the cube are parallel to the
coordinate axis. So the x-coordinate of the the center equals
5+5
2=2.5. Similarly, the y-coordinate of the center equals
2+5
2=0.5, and the z-coordinate of the center equals 2 5
2=
0.5.
Correct Answers:
(-2.5,0.5,-0.5)
3. (1 point) On a separate page, sketch a graph of the equa-
tion x=1 in 3-space.
Suppose you are observing the coordinate system as shown
below.
If your graph represents a solid (opaque) wall, can you see the
point (1,3,3)? [?/yes/no]
Which of the following is your graph (note that by clicking
on any graph you will get a larger image)?
graph [?/1/2/3/4/5/6]
1. 2. 3. 4. 5. 6.
Solution:
SOLUTION
The graph is a plane parallel to the yz-plane, and passing
through the point . Accordingly, it looks like graph 1, and if
this is a solid wall, we can see the point (1,3,3).
Correct Answers:
yes
1
4. (1 point) Without a calculator or computer, match the
functions with their graphs in the figures below. Note that two
of the functions do not have a matching graph.
(a) z=2+x2+y2
?
1
2
3
4
none of these graphs
(b) z=2x2y2
?
1
2
3
4
none of these graphs
(c) z=2
?
1
2
3
4
none of these graphs
(d) z=2+x2y
1
pf2

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Download Math 3012 Mathematics and more Lecture notes Engineering Mathematics in PDF only on Docsity!

Yilin Wang MTH212 Fall

Assignment 01 Intro 12.1 12.2 due 09/19/2016 at 11:59pm EDT

  1. (1 point) Find the derivative of h(w) = − 2 w−^3 + 14

w. h′(w) = Solution: SOLUTION Recall that

w = w^1 /^2 , so that

h′(w) = (− 2 ) · (− 3 )w−^3 −^1 +

w−^1 /^2 = 6 w−^4 + 7 w−^1 /^2.

Correct Answers:

  • 23wˆ[-1(3+1)]+14/2wˆ(-1/2)
  1. (1 point) A cube is located such that its top four cor- ners have the coordinates (− 5 , − 2 , 2 ), (− 5 , 3 , 2 ), ( 0 , − 2 , 2 ) and ( 0 , 3 , 2 ). Give the coordinates of the center of the cube. center = Solution: SOLUTION By drawing the top four corners, we find that the length of the edge of the cube is 5. See the figure below.

We also notice that the edges of the cube are parallel to the coordinate axis. So the x-coordinate of the the center equals − 5 + 52 = − 2 .5. Similarly, the y-coordinate of the center equals − 2 + 52 = 0 .5, and the z-coordinate of the center equals 2 − 52 = − 0 .5. Correct Answers:

  • (-2.5,0.5,-0.5)
  1. (1 point) On a separate page, sketch a graph of the equa- tion x = −1 in 3-space. Suppose you are observing the coordinate system as shown below.

If your graph represents a solid (opaque) wall, can you see the point ( 1 , − 3 , 3 )? [?/yes/no]

Which of the following is your graph (note that by clicking on any graph you will get a larger image)? graph [?/1/2/3/4/5/6]

Solution: SOLUTION The graph is a plane parallel to the yz-plane, and passing through the point. Accordingly, it looks like graph 1, and if this is a solid wall, we can see the point ( 1 , − 3 , 3 ). Correct Answers:

  • yes
  • 1
  1. (1 point) Without a calculator or computer, match the functions with their graphs in the figures below. Note that two of the functions do not have a matching graph. (a) z = 2 + x^2 + y^2 -?
  • 1
  • 2
  • 3
  • 4
  • none of these graphs

(b) z = 2 − x^2 − y^2

-?

  • 1
  • 2
  • 3
  • 4
  • none of these graphs

(c) z = 2

-?

  • 1
  • 2
  • 3
  • 4
  • none of these graphs

(d) z = 2 + x − 2 y 1

  • none of these graphs

(e) z = 2 − x + 2 y

-?

  • 1
  • 2
  • 3
  • 4
  • none of these graphs

(f) z = 2 − x

-?

  • 1
  • 2
  • 3
  • 4
  • none of these graphs

Solution: SOLUTION (a) z = 2 + x^2 + y^2 is an upward opening parabaloid with pos- itive z intercept, centered on the z-axis, and so is none of these graphs. (b) z = 2 − x^2 − y^2 is a downward opening parabaloid with pos- itive z intercept, centered on the z-axis, and so is 3. (c) z = 2 is a plane parallel to the x- and y-axes, with positive z intercept., and so is 1. (d) z = 2 + x − 2 y is a plane with positive y and z intercepts, and negative x intercept, and so is none of these graphs. (e) z = 2 − x + 2 y is a plane with positive x and z intercepts, and negative y intercept, and so is 2. (f) z = 2 − x is a plane parallel to the y-axis, with positive x and z intercepts., and so is 4. Correct Answers:

  • none of these graphs
  • 3
  • 1
  • none of these graphs
  • 2
  • 4
  1. (1 point) You like pizza and you like cola. Which of the graphs in the figure below represents your happiness h as a func- tion of how many pizzas p and how much cola c you have if (a) There is such a thing as too many pizzas and too much cola? figure [?/1/2/3/4] (b) There is such a thing as too much cola but no such thing as too many pizzas? figure [?/1/2/3/4]

Solution: SOLUTION Let’s look at what each of the graphs tells us:

  1. If we get sick upon eating too many pizzas or drinking too much cola, then we expect our happiness to decrease once ei- ther or both of those quantities grows past some optimum value. This is depicted in graph (1) which increases along both axes until a peak is reached, and then decreases along both axes.
  2. If we do get sick eating too much pizza, but are always able to drink more cola, then we expect our happiness to decrease after we eat some optimum amount of pizza, but continue to in- crease as we get more cola. This is shown by this figure, which increases continuously along the cola axis but, after reaching a maximum, begins to decrease along the pizza axis.
  3. If we do get sick after too much cola, but are always able to eat more pizza, then we expect our happiness to decrease af- ter we drink some optimum amount of cola, but continue to in- crease as we get more pizza. This is shown by graph (3) which increases continuously along the pizza axis but, after reaching a maximum, begins to decrease along the cola axis.
  4. If we have iron stomachs and can consume cola and pizza endlessly without ill effects, then we expect our happiness to increase without bound as we get more cola and pizza. This is what is shown in this graph, since happiness increases along both the pizza and cola axes throughout. Thus, our first answer is 1, and the second is 3. Correct Answers:
  • 1
  • 3

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