The "assigned" dates
for HW on this website are approximate.
The
rule as to when each HW is actually assigned is stated in the Syllabus:
HW
for a section is assigned on the day when we have finished covering
that section in class.
HW # 1
Assigned: 08/31 (All problems except
those marked in bold
red
must
be done via WebAssign.)
Sec. 12.1: ## 3, 12,
15, 23, 25, 29.
HW
# 2
Assigned: 08/31
(All problems except
those marked in bold
red
must
be done via WebAssign.)
Sec. 12.2: ## 3,
9, 13, 43, 15, 16, 25, 26, 31, 33, 35, 45(only a-c in book), 47.
Notes for #
45: - For your own
benefit, you should do the separate step (b)
described in the book even
though WA combined this step with (c).
- To find approximate
values of s and t, you need to sketch the
given vectors a, b, and c, as well as the resulting
parallelogram,
accurately. Then,
measure the lengths of vectors and the parallelogram's
sides with a ruler,
preferably one with the millimeter scale.
HW
# 3
Assigned: 09/01
(All problems except
those marked in bold
red
must
be done via WebAssign.)
T/F Quiz at the end of Chap. 12 (p. 842):
## 1, 3,
19, 8;
Note: These are
problems from the True/False
(T/F) Quiz, NOT
from Concept Check. Here is an easy way to tell that you are doing
a problem from a wrong part, e.g., from Concept check intsead of T/F
quiz or Review Exercises, or vice versa. For example, if you have
started, by mistake, to do # 3 in Review Exercises, you notice that it
is
asking a question that is clearly not covered in Sec. 12.3, which is
what this
HW is on. Then, instead of trying to find an answer in Sec. 12.4 (if
you can guess it) or give up, you should realize that you are
attempting a problem that was not actually assigned.
Sec. 12.3: ## 1, 7,
9, 14,
20, 55,
23, 40,
41, 45,
46, 48;
Review Exercises at the end of Chap. 12 (p. 843):
# 9.
Note for ## 40, 41: In
addition to finding the numbers for
the answers,
sketch the two given vectors in the same plane
and also sketch the required projection.
(Consult figures in pp. 2-3, 2-4 of posted notes
as well as Figs. 4, 6 in the textbook.)
Clarification:
Your sketch should be a simple 2D drawing;
so please disregard the z-coordinate in your sketch
even though it may be provided in the
problem.
Answer for # 40:
component = 14/sqrt[17]; projection = 14/17
*<1,4>.
Note for #
45: Note that the given formula, orthab =
b - projab, defines
the
orthogonal projection orthab
in terms of the
"usual" projection projab .
Hints for #
45: - Use the formula that allows one
to determine if
two vectors are perpendicular.
- Do not use the component form of a vector,
like a = <a1,a2>.
Instead, use the formula for projab that was given in class
(or can be found in the book).
Hint for # 46: For the sketch, again,
consult the figures mentioned in
the Note for ## 40, 41. Also, for the sketch of orthab,
recall the geometric definition of the difference of two vectors:
see p. 1-3 of the posted notes for Secs. 12.1 and 12.2.
You can check your answer geometrically, by verifying
that projab
+ orthab = b, and algebraically, by verifying that
orthab
_|_ a.
Answer for # 46: orthab = <20, -5>/17.
Hint for # 48: See the definitions before Example 6 in the
book.
Also, try to visualize the
problem by making a sketch.
Extra
credit # 1 (worth
0.2% added to your final
grade)
Assigned: 09/01
Due:
09/08
Note: Before
you attempt this
extra-credit assignment, please read these
instructions!
Sec. 12.3: # 58.
Hint:
Use the
dot product.
Clarification:
You should not assume any specific components
for a and b.
(In fact, I suggest that you
not use the component form of vectors at all.)
An additional 0.1% will be
added to your score if you explain the fact that you have proved
using
elementary
geometry. Hints for this "additional extra
credit":
(i) What are the lengths of each term in the vector equation for c?
(ii) Draw the parallelogram representing c
as the vector sum of
those two terms. This parallelogram is special (because of
your answer in (i)).
(iii) What properties do the diagonals in this special
parallologram have?
HW
# 4
Assigned: 09/02
(All problems except
those marked in bold
red
must
be done via WebAssign.)
T/F Quiz at the end of Chap. 12 (p. 842):
## 4, 6,
7, 9, 13, 14, 20, 21 (see
the Note for HW 3);
Sec. 12.4: ## 13, 3, 16, 29,
19, 22,
38, 11, 37.
Answer for T/F # 4:
False.
Note for # 22:
Use properties of the cross and dot products, not
a direct calculation.
Hint for # 29(b): The
area of a triangle equals half the area of the parallelogram made
by two of the triangle's sides.
Hint for # 38: Begin by computing vectors AB, AC,
AD.
Note 1 for #
11: Do as
the
assignment says: Find the answers
by using the
properties
of cross product,
not by a calculation via a determinant!
Note 2 for #
11: If you haven't done so already, make sure that you read
the
"Common syntactic errors and issues" document found right below
the Homework link on the course webpage.
HW
# 5-A
Assigned: 09/08
(All problems except
those marked in bold
red
must
be done via WebAssign.)
T/F Quiz at the end of Chap. 12 (p. 842):
# 16;
Sec. 12.5: ## 3, 7, 11, 13, 17, 18, Word problem
(below), 81.
Word problem:
Point Q is on segment P1P2
and is 3 times closer to P2 than to P1.
If P1 =
(1, 2, 3) and P2 = (4, 5, 6),
find the coordinates of Q.
Hint for # 11: Recall that if you have an equation of a line,
you are given
the two "ingredients" of that line.
Hint for # 13: See p. 3-1 of posted Notes about how to tell if
two vectors
are parallel or not.
Hint for WP: What value of t corresponds
to point Q?
Answer for WP: Q = (13/4, 17/4, 21/4).
Hint for # 81: See topic 5 (Segment connecting two points) in
Lec. 4; especially
where it handles t-values with 0 < t < 1.
Extra credit # 2 (worth
0.1% added to your final
grade)
Assigned: 09/09
Due:
09/16
Note: Before
you attempt this
extra-credit assignment, please read these
instructions!
Word
problem:
Use a formula from the Notes for Sec.
12.5(Lines) to show that the coordinates of
a segment's midpoint computed by that formula
coincide with the coordinates
stated in Sec. 12.1 (in the Notes or in the
textbook).
HW
# 5-B
Assigned: 09/09
(All problems except
those marked in bold
red
must
be done via WebAssign.)
Sec. 12.5: ## 24, 27, 33, 63, 45, 53,
55,
71, 73, 74, 1.
Extra credit # 3
Assigned: 09/11
Due:
09/18
Note: Before
you attempt this
extra-credit assignment, please read these
instructions!
1) Sec. 12.5: # 66 (worth
0.25% added to your final
grade).
Clarification:
The point given in the problem is not on the line given in the problem.
2) Word
problem:
(worth
0.15% added to your final
grade).
Show that the formula for distance between two
parallel planes ( D = |ax0 +
by0 + cz0 + d| / Sqrt[a^2+b^2+c^2] )
always gives the same answer regardless of which of
the two planes one picks the point
(x0, y0, z0)
on to compute that distance (see topic 6b in the
Notes for Sec. 12.5(Planes)).
You must provide
a clear explanation in order to receive credit.
HW # 6
Assigned: 09/11
(All problems except
those marked in bold
red
must
be done via WebAssign.)
T/F Quiz at the end of Chap. 12 (p. 842):
# 18
(see the Note for HW 3);
Sec. 12.6: ## 1, 2, 4,
5, 6, 7, 41, 13 (see
Notes below for both of these problems);
Word Problem 1:
Describe and sketch the surfaces: (a)
x^2 - z^2 = 4;
(b) x^2 - z^2 = 3.
In what way do the constants "4" and "3"
make
these
surfaces different?
Word Problem 2:
Describe and sketch the
surfaces: (a)
2x^2 + y^2 = 1;
(b) y^2 +3 z^2 = 4.
See the General Note
and Notes and Answers to specific problems, below.
Now
let us get back to problems
at the end of Sec. 12.6.
Very soon we will start
extensively referring to a surface called "cone".
You all know, of course, what
a cone looks like, but this is a good time to get familiar
with its equation. To that
end, do the following.
# 41:
Solve the given equation
for z (you should obtain two
answers that differ by +/-)
and plot it in Mathematica using command Plot3D. If you are not
familiar with its
syntax, go to Help in Mathematica's menu, select Wolfram Documentation,
and type
"Plot3D" without quotation signs in the search box. When you plot your
function,
make sure to use
the plotting option
RegionFunction
-> Function[{x, y, z}, x^2 + y^2 <= 4]
(it is explained
in examples for Plot3D). The number "4" above can be changed to
any other number
depending on what rang of x,y-values
you want to show in your plot.
The coefficients
in front of x
and y should also
mimic those of the surface in question.
Notice which coordinate axis is the axis of this cone.
Also notice and
explain, based on the
material of Sec. 12.6 and Mathematica Lab 1,
which dimension, x or y, of the cone's base is greater and
by what factor.
# 13:
Sketch this surface by hand,
based on your experience with # 41. (Plot3D will not
work in this case; we will learn how to sketch such surfaces in
Mathematica, but only
at the end
of this course.)
To make a correct sketch, you'll first need to answer the two questions
asked about # 41:
(i) What is the axis of this cone? and
(ii) Which
dimension of the
cone's base is larger and
by what factor?
General
Note for all
problems related to ellipses and hyperbolas:
Review Part 1 of
Lab 1 and Appendix C in your textbook (pp. A19--A22)
on how to tell
the parameters (e.g., semi-axes) of these curves from their equations.
Answer for T/F # 18:
False.
Answer for # 2: To make a sketch in (b) and (c), use the Plot3D
command in Mathematica.
Also, in (c) you have an "exponential cylinder", defined similarly
to the parbolic cylinder described in the next Answer.
Answer for # 6: Parabolic cylinder obtined by extending the
parabola y=z^2
in the yz-plane along the x-axis.
Note for #
7: I am hoping that you will
combine your knowledge of the curve xy=1,
which you learned in high school, with the material about
surfaces that
you learned in this section.
To
verify your sketches
for the Word Problems, solve the equations for z
and then use the Plot3D
command in Mathematica.
HW
# 7
Assigned: 09/15
(All problems except
those marked in bold
red
must
be done via WebAssign.)
T/F Quiz at the end of Chap. 13:
## 1, 2, 3
(you may want to do it along
with ##
7, 15, and 21, 25 listed below);
Sec. 13.1: ## 1; 18; 8,
7, 15, 10,
12, 27,
21, 25, 31,
32, 42, 43, 44, 49;
Review Exercises at the end of Chap. 13 (p. 882):
# 1(a) (along
with ## 7, 15, and 21, 25 above), 6(a)
(along
with ## 29, 30 above).
Answer for T/F # 2:
True.
Note for ##
8,7: I expect you
to recognize these as very familiar cartesian curves,
i.e. curves of the form y = f(x) or x=f(y).
Remember that, conversely, any
cartesian curve y
= f(x)
can be written in parametric form as x = t, y =
f(t).
So, to go from parametric to cartesian, you need to reverse this step.
Note for ## 10, 12: While
you must sketch the curves by hand (following the examples
presented in class/posted notes, you should verify your answers
with Mathematica's command ParametricPlot3D (see Lab 1).
Notes for ## 27, 21, 25: 1. Recall the parametric
equation of a certain curve from Sec. 12.6
which you were required to memorize.
2. Use the knowledge
of some of the surfaces which you sketched
in HW # 6 (Sec. 12.6).
Answer for # 6(a) on p. 882: (15/8, 0, -ln(2)).
HW
# 8
Assigned: 09/18
(All problems except
those marked in bold
red
must
be done via WebAssign.)
T/F Quiz at the end of Chap. 13: 4, 5;
Sec. 13.1: # 5;
Sec. 13.2: ## 3,
5, 13, 14, 17, 19, 23,
25, 26,
33, 36,
37, 41;
Review Exercises at the end of Chap. 13 (p. 882): # 5.
Hint
for # 5 of Sec. 13.1: Use a method from Sec. 2.6 for one
components;
for
the others, use the limiting values of the functions involved.
Note for ## 3,5 of Sec.
13.2: I expect
you to recognize these as very familiar cartesian curves,
i.e. curves of the form y = f(x) or x=f(y).
Hint
for # 5 of Sec. 13.2: This cartesian curve has nothing to do
with the expenential curve.
Note for ## 13, 14: Review the Chain Rule.
Note for # 13: The
expression e^t^n should be interpreted as e^(t^n).
Answer for # 26: x=2+t/2, y=ln4 + t/2, z=1+t.
Note for # 36: Review formula 10 and Ex.
5 in Sec. 7.4
(don't pay attention to details of Partial Fraction Expansion,
but note how the integrals are done).
Note for # 37: Review Exs. 1 and 2 of Sec. 7.2.
Note for # 5 on p. 882:
Review Ex. 4
in Sec. 5.5.
Extra
credit # 4 (worth
0.2% added to your final
grade)
Assigned: 09/18
Due: 09/25
Note: Before
you attempt this
extra-credit assignment, please read these
instructions!
Sec. 13.2: # 28.
HW
# 9
Assigned: 09/23
(All problems except
those marked in bold
red
must
be done via WebAssign.)
T/F Quiz at the end of Chap. 13:
## 12, 8, 7, 10, 13;
Review Exercises at the end of Chap. 13 (p. 882):
# 12.
Sec. 13.3: ## 1, 2, 3, 11, 13, 15,
47, 48,
49, 53, 17, 19, 21,
33.
Note for ## 2, 11: Use identity
a^2+2ab+b^2 =
(a+b)^2 and then
Sqrt[(a+b)^2] = (a+b).
Also, for # 11, use x=t for parametrization of the curve.
Note for # 3: Use the main hyperbolic identity about cosh^2 and
sinh^2
(see p. 260
of Sec. 3.11 or
Lab 1),
or, instead, identity (x + 1/x)^2 = x^2 + 2 + 1/x^2.
Note for # 47: Use Mathematica to calculate T' and its
length! (See also the Hint for ## 2,11.)
The expression for the length is fairly simple.
Note for # 48: Use the Chain Rule to find r' and then a trig
identity from
the middle of p. A28 (Appendix D) to find its length.
Note for ## 17(a), 19(a):
You do not
need to sketch these curves in 3D!
Pretend that your curve is in
2D and sketch both T and N for it.
Focus on how these vectors are
oriented relative to the curve.
Note for # 19(a): When finding the length of r'(t), use the following observation:
e^(2x) + 2 + e^(-2x) = (e^x +
e^(-x))^2 for any x.
If you use Mathematica to do
calculations for 19(a), you will see
that denominators in some of your terms look like the above expression.
Unfortunately, I don't know of a robust way to make Mathematica
recognize this, so you will need to replace that expression by a
complete square by hand.
Note for ## 17(b), 19(b): Contrary
to the direction given by the textbook and WA,
use
Theorem 10 instead of Formula
9.
Also, when computing the cross
product
between vectors
with only two components (with the third being zero,
as
it is for vectors in a plane), you must use the original
definition of the cross product as a 3x3 determinant,
rather than its 2x2 shorcut version that may be
presented in WebAssign solutions or elsewhere online.
Note for # 12 on p. 882: Find a very similar
example in
the Notes.
Extra credit #
5 (the amount
of credit added
to your
final grade is stated next to each problem)
Assigned: 09/23
Due:
09/30
Note: Before
you attempt this
extra-credit assignment, please read these
instructions!
Sec. 13.3: ## 67 (0.1%),
68 (0.3%).
Note that # 67 is worked out in a WA tutor video. You may watch it and
follow its guidelines.
However, credit will be given if you include in your work absolutely
all the derivations,
especially those skipped or glanced over in the video.
HW
# 10
Assigned: 09/25
(All problems except
those marked in bold
red
must
be done via WebAssign.)
Sec. 13.4: ## 1(a), 3, 6, 5,
7, 10, 36, "37", 39,
43,
the 2 Word Problems below (yes, in
this order),
22,
16, 17(a);
Concept Check at the end of Chap. 13: # 8(b).
Word
Problems: (a)
Sketch
the following parametric curves and then, for each curve, sketch
the unit tangent and
unit normal vectors at
t=0 and t=1.
(b) Sketch the acceleration vector and its tangential and normal
components
at t=0 and t=1 for each of these curves:
1. r = < t, t^2 - 2t >;
2. r = < 3t-t^3, 3t^2 >.
To
emphasize: In
these problems you should focus on sketching rather than on
a calculation of a_T and a_N.
Note for ## 3,6: See the Note for ## 7,8 for Sec.
13.1 (HW # 7).
Hint
for # 6: This cartesian curve has
nothing to do with the exponential curve.
Note for #
5: You should recognize this
curve from an earlier section in this course.
Answer
for # 6: To sketch the path, use ParametricPlot in
Mathematica;
v(0)
= <1,2>, a(0)
= <1,4>, |v(0)|
= Sqrt[5].
Make sure that you still sketch the path and v
and a
in the same
figure (as you should also do for ## 5 and 7).
Answers
for # 36: (a) a_N=0, (b)
a_T=0.
Note for # "37": This isn't quite
the # 37 from the textbook, but instead # 5 on WA,
which is the closest problem on WA to what I actually want to assign.
Note for ## "37", 39: Use Mathematica to calculate the
cross
and
dot products,
as well as to do
any other
calculations.
Note for ## 43 and for 8(b) (Concept Check on p. 881):
See Fig. 7 in
Sec. 13.4
that shows the acceleration vector
as a vector
sum of its projections on T
and N.
Note for Word Problem 1: Recognize this curve as a familiar
cartesian curve;
see the Note for ## 3, 6 above.
If in doubt, plot it using the ParametricPlot command
in Mathematica
(see Lab 1 or Mathematica's Help).
Note for Word Problem 2: This is essentially the same curve as in
# "37" ( = WA # 5).
However, unlike in # "37", here you should focus on sketching,
not on a calculation.
Answers for Word Problem 1:
a = <0,2>;
@t=0: T =
<1,-2>/sqrt(5); N=<2,1>/sqrt(5)
(you are not asked to compute N,
but can verify
from your sketch that this N
works);
a_T < 0, a_N > 0;
@t=1: T =
<1,0>; N=<0,1>
(you are not asked to compute N,
but can verify
from your sketch that this N
works);
a_T = 0, a_N = 2.
Answers for Word Problem 2: a
= < -6t, 6 >;
@t=0: T =
<1,0>; N=<0,1>
(you are not asked to compute N,
but can verify
from your sketch that this N
works);
a_T = 0, a_N = 6.
@t=1: T =
<0,1>; N=<-1,0>
(you are not asked to compute N,
but can verify
from your sketch that this N
works);
a_T > 0, a_N > 0 (and both = 6/sqrt(2)).
Hint for # 22:
Review Example 4 in the book for Sec. 13.2. The conceptual part
of this Example was previewed in class, and the technical
part
was assigned as a must-do independent reading.
Extra credit #
6 (value of
each problem (in %
added to your
final grade) is marked next to it)
This EC assignment is unusual
in that part of it (## 28, 29, 32) is based on the material
(projectile motion) not covered in class. (We needed to skip this topic in
order to save time for
the material on Vector Calculus, to be covered at the end of the
course.) The material on
projectile motion is covered in Examples 5 and 6 in the textbook.
Assigned: 09/28
Due:
10/05
Note: Before
you attempt this
extra-credit assignment, please read these
instructions!
Sec. 13.4: ## 35 (0.3%), 28
(0.2%), 29
(0.15%), 32 (0.35%), 33 (0.1% for each
part), 45 (0.1%).
Common
note:
Remember: I grade your clearly
presented
solutions, with detailed explanations,
including sketches whenever appropriate.
I will not give credit for
solutions with scarce explanations,
even if your answers are correct.
Note for ## 29, 33, 45: These problems are worked
out in a WA video tutor.
You may watch it and follow its guidelines. However,
credit will be given if you present absolutely all the derivations and
explanations, especially those skipped or glanced over in the video.
Hints for # 35:
This problem is actually on the material of:
Sec. 13.2
(specifically, on an Example which is mentioned above and
which I asked you to review on
your
own when we covered that section)
and on Sec. 12.4.
Sketch c and r (as some two vectors in the
same plane), and
decompose r as: r = r_par +
r_perp,
where r_par and r_perp are parallel and
perpendicular to c,
respectively.
Then from the equation for r',
determine the evolution of r_par
and r_perp.
Note for # 29: Assume y0 = 0.
Note for # 33: The "constant speed" referred to in
this
problem is the own speed
of the boat. The total velocity = own velocity + current's
velocity.
Hins for # 45: 1) When the engine is shut
off, the
ship leaves the trajectory
along the tangent line. So, review Sec. 13.2.
2) The equations for the shut-off time that you'll obtain are
too
complicated
to be solved by any systematic method. However, they can be solved
by inspection. To that end, think for which rational (i.e., not
transcendental)
value of t0, ln(t0) is also a rational
number.
HW # 11
Assigned: 09/28
(All problems except
those marked in bold
red
must
be done via WebAssign.)
Sec. 14.1: ## 2, 9, 11, 12, 24, 25, 29, 32,
33, 34(b),
36, 37, 41, 45,
14.1.509.xp (this is # 10 on the WA list; unfortunately, it is not in
the 8-th edition of e-book),
49, 55, 67, 68, 69, 70, 71, 72.
General
notes: 1) Review the surfaces studied in
Sec.
12.6.
2) You may use Mathematica to do 3D plots.
The syntax of Plot3D was introduced in Lab 3.
Answer for # 12(b): The half-space below the plane z = 10 - x - y.
Notes for # 32: 1) Review a class example.
2) When f(x,y) depends on the combination (x^2+y^2), the graph
has the circular symmetry (because the function depends on
k = x^2 + y^2, which is the equation of a circle).
When f(x,y) depends on the combination (xy), then the graph
does not have the
circular symmetry. Instead, to visualize it,
set, e.g., x=k, and then the graph's trace in the plane x=k
should be some familiar curve z = f(ky).
Note for # 33: Make sure to answer the question
about the shape. Comment on
where it is steeper and where it is gentler. Review a class example
about that.
Hints for # 34(b): 1) The wind speed is higher where the pressure
changes more rapidly
over the terrain (not over time!). So, again,
you are asked to find
where the graph pressure(x,y)
is the steepest.
2) Somehow, the answer does not
agree with Chicago's nickname.
Note for # 36: Review a class example.
Note for # 37: Again, this is the question
about the slope of the terrain (i.e., surface).
Note for # 45: Review Labs 1 and 3 (and pp.
A19--A22 in Appendix C of e-book, if needed).
Note for # 55 just
FYI: The surface T(x,y)
looks similar to that in Fig. 11.
Note for ## 68, 70: See the Table of surfaces in Sec.
12.6.
(You created a particular example of one of these surfaces in Lab 3.)
Note for ## 71, 72: Review pp. 37--39
(Sec. 1.3) in
the textbook.
Extra credit #
7 (this problem is worth 0.3% added
to your
final grade)
Assigned: 09/28
Due: 10/06
Note: Before
you attempt this
extra-credit assignment, please read these
instructions!
Sec. 14.1: # 79 (see clarifications below).
1) As the problem says, there will be two intervals for c.
Makes plots for two values of c
from each interval, showing how
the graphs change with c.
Also, make a graph for the critical value of c.
2) Explain the existence of two intervals for c by completing the
square.
Name the sirface obtained in each of these
intervals. (For an explanation,
you must complete the square not only when c = 2 or -2, but for an
arbitrary c.
Then discuss a relation of your result with equations of
some two familiar surfaces.)
HW
# 12
Assigned: 09/30
(All problems except
those marked in bold
red
must
be done via WebAssign.)
Sec. 14.2: ## 5, 6 (this number in the book is the
closest match to # 2 on the WA list),
9, 10, 11, 13,
14, 17,
20,
21 (this number in the book is the closest match to # 7 on
the WA list),
23,
24, 39,
40, 41, 42,
43.
Note for # 10:
Use the fact that as x-->0,
cos(x)
approaches 1.
Note for # 11:
Use the fact that as x-->0,
sin(x)
approaches x.
Note for # 13: Use polar coordinates.
Note for # 17: Use a method from Sec. 2.3
or
L'Hospital's Rule.
Note for ## 40, 41: Review Sec.
4.4 to
find the correct form of L'Hospital's Rule
to be
used in each of these
problems.
Note for ## 23, 24: Use the option
PlotPoints->200 (or
higher) in Plot3D.
This will make your graphics run slow, but will show
what is going on near the discontinuity point.
HW
# 13
Assigned: 10/02
(All problems except
those marked in bold
red
must
be done via WebAssign.)
Sec. 14.3: ## TF1, 1, 3,
5, 6,
7, 8, 10, 82, 15, 17, 18,
20, 33, 32, 41, 53, 55,
60,
61, TF2,
TF3, 97,
73,
63, 65.
Review Exercises at the end of Chap. 14:
# 11(a,c).
Note: TF1 etc. refers to True/False questions at
the end of Chap. 14.
Note for ## 5-8, 10: Interpret PDs as
slopes, or, equivalently,
rise-over-runs.
In ## 7, 8, where you have to
find second derivatives, interpret
them as rate of change in direction "A" of a slope in direction "B",
where "A" and "B" can be x
and y in any
combination.
Note for ## 10, 73: In 10 and 73(a,b), round the
answer to 1
decimal
place.
In 73(c), round to 2 decimal
places.
Note for #
10:
You will need a ruler to estimate the "run"
in the "rise over run" formula for the slope.
Answer for # 18:
(3/2)/Sqrt[3x+4t], 2/Sqrt[3x+4t]
Note for ## 8, 73, and # 11 on
p. 982:
Recall that f_xy = (f_x)_y. The f_x here is the slope in the
x-direction,
and then (f_x)_y is the rate of change of this slope along the
y-direction.
Thus, you need to find how the slope f_x changes when you change y.
HW
# 14
Assigned: 10/
(All problems except
those marked in bold
red
must
be done via WebAssign.)
Sec. 14.4: ## 4, 5, 6,
7,
11, 12, 13,
18 (this number in the book is the closest match to # 3 on the WA
list),
19, 20,
21, 38, 22, 25, 27, 30,
31, 33, 35, 34,
40 (I suggest that you do this problem after the Word Problems
below), TF6.
Note: TF6 refers to the
True/False
question 6 on p.
982.
Answer for
# 4: z = -1 -(1/4) (x+4) + 1*(y-2).
Note for ## 7, 20: Use Mathematica's
command Plot3D for plotting.
Answer for # 12: f_x = Sqrt[y/x]/2,
f_y=Sqrt[x/y]/2 are continuous
at (1,4); hence f is differentiable at that point.
L = 2 + (x-1) + (1/4)(y-4)
Note
for # 20: In deviation from
the "'near' versus 'at'" rule which I stated in class,
here `at' indicates the point (x0,y0).
Answer for # 20:
L = 2
+ (x-1) + (y-1); then use x=1.02, y=0.97.
Note for # 22:
Note that "delta t" between two adjacent columns changes
from
one pair of columns to another somewhere in the middle of the table.
This will affect your estimate for the partial derivative with respect
to t,
which WA wants you to find as the average of the left
and right quotients.
Note for # 27:
In addition to the book's assignment, also find
the
relative change
of the function,
using the formula
for f=x^m*y^n derived in class.
Answer:
5*(dp/p) + 3*(dq/q).
Answer for # 30:
Denote EE =
E^(-y^2-z^2). Then:
df = z*EE*dx -
2xyz*EE*dy + (x-2xz^2)*EE*dz.
Word Problem 1
Suppose that certain measured quantities, x and y, have errors of at
most
r % and s %, respectively. Use differentials to approximate the
maximum percentage error in each of the following:
(a) xy, (b) x/y, (c) 7 x^2 y^3, (d)
x/Sqrt[y].
Answers: (a) (r+s) %, (b)
(r+s) %, (c)
(2r+3s) %, (d) (r+(s/2)) %.
Word Problem 2
The period T of a pendulum undergoing small oscillations is
related
to its length L
and specific
gravity g by the formula:
T = 2*Pi *Sqrt[L/g].
(a) Use differentials to approximate the percentage change in
the
period if
the length of a pendulum is decreased by 4%, and the pendulum
is
raised
very high above the ground, which
decreases g
by 1%.
(b) Use differentials to approximate the
maximum percentage
error in
calculating the period if the length and specific gravity are measured
with
percentage errors of at most 4% and 1%, respectively.
Answers: (a) -1.5%, (b) 2.5%.
Word Problem 3
The length and width of a rectangle are measured with errors of at most
3% and 5%, respectively.
(a) Use differentials to approximate the maximum percentage
error
in the
calculated area.
(b) Is the information given sifficient to determine the
maximum
percentage
error in the perimeter of this rectangle? If 'yes', state the
corresponding
maximum percentage error.
Answers: (a) 8%, (b)
No.
Word Problem 4
Four positive numbers, each between 2 and 20, are rounded to the first
decimal place and then multiplied together. Use differentials to
estimate the following:
a) The maximum possible absolute
error in the computed product that may
result from the rounding;
b) The maximum possible percentage
error of the same.
Hints for (a) and (b):
1) The maximum error introduced by rounding to one decimal place
is 0.05.
2) Use straightforward generalizations of the formula
for f=x^m*y^n derived in class.
Answers: (a) 4*0.05*20^3, (b)
4*(0.025) = 10%.
As you may notice, |max abs error| / (max product) is not 10%.
HW # 15
Assigned: 10/
(All problems except
those marked in bold
red
must
be done via WebAssign.)
Sec. 14.5: ## 1, 3, 5, 13,
7, 9, 10,
14, 45(a),
17,
19,
24,
23, 27, 31, 33, 49,
47,
35, 39, 40, 41, 43.
Review Exercises at the end of Chap. 14:
# 40.
Word Problems:
1. Find df/dt, where f(t,x,y) = x*t +
exp(y)/t, and x=sin(t), y=t^2.
2. Find df/dt, where f(t,x,y) =
x*exp(x*y/t), and x=sin(t), y=t^3.
Answer for # 24: P_x = (u^2/x + v^2 + y*w^2)/S,
P_y =
(u^2 + v^2/y + x*w^2)/S,
where S = Sqrt[u^2 + v^2 + w^2].
Hint for # 43 and # 40
(p. 983): Use the formula for the area of triangle from Sec.
12.4;
see the Hint for # 29(b) in HW
# 4 (Sec. 12.4).
Answer for # 40 (p. 983): 75 - 40 +
2000*Sqrt[3]/2*0.05.
Notes for # 49: 1) Do this problem only for
z=f(x+at) to save time;
for the other piece it is similar.
2) Do this problem before
47 (as indicated).
Note for # 47: Do this problem only
for
z = (1/x)*f(x-y) to save time;
for the other piece it is similar.
Answer for Word
Problems: 1. (x -
exp(y)/t^2) + t*cos(t) + 2*exp(y).
2. ( -x^2*y/t^2 + (1 +
y/t)*cos(t) +3*x^2*t ) *
exp(x*y/t)
Extra
credit # 8 (value
of each problem (in %
added to your
final grade) is marked next to it)
Assigned: 10/08
Due:
10/15
Note: Before
you attempt this
extra-credit assignment, please read these
instructions!
Sec. 14.5: # 52(b & c) (0.1%
for both parts, not for
each part), 53 (0.2%).
Note for # 53:
This formula is called "the Laplacian in polar coordinates".
It plays a central role in
many Physics and Engineering courses.
The easiest way to
verify this formula is to compute its r.h.s.
and show that it equals the
l.h.s..
HW
# 16
Assigned: 10/09
(All problems except
those marked in bold
red
must
be done via WebAssign.)
Concept Check at the end of Chap. 14: # 14;
T/F Quiz at the end of Chap. 14:
# 9;
Sec. 14.6: ## 5, 7, 9, 11,
13, 15, 17, 19,
21, 22,
23,
24, 25, 28,
29, 30, 34,
41, 42, 45, 54, 55.
Review Exercises at the end of Chap. 14:
# 11(b).
Answer for # 22:
The direction is that of <4,1>; the max rate of
change is Sqrt[17].
Note for # 29: The condition you need to impose is
slightly
more general than
grad f = <1, 1>.
Answer for # 28: <cosT, sinT>,
where 5cosT +
6sinT = 2;
you don't need to go any further.
Answer for # 30: Directional derivative =
(-4*0.04*80
+ 3*0.003*60^2)/5 > 0,
hence the depth increases.
Extra
credit # 9 (value
of each problem (in %
added to your
final grade) is marked next to it)
Assigned: 10/09
Due:
10/16
Note: Before
you attempt this
extra-credit assignment, please read these
instructions!
Sec. 14.6: ## 51 (0.1%), 53 (0.1%), 58 (0.15%), 60
(0.2%).
HW
# 17
Assigned: 10/18
(All problems except
those marked in bold
red
must
be done via WebAssign.)
T/F Quiz at the end of Chap. 14:
# 7;
Sec. 14.7: ## 1, 3,
5, 9,
Review
Exercises at the end of Chap. 14 (p. 984):
## 52, 53 (these numbers in the book are the closest match
to ## 3, 4 on the WA list),
Sec. 14.7: ## 21, 41, 43, 49,
52, 33, 34, 35, 36, 37;
Word Problem:
Find the absolute extrema of f(x,y) = x*y - x
inside
the closed disk x^2 + y^2 <= 1.
Suggestion for all problems
involving the Second Derivative Test:
Use Mathematica to
compute derivatives and simplify the expression,
as in Lab 5.
Hint for # 21: To answer the last question ("Then
show..."), complete the square.
(If you forgot how to complete a square, google it.)
Note 1 for ## 33 - 37:
Do NOT mimic the treatments of the extrema
on the boundaries
as found in WA's solutions.
For rectangular regions, this treatment is very sketchy and will not
help you
do similar problems on a Quiz or Test.
For circular regions, WA incorrectly suggests that you use
Cartesian coordinates (x,y) instead of the proper, parametric
coordinates,
as shown in a Class Example. Again, following WA's approach on a
Quiz or Test will likely put you in trouble.
Use the approach of the Class Example.
Note 2 for ## 33--37 and Word Problem:
Study and USE the Mathematica notebook posted next to the
Lecture Notes for Sec. 14.7.
It is there for a very good reason...
For the HW problems, you should also plot the surfaces,
as shown in that notebook, and then verify whether your
answer makes sense by comparing them with those plots.
Hint for # 37 and Word Problem:
When finding critical points on
the boundary, you will arrive
at an equation that has a mixture of sin and cos.
You need to exprerss one of them via the other
(e.g., cos^2 = 1-sin^2), obtain a quadratic equation
for the remaining function (which would be sin(t) in the above example),
denote sin(t)=u (or cos(t)=u), and solve the quadratic equation
for u.
Answer for Word
Problem: Abs. min. at (Sqrt[3]/2, 1/2): f =
-3Sqrt[3]/4;
Abs. max at (
-Sqrt[3]/2, 1/2): f = 3Sqrt[3]/4.
Extra
credit # 10 (this problem is worth 0.25% added
to your
final grade)
Assigned: 10/18
Due: 10/25
Note: Before
you attempt this
extra-credit assignment, please read these
instructions!
Sec. 14.7: # 60.
Section
14.8
(Lagrange multipliers) will be covered in Lab 6.
HW
# 18-A
Assigned: 10/19
(All problems except
those marked in bold
red
must
be done via WebAssign.)
Sec. 15.1: ## 9, 11.
HW
# 18-B
Assigned: 10/23
(All problems except
those marked in bold
red
must
be done via WebAssign.)
Here you continue to work on problems from Sec. 15.1
(so , do NOT do
problems from Sec. 15.2 in this HW).
Sec. 15.1: ## 15, 17, 21, 25, 24, 23, 29, 32, 33, 35, 37,
39, 43, 44.
Hint for # 25:
Using one integration order is much eaither than using the other.
Hint for
# 23: Use the method of Ex. 1 in Sec. 7.2.
Hint for ## 32, 33: Decide which is the more
convenient
variable to integrate over first.
Note for # 44: You do not need to sketch the surfaces, but
you do need to determine
which of the two is higher. (They do not cross within the given
rectangle.)
Answer for # 44: 8*(5-Ln(5)).
Extra
credit # 11 (this problem is worth 0.15% added
to your
final grade)
Assigned: 10/23
Due: 10/30
Note: Before
you attempt this
extra-credit assignment, please read these
instructions!
Sec. 15.1: # 34.
Hint: You need to use integration by parts at some
point.
HW
# 19
Assigned: 10/27
(All problems except
those marked in bold
red
must
be done via WebAssign.)
Sec. 15.2: ## TF2, 5, 7, 46,
48, 49, 51, 53, 54,
16, 14, 21,
TF7, 39, 66,
27, 28, 29, TF4, 69.
Word
Problem:
Express the given integral as
an equivalent integral with the order of integration reversed:
Integrate[ Integrate[f[x,y],
{y, 3x, Sqrt[x]}], {x,0, 1/2} ].
(If you are unsure what the
above syntax means, see "Help" for command "Integrate" in Mathematica.)
General Note 1: You can always check your answer
with Mathematica.
Example:
Integrate[x y, {x, 0, 1}, {y, 0, x}].
General Note 2: This concerns sketching
curves y = x^2 and y = Sqrt[x].
The parabola y = x^2 has the zero slope at x=0.
This implies that in the first quadrant between x=0 and their
intersection point,
this parabola passes below any straight line y=kx, k>0,
no matter how small k is (as long as k>0).
Similarly, the sideways parabola
y = Sqrt[x] has the infinite
slope at x=0.
This implies that in the first quadrant
between x=0 and their intersection point,
this parabola passes above
any straight line y=kx, k>0,
no matter how large k is.
Note: TF2 refers
to the True/False
question 2 at the end of Chap. 15 (p. 1061),
etc.
Answer for # 46: Limits over x are 0 to
Sqrt[y]; limits
over y are from 0 to 4.
Answer for # 48: Limits over
y are -Sqrt[4-x^2] to Sqrt[4-x^2]; limits over x are from 0 to 2.
Answer for # 16: (E^16 - 17)/2.
Answer for # 14: 3^5/8; Type I limits
are: 0<= x<=3, x^2 <=
y<= 3x;
Type II limits are: y/3
<=x<=Sqrt[y], 0 <=y<= 9.
Hint for # 39:
Sketching planes with equation z = a*x+b*y+c was
considered
in Ex. 1(a) (Notes) and Ex. 5 (book) for Sec. 14.1;
a related example is Ex. 5 (book) for Sec. 12.5.
Hint
for # 66: Recognize the integral as the volume of a familiar
object.
Answer
for # 66: (2*Pi/3)*R^3.
Hint for ## 69: The integrand has three
terms. To evaluate each of the resulting
integrals WITHOUT actually computing them, use
symmetry
on
the first two terms and geometry (as I showed in class) on the third
term.
Answer for Word
Problem: Integrate[
Integrate[f[x,y], {x,
y^2, y/3}], {y,0, 3/2} ]
HW
# 20
Assigned: 10/28
(All problems except
those marked in bold
red
must
be done via WebAssign.)
Sec. 15.3: ## 1, 2, 5, 6, 8,
11, 14, 17, 24, 23,
25, 26, 29,
31, 32,
35, 40.
Hint for #
6: Review
Ex. 3 in
Sec. 7.2.
Answer for # 6:
Pi/2. (The region is the left half of the circle
x^2 + (y-1)^2 = 1.)
Note for #
17: To determine the limits of integration over
theta, you need to find those
values of theta where the two circles intersect. To do so, you need to
solve f1(theta) = f2(theta), where r = f1(theta) and
r = f2(theta)
are the polar equations of the two circles.
Hint for # 40(a): See # 11 above.
Note for # 40(c,d): This integral is called the Poisson
integral and arises in many areas
of mathematics, not just in probability and statistics.
HW
# 21
Assigned: 11/02
(All problems except
those marked in bold
red
must
be done via WebAssign.)
Sec. 15.4: ## 3, 8, 16, 27, 28(a,b),
29(a,b);
Review Exercises at the end of Chap. 15 (p. 1063):
# 51.
Answers for #
8: M = 63k/20
xCoM = k*Integrate[x^3*(x+2-x^2), {x,-1,2}] / M;
yCoM = k*Integrate[x^2*((x+2)^2
- x^4)/2, {x,-1,2}] / M
Answer for 28(b): (i) 3/4; (ii)
3/16.
Hints for #
16: 1) Use polar coordinates. Review Exs. 1
and 2 in the Notes for Sec. 15.3.
2) For one
of the integrals you will need to review
Ex. 1 in
Sec. 7.2.
Extra
credit # 12 (each part of this problem is worth 0.25%
added
to your
final grade)
Assigned: 11/02
Due: 11/09
Note: Before
you attempt this
extra-credit assignment, please read these
instructions!
Sec. 15.4: # 30.
Note: See the formula
before Example 7 for the exponential
probability density function with a given mean value.
HW
# 22
Assigned: 11/04
(All problems except
those marked in bold
red
must
be done via WebAssign.)
Sec. 15.6: ## 4,
9, 12, 13,
16, 14, 18, 23(a),
27,
28, 29, 30, 31,
34, 35.
Hint for # 12: When you come to the integration
over
x, you need to do
integration by parts. A
small simplification will occur if you
also use
a trig identity cos(Pi-x)=-cos(x).
Answer for # 12: (Pi^2)/2 -
2.
Hint for # 16: To obtain the equation of the plane
making one of the faces of the
tetrahedron, use the method of Ex. 5 of Sec. 12.5 in the textbook.
Answer for # 28: It's a solid bounded at
bottom and top by z=0 and z=2-y and
on the sides by planes x=0, y=0, and cylinder x=4-y^2.
Its projections on the coordinate planes are:
{xy-plane: 0 < x < 4-y^2,
0<y<2},
{xz-plane: 0<x<4,
0<z<2},
{yz-plane: 0<z<2-y, 0<y<2}.
Answer for # 30:
First integration over
z: 0 < z < sqrt(9-y^2),
-2 < x < 2, -3 < y <
3; or -3
< y < 3, -2 < x
< 2;
First integration over x: -2
< x < 2,
0<z<sqrt(9-y^2),
-3<y<3; or -sqrt(9-z^2)<y<sqrt(9-z^2),
0<z<3;
First integration over y: -sqrt(9-z^2)<y<sqrt(9-z^2),
-2 < x < 2, 0 < z
< 3,
or 0 < z < 3, -2 <
x < 2.
HW
# 23
Assigned: 11/06
(All problems except
those marked in bold
red
must
be done via WebAssign.)
Sec. 15.7: ## 1,
4, 5,
6, 7, 10,
11, 12, 15,
16, 19, 20, 21, 22, 23, 24, 29.
Note for ## 19, 20, as well as in
general:
The point of this section's assignment is to teach
you to set up
integrals in cylindrical coordinates, not to evaluate integrals.
Therefore, you SHOULD use
Mathematica to evaluate
integrals that look complicated.
For example, for # 20, I recommend that you integrate
first over z and Theta, and only then over r, with the latter integration
being done by Mathematica. When integrating over Theta, note
the identity: Integrate[Sin[t],{t,0,2Pi}]=0;
the same result holds if Sin is replaced with Cos or Sin[t]*Cos[t]
(note, however, that this won't be true if the limits are different).
To further simplify your work on integration, you may
look up the syntax for double integrals under Help for "Integrate".
I
recommend that you follow the general guidelines of this Note
for all subsequent assignments, unless otherwise
noted.
Answer for # 16: This solid is inside the cylinder
r=2, above the xy-plane,
and below the cone
z=r. Its volume is
(16/3)*Pi.
Hints for # 24: 1. When
determining the boundary of region R in the xy-plane,
follow the steps of Ex. 1 in the Notes for sec. 15.3.
2. In this way, you'll
arrive at an equation for r that looks like this:
a*r^4 + b*r^2 + c = 0,
where a, b, c are some constants. To solve it, denote r^2=u
and solve the quadratic equation for u.
Note for # 29: This problem's version
on WA is more complicated
than the book's version.
See the Note for this on WA.
When working on this problem in preparation for a test, use the book's
version
rather than WA's version.
HW
# 24
Assigned: 11/09
(All problems except
those marked in bold
red
must
be done via WebAssign.)
Sec. 15.8: ## 1,
3, 4,
5, 13, 12,
15,
17, 20, 24,
25, 26, 27,
30, 41.
Hint for # 3(b): Review
Ex. 1(c,d) for Sec. 10.3.
Answer for # 4: (a) (2, theta=0, phi=Pi/6);
(b) (4,
theta=11Pi/6, phi=Pi/6).
Answer for # 12: The part of the space between the
spheres
centered at the origin and
with radii 1 and 2, below the z=0 plane.
Hint for # 15: Look at Ex. 4 in
Sec.
15.8;
a similar problem, albeit in a
different coordinate system,
was done in
Ex. 1 in the Notes for Sec. 15.3.
Note for # 17: This is the closest match
to WA's # 6.
When preparing for a test, you need to use the WA's problem,
not # 17 from the textbook.
Extra
credit # 13 (this problem is worth 0.2%
added
to your
final grade)
Assigned: 11/09
Due: 11/16
Note: Before
you attempt this
extra-credit assignment, please read these
instructions!
Sec. 15.8: # 46.
Hint: Essentially, you need to find the angle between two
vectors,
whose terminal points are the indicated cities. You did such problems
in Chap. 12.
HW
# 25
Assigned: 11/13
(All problems except
those marked in bold
red
must
be done via WebAssign.)
Sec. 15.9: ## 1,
2, 6, 11,
15, 16, 23, 24, 13, 14,
10,
21(a).
Review Exercises at
the end of Chap. 15 (p.
1063):
# 55.
General Note about
computing
integrals:
In this and the next section, the primary focus is not on
computing double or
triple integrals, but on setting up integration regions.
Therefore, you may reduce the amount of work for yourself
if you use Mathematica to do the integrations.
Look up the syntax for multiple (e.g., double) integration in the Help
menu
for the command Integrate. (This has already been noted in Sec.
15.8's assignment.)
Note
for ## 11, 15, 16, 23, 24, 13, 14, 55: These
are
problems similar to Ex. 2 in the Notes.
Hint for # 13: Here (u,v) are polar coordinates.
Hint for # 14: For hyperbolas, think what combination of x
and y is a
constant along each of the hyperbolas. This will be your u.
Similarly, for the straight lines, think what combination of
x and y is a
constant along each of the straight lines. This will be your v.
Answer for # 14:
u=xy, v=y/x; 1 <= u <=4, 1
<= v <= 4;
Inverse transformation is: x=Sqrt[u/v],
y=Sqrt[uv]
(the positive signs are chosen because x,y >
0).
Note for # 10: Find the answer in one of the class
examples.
Extra
credit # 14 (this problem is worth 0.15% added
to your
final grade)
Assigned: 11/13
Due:
11/25
Note: Before
you attempt this
extra-credit assignment, please read these
instructions!
Sec. 15.9: # 22.
HW
# 26
Assigned: 11/13
(All problems except
those marked in bold
red
must
be done via WebAssign.)
Sec. 16.6: ## 33,
39, 40, 48;
3, 6, 5, 4, 19, 23, 24, 59, 58(a), 58(b), 60(a).
See the general Note for
Sec. 15.9
about computing integrals.
Use the same approach when
computing partial derivatives and/or
cross-product of vector functions. (Define the functions first using
correct Mathematica syntax, then compute partial derivatives, etc.)
Also, you may visualize parametric
surfaces using Mathematica. E.g., try:
ParametricPlot3D[{(3 + Cos[u])*Sin[v], (3 + Cos[u])*Cos[v],
Sin[u] + v}, {u, 0, Pi}, {v, 0, 3.5*Pi}]
See other examples in the Help browser for the command ParametricPlot3D.
Answer
for # 40: Sqrt[22]*2*2.
Note for ## 6, 5, 4: Review Examples 1, 2, 3(a), and 5 in the
Notes before doing these problems.
Note for ## 23,
24:
Use the method described
in the notes,
NOT the method presented
in Examples 2
and 4--7 of the book.
Namely, you need to identify what curve, lying in which plane,
needs to be rotated about which axis to produce the specified surface.
Note for # 23:
See Ex. 2 in the Notes for Sec. 15.8 and Ex. 2 in the Notes for
16.6.
Answer for # 23: x =
2*sin(u)*cos(v), y = 2*sin(u)*sin(v),
z = 2*sin(u);
0 <= u <= Pi/4, 0 <= v
<= 2*Pi.
Answer for # 24: x =
3*cos(u), y = v,
z = 3*sin(u);
0 <= u <= Pi, -4 <= v
<= 4.
Note for # 58(b): Do not
set up the
integral for the
surface in (x,y)-coordinates;
only eliminate the parameters to show that the surface is a paraboloid.
Answer for # 58(b): z = (x/a)^2 + (y/b)^2
Answer for # 60(a): (x/a)^2 +
(y/b)^2 -
(z/c)^2 = 1
Extra
credit # 15 (this problem is worth 0.15% added
to your
final grade)
Assigned: 11/13
Due:
11/20
Note: Before
you attempt this
extra-credit assignment, please read these
instructions!
Sec. 16.6: # 32. Credit will be
given only if you
present a coherent answer to the last question
of this problem. Feel free to look for an answer online or in Sec. 16.7.
Extra-credit Mathematical
Lab (this Lab is worth 2% added
to your
final grade)
Dumbbells, Squashed spheres,
Hearts, and Surfaces with petals:
From revolution to “generalized revolution”
- The notebook
mentioned in the Lab, which is intended to help you do this Lab:
EC_SurfOfRevol_ParamCurves.nb
- Note: Even if you do not
intend to do this extra-credit Lab, feel free to browse through it for
cool
images of surfaces of revolution (and beyond!).
Assigned: 11/13
Due:
12/06
HW
# 27
Assigned: 11/15
(All problems except
those marked in bold
red
must
be done via WebAssign.)
Sec. 16.1: ## 1, 6, 7, 23,
15 -- 18 (these problems in the book are the closest match to WA's ##
7,8
and are listed here only as a reference; you should do WA's problems
7,8),
27,
33, 34.
Note for # 6: Have
you read the "Must-Read" Example in the textbook?
Note for # 27: The
Mathematica command
to plot a
vector field is VectorPlot.
You learned the command for contour plots in Lab 4.
You learned the command to plot two figures together in Lab 1
and then used it in all Labs except Lab 2.
The emphasis of this problem is on visualizing the relation between
the gradient and level curves, which you learned in Sec. 14.6.
Hint for ## 33, 34: Recall from Sec. 13.4 that
velocity is the derivative of the
location of a particle: dr/dt
= v. If dt is
small but still finite,
one can write it as r(t+dt)
- r(t) = v dt. Then r(t+dt) can be found
when the other ingredients of the formula have been given.
HW
# 28
Assigned: 11/18
(All problems except
those marked in bold
red
must
be done via WebAssign.)
p. 1148 of the book, T/F
Quiz: 1, 2, 3, 7,
9, 10,
12.
Sec. 16.5: ## 1, 3, 5,
9,
10, 11, 12, 19, 20,
21,
22, 31(a-c).
Note for
all problems where tedious calculation of partial derivatives is
required:
To compute
partial
derivatives of complicated functions,
use Mathematica's command D (which you used in
Labs 4 and
5).
Moreover, you may compute Div and Curl using these commands
in Mathematica.
Note, however, that this should be used only to check your answers,
not to obtain them. Mathematica will not help
you do similar
calculations on a quiz or test.
Note about # 5: This is a placeholder for WA's # 10.
They require similar calculations,
but the WA's problem has a physical meaning (see the Note for # 31)
while the # 5 from the 8th edition does not.
So, just do WA's # 10 and don't worry abou this placeholder
from the textbook.
Note about # 31: These identities are very important in the
theory of Electromagnetism
and in Fluid dynamics.
Extra
credit # 16 (value
of each problem (in %
added to your
final grade) is marked next to it)
Assigned: 11/18
Due:
12/02
Note: Before
you attempt this
extra-credit assignment, please read these
instructions!
Sec. 16.5: ## 26
(0.15%), 29 (0.1%).
Note for # 29: No credit
will be given for a solution similar to the one
given by WA online tutor.
Instead, to receive credit, you must solve this problem
by interpreting curl F
as (nabla x F), where x denotes the
cross-product, and then using properties of the cross-product.
In particular, you should figure out how to use the formula
for a x b x c
in this case.
HW
# 29
Assigned: 11/20
(All problems except
those marked in bold
red
must
be done via WebAssign.)
Sec. 16.2: ## 1, 3, 7, 10,
15, 17, 20,
21,
41, 45, 47(a),
34, 36.
p. 1148 of book, T/F Quiz:
6.
General
Note: Read the
Observation on p. 30-5 of the Notes for
Sec. 16.2 about a shortcut of doing some of the integrals.
This Observation applies to the integrals (or parts thereof)
in ## 7, 21, 41.
Note for ## 10, 41: You
need to use the
parametric equations of
the segment
connecting two given points.
See notes and/or
book for Sec. 12.5.
Note for ## 34, 36: Do only the part about finding
the mass
(i.e.,
you do not
need to find the center
of mass).
HW
# 30
Assigned: 11/
(All problems except
those marked in bold
red
must
be done via WebAssign.)
p. 1148 of the book, T/F Quiz: ## 4, 8;
Sec. 16.3: ## 21, 22,
25, 3,
4, 7, 13, 14,
15,
11, 19, 35.
Note for # 25: What do we know about the curl
of a conservative field?
Also, review
## 9(b), 10(b),
11(b) in Sec. 16.5.
Answer for # 22: The work depends on the path,
hence F is
not conservative.
Note for # 14: You'll need to use integration
by parts:
u = x, dv = y*exp[xy]*dx, etc.
Answer for # 14: (a) f = x*exp[xy] + K0,
(b) f(0,2) - f(1,0) = -1.
HW
# 31
Assigned: 12/
(All problems except
those marked in bold
red
must
be done via WebAssign.)
Sec. 16.4: ## 1, 2, 3, 5, 6, 11,
13, 17, 27;
Review Exercises at the end of Chap. 16 (p. 1150 of the book): #
38.
Notes for # 27:
(i) Use Mathematica to evaluate partial derivatives.
(ii) At a final stage of the
calculation, you will need to do a
trigonometric integral. See
Example 1 in Sec. 7.2.
Note for # 38: See # 35 for Sec. 16.3 and Example
5 in the book for
Sec. 16.4.
Also, as a technical convenience, use the result of # 21 on p. 1149.
(That problem was not assigned but
was considered in class
at the
end of Sec. 16.2.)
Answer for # 38: 4*Pi.
Extra
credit # 17 (value
of each problem (in %
added to your
final grade) is marked next to it)
Assigned: 12/
Due:
12/
Note: Before
you attempt this
extra-credit assignment, please read these
instructions!
Sec. 16.4: ## 21 + verify 21(c) by direct calculation
from the sketch (0.15%),
22
(0.2%), 31 (0.15%).
Note for ## 21 and
31: These problems are worked out in WebAssign. You
are
welcome to
watch their
video. However, your
solution will receive credit
only if all
steps
of the derivation, including those that could have
been skipped in the video, are present. That is, if you
skip any explanation
that I deem essential, I will not give any credit
to your solution.
So, if you are unsure whether you need to include a
particular detail
in your solution - just
include it (or ask me).
HW
# 32
Assigned: 12/
(All problems except
those marked in bold
red
must
be done via WebAssign.)
Sec. 16.7: ## 10, 6, 14, 13, 24, 27, 25, 40,
43.
General Note 1:
The focus of this HW is on setting up the double integrals,
not
on computing them.
Therefore, once you have set up a double integral,
evaluate it using Mathematica. This will save you time and frustration
of
dealing with minor arithmetic errors.
General
Note 2:
Review
parametrization of a cylinder, sphere, and cone
from Notes for Sec. 16.6. The parametric equation of a paraboloid,
which
you will also need in a one problem, is very similar to that of
the cone.
For example, for z=x^2+y^2, the parametric
equations are:
x = u*cos(v), y = u*sin(v), z = u^2.
Note
for all problems involving a cone, a cylinder, or a
sphere:
Do NOT
use the Cartesian form of the surface!
Instead, use its
parametric representation in suitable
cylindrical (or spherical -- for a sphere) coordinates.
It is this parametric
form that
you will need to use
in a corresponding problem on the Final Exam.
Answer for # 10: 3*Integrate[4x-2x^2-2x*y,
{y,0,2-x},{x,0,2}]
Answer for # 6:
Integrate[u^3*Sin[v]*Cos[v]*Sqrt[2]*u,{v,0,Pi/2},{u,0,1}]
Note for # 14: For a trig integral that
you
will need to compute, see
Ex.
3 in
Sec. 7.2.
Note for # 13:
Decide which pair out of (x,y,z) is convenient to replace by
a pair (u,v). See, e.g., Ex. 3(a) in Notes for Sec. 16.6.
Note for # 25: Use the
parametrization of a sphere.
Note for ## 43: Parametrize the cylinder
similarly to
Ex. 3 in the textbook (Sec. 16.7)
or Ex. 1 in the Notes for Sec. 16.6. However, make sure to pick
the correct variable (x, y,
or z) as the cylinder's axis.
Answer for # 24: -
Integrate[(r+r^3)*r,{r,1,3},{theta,0,2*Pi}]
HW
# 33
Assigned: 12/
(All problems except
those marked in bold
red
must
be done via WebAssign.)
Sec. 16.9: ## 5, 7,
11, 12, 13,
19.
General
Note 1:
Use appropriate coordinate systems when setting
up the
triple integrals.
General Note 2:
The focus of this HW is on setting up triple integrals,
not
on computing them.
Therefore, once you have set up a triple
integral,
evaluate it using Mathematica. You may use Mathematica to find
partial derivatives, too.
Hint for # 19: Think whether the field flows mostly in or out of the given point.
Relate this with the meaning of the Divergence theorem.
Good
luck in your finals,
and
Have
a safe and enjoyable * winter * break* !