As stated in the
Syllabus:
HW
for a section is assigned on the day when we have finished covering
that section in class.
HW # 1
(All problems except
those marked in bold
red
must
be done via WebAssign (WA).
There are no non-WA problems in this HW, but there will be one later
on, starting with HW 3.)
Sec. 12.1: ## 3, 14,
17, 25, 27, 31.
HW
# 2
(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, 39, 45(only a-c in book), 47.
Note for # 33: The angle that a vector <a,b> makes with the positive x-axis is:
arctan(b/a) if a >0 and arctan(b/a) +
Pi if a < 0.
Notes for # 39: - Notation "in the direction N 45o W" means the direction
that is 45o West of the Northerly direction. That is, the pilot is steering the plane at
135o to the positive x-axis. Similarly, the notation "in the S 30o E direction"
means the direction that is 30o East of the Southerly direction.
That is, the wind blows in the direction that is minus 60o from the positive x-axis.
- "Ground speed" is the speed relative to
the ground.
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
(All problems except
those marked in bold
red
must
be done via WebAssign.)
T/F Quiz at the end of Chap. 12 (p. 884):
## 1, 3,
19, 8;
Note: These are
problems from the True/False
(T/F) Quiz, NOT
from Review Exercises on the same page. Here is an easy way to tell that you are doing
a problem from a wrong part, e.g., from Review Exercises intsead of T/F
quiz, 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 about the cross-product, which 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 which section that material is in) 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. 884):
# 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: 08/
Due:
09/
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
(All problems except
those marked in bold
red
must
be done via WebAssign.)
T/F Quiz at the end of Chap. 12 (p. 884):
## 4, 6,
7, 9, 13, 14, 20, 21 (see
the Note for HW 3);
Sec. 12.4: ## 13, 3, 16, 29,
19, 22,
38, 10, 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 ##
10,11: 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 ##
10,11: 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
(All problems except
those marked in bold
red
must
be done via WebAssign.)
T/F Quiz at the end of Chap. 12 (p. 884):
# 16;
Sec. 12.5: ## 2, 6, 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/
Due:
09/
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
(All problems except
those marked in bold
red
must
be done via WebAssign.)
Sec. 12.5: ## 24, 29, 33, 63, 45, 53,
55,
71, 73, 74, 1.
Note for ## 73, 74: You must use the formula presented in class
(which is the same
as Eq. 9 on p. 871 of the textbook). You are not allowed to use the
formula shown in # 75, because it was not presented in class and
hence is considered an "outside" material.
Extra credit # 3
Assigned: 09/
Due:
09/
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
(i.e., 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
(All problems except
those marked in bold
red
must
be done via WebAssign.)
T/F Quiz at the end of Chap. 12 (p. 884):
# 18
(see the Note for HW 3);
Sec. 12.6: ## 1, 2, 4,
5, 6, 7, 43, 15 (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 (if unsure, look at the Table of Surfaces
in Sec. 12.6). It is now a time to get
hands-on experience with a
cone and its equation.
To that
end, do the following.
# 43:
1) Solve the given equation
for z (you should obtain two
answers that differ by +/-).
2) Plot each of these surfaces 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}, 4*x^2 + y^2 <= 10]
(it is explained
in examples for Plot3D).
* The number "10" above can be changed to any other number
depending on
what range of x,y-values
you want to show in your plot.
* The coefficients
in front of x^2
and y^2 (i.e., 4 and 1)
match those of the
surface in question.
3) Notice which coordinate axis is the axis of this cone.
4) 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.
# 15:
Sketch this surface by hand,
based on your experience with # 43. (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 # 43:
(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: Lower half (since z<0) of the parabolic
cylinder obtained by extending the
half-parabola z=-Sqrt[x] in the xz-plane along the y-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.
Answer for # 43(4): The cone's base along x is twice as large as that along y.
Answer for # 15(i): The cone's axis is x.
To
verify your sketches
for the Word Problems, solve the equations for z
and then use the Plot3D
command in Mathematica.
HW
# 7
(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
(I recommend that you do it along
with ##
7, 19, and 25, 29 listed below);
Sec. 13.1: ## 1; 21; 8,
7, 19, 12,
14, 35,
25, 29, 39, 40, 57;
Review Exercises at the end of Chap. 13 (p. 928):
# 1(a) (along
with ## 7, 19, and 25, 29 above), 6(a).
Word Problems:
1) Show that the parametric curve x = t, y = 2t, z = Sqrt[3 -
5t^2] lies on a sphere.
What is the radius of this sphere?
2) Show that the parametric curve x = Sin[t], y = Sqrt[2]*Sin[t], z = Sqrt[3]*Cos[t] lies on a sphere.
What is the radius of this sphere?
3) Show that the parametric curve x = e2t Cos[t], y = e2t, z = e2t Sin[t] lies on a cone.
(See ## 43, 15 in HW 6 if you forgot what the equation of a cone is.)
(i) What is the axis of this cone?
(ii) As you remember from HW 6, any cone has two "halves."
Does the curve in this problem
lie on both halves of the cone or just one one half?
(iii) Use the above information to sketch this curve.
General note:
While this section is about vector functions, which, in general, are curved lines, some of those lines may be straight.
We learned about them in Sec. 12.5A. I expect you to review that section, as some of the HW problems are about
straight lines (even if they don't say that explicitly).
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 ## 12, 14: 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).
Note for ## 35, 25,
29: Recall the
parametric
equation of a certain curve from this Section
about which you needed to read on your own and
which you were required to memorize.
Note for ## 35, 25: Recall the equation
of a cone from HW 6. If you have not done
what was asked for ## 43, 15 there, go back and do it now.
Only then can you correctly do ## 35 and 25.
Answer for # 6(a) on p. 928: (15/8, 0, -ln(2)).
Answers for the Word Problems:
1) & 2) Sqrt[3].
3) (i) y; (ii) On the right half; (iii) Use Mathematica's
ParametricPlot3D.
HW
# 8
(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, 11, 14, 17, 19, 25,
27, 28,
35, 38,
39, 43;
Review Exercises at the end of Chap. 13 (p. 928): # 5.
Hint
for # 5 of Sec. 13.1: Use the method of Example 3 from Sec. 2.6
for the 1st
components;
for
the others, use the limiting values of the functions involved.
Note for ## 3,5 of Sec.
13.2: Find a (simple!) relation between x and y (by excluding t) and
thereby 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 ## 11, 14: Review the Chain Rule.
Note for # 11: The
expression e^t^n should be interpreted as e^(t^n).
Note for ## 25, 27: These problems give you the coordinates
of the point on the curve but,
unlike earlier problems, do not give you the value of t. To find
the latter,
equate one of the component of the curve to the given component of the
point.
Pick the easieest of the
expressions. E.g., for # 25, do:
6*Sqrt[t]=6 => t=1.
The remaining two components should automatically match (verify that
they do).
Answer for # 28: x=2+t/2, y=ln4 + t/2, z=1+t.
Note for # 39: Review formula 10 and Ex.
5 in Sec. 7.4
(don't pay attention to details of Partial Fraction Expansion,
but only note how the integrals are done at the very end).
Note for # 5 on p. 928:
Review Example 4
in Sec. 5.5.
Extra
credit # 4 (worth
0.2% added to your final
grade)
Assigned: 09/
Due: 09/
Note: Before
you attempt this
extra-credit assignment, please read these
instructions!
Sec. 13.2: # 30.
HW
# 9
(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. 928):
# 12.
Sec. 13.3: ## 3, 4, 5, 15, 17, 51, 52,
53, 57, 20, 24, 25,
37.
Note for # 4: Use identity
a^2+2ab+b^2 =
(a+b)^2 and then
Sqrt[(a+b)^2] = (a+b).
Note for # 5: Use the identity (x + 1/x)^2 = x^2 + 2 +
1/x^2, where your x is somehow related to t.
Note for # 51: Use Mathematica
to calculate T' and its
length.
When computing T', recall that T is a vector, and then recall from Lab 2
how one enters vectors in Mathematica. You may also do
Tsimple[t_] = Simplify[T'[t]].
To find the length, you can review Lab 2 as well and/or see the Hint for # 4.
The expression for the length (which is a scalar!) should be fairly
simple.
Finally, to find T'/|T'| at a specific value of t = t0, do this:
f[t_] = Tsimple'[t] / |Tsimple'[t]| (where you find the denominator as explained above)
and then type f[t0].
Note for # 52: Use the Chain Rule to find r' and then a trig
identity for tan^2 from
the middle of p. A28 (Appendix D) to find its length.
Note for ## 20(a), 24(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 # 24(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.
(This is the same idea as expressed in the Hints for ##4,5,51 above.)
If you use Mathematica to do
calculations for 24(a), you will see
that, for a reason unknown to me, Mathematica refuses to take
the square root of (e^x +
e^(-x))^2. You can just do so by hand
and proceed using Mathematica to do the rest of the calculations.
Note for ## 20(b), 24(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. 928: 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/
Due:
09/
Note: Before
you attempt this
extra-credit assignment, please read these
instructions!
Sec. 13.3: ## 77 (0.1%).
Note that it 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.
Make sure to express the answer in more familiar length units (i.e.,
in feet/meters/inches/centimeters as opposed to Angstroms).
HW
# 10
(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 ## 8,7 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. 927):
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 Theorem 4 and its Proof in the book for Sec. 13.2. The
conceptual
part of that Theorem 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/
Due:
10/
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 Theorem 4, 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.
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