Below is a tentative list of homework problems for Math 105. In general, HW is due
at the start of each class, and we will typically cover on the order of one
section a day. All problems are worth 10 points; I will drop whatever assignment
helps your HW average the most.
Homework solutions are available
here (do not look at a week's solution until you've done the problems)..
HOMEWORK: Homework problems listed below;
suggested problems collected together at the end.
- Due Monday, Feb 4:
Handout from first day of class is here.
- Read: Section 11.1, 11.2. Use that time to
read the material and make sure your Calc I/II is fresh. Feel free to check
out the review videos: part 1
part 2.
-
Video of the week: Fibonacci numbers (click
here for more on the Fibonacci numbers).
-
Slides on the course mechanics.
- HW problems:
(1) What is wrong with the following argument
(from Mathematical
Fallacies, Flaws, and Flimflam - by
Edward Barbeau): There is no point
on the parabola 16y = x2 closest
to (0,5). This is because the distance-squared from (0,5) to a point (x,y) on
the parabola is x2 +
(y-5)2. As 16y = x2 the distance-squared is f(y) = 16y +
(y-5)2. As df / dy = 2y+6, there is only one critical point, at y =
-3; however, there is no x such that (x,-3) is on the parabola. Thus there is
no shortest distance! (2) Compute the derivative of cos(sin(3x2 +
2x ln x)). Note that if you can do this derivative correctly,
your knowledge of derivatives should
be
fine for the course. (3) Let f(x)
= x2 + 8x + 16 and g(x)
= x2+2x-8. Compute the limits as x goes to 0, 3 and ∞
of f(x)+g(x), f(x)g(x) and f(x)/g(x).
-
Extra
Credit (due 2/8/13): Assume team A wins p percent of their games, and
team B wins q percent of their games. Which formula do you think does a good
job of predicting the probability that team A beats team B? Why? (a) (p+pq) /
(p+q+2pq), (b) (p+pq) / (p+q-2pq), (c) (p-pq) / (p+q+2pq), or (d)
(p-pq) / (p+q-2pq).
-
Due Wednesday, February 6:
-
Read: 11.2, 11.3,
11.4. NOTE: we will skip sections 11.5, 11.6, 11.7
-
HW
problems: Section 11.1: Page 823: #9, #18, #38, #42. Section
11.2: Page 833: #1, #39, and also find the cosine of the angle between a
= <2, 5, -4> and b = <1, -2, -3>.
- Due Friday, February 8:
- Read:
11.4, 11.8.
NOTE: we will skip sections 11.5, 11.6, 11.7
-
HW problems: Section 11.2: Question 1: The corollary on page 830 states two
vectors are perpendicular if and only if their dot product is zero. Find a
non-zero vector, say u, that is perpendicular to <1,1,1>. (Extra
credit: find another vector perpendicular to <1,1,1> and the vector u
that you just found.) Question 2: Consider a triangle with sides of length 4,
5 and 6. Which two sides surround the largest angle, and what is the cosine of
that angle? Section 11.3: Question 3: Find the determinant of the 2x2 matrix
; in other
words, we filled in the entries with the numbers 1, 2, 3 and 4 in that order,
row by row. Similarly, find the determinant of the 3x3 matrix
; in other words, we
fill in the numbers by 1, 2, 3, 4, 5, 6, 7, 8, 9.(Extra credit: find a nice
formula for the determinant of the n x n matrix where the entries are 1, 2,
..., n2 filled as above, and prove your claim.) Question 4: Find
the area of the parallelogram with vertices (0,0), (2,4), (1,6), (3,10).
Question 5:
Here is a great website with 10 excellent commencement speeches.
It's worth the time reading these; I
particularly liked the one by Uslan (on
how it's not enough to just have a good idea, but how to get noticed). Read
Uslan's graduation speech, and write a sentence or two on what you take away
from it (any reasonable answer will receive full credit).
- Due Monday, February 11:
-
Read: 11.8,
12.1, 12.2.
-
Video of
the week:
light
cycle scene from Tron (the original).
-
Pictures of
the week: lines and art.
-
HW problems:
Section 11.3: Page 842: #1, #5, #11, #12. Section 11.4: Page 849: #1,
#2, #3, #22.
-
Due Wednesday, February 13:
-
Read: 12.3,
12.4, 12.5.
-
HW: Section 11.8: Page 893: #1, #26.
Section 12.2: Page 908: #2, #4, #5, #27, #32. (Extra Credit: Section 11.8:
#55.)
-
Due Monday, February 18:
-
Read 12.4,
12.5, 12.6, 12.7.
-
Section 12.3: Page 917:
#1, #8, #10, #24, #38, #54. Section 12.4: Page 928: #1, #4, #5.
-
Due Monday, February 25:
-
Due Wednesday, February 27:
-
Due Friday, March 1:
-
Due Monday, March 4:
-
Read 12.7, 12.8
-
Homework: (1) Page 949: #18: Use the exact value of \(f(P)\) and the
differential \(df\) to approximate the value \(f(Q)\), where \(f(x,y)=\sqrt{x^2-y^2}\),
with points \(P(13,5)\) and \(Q(13.2,4.9)\).
(2) Page 949: #23: Use the exact value of \(f(P)\) and the differential \(df\)
to approximate the value \(f(Q)\), where \(f(x,y,z)= e^{-xyz}\) with the
points \(P=(1,0,-2)\) and \(Q=(1.02, 0.03, -2.02)\). (3) Briefly describe what
Newton's Method is used for, and roughly how it works.
-
Extra Credit: to be handed in on a separate
paper: Let \(f(x) = \exp(-1/x^2)\) if \(|x| > 0\) and 0 if \(x = 0\). Prove
that \(f^{(n)}(0) = 0\) (i.e., that all the derivatives at the origin are
zero). This implies the Taylor series approximation to \(f(x)\) is the
function which is identically zero. As \(f(x) = 0\) only for \(x=0\), this
means the Taylor series (which converges for all \(x\)) only agrees with the
function at \(x=0\), a very unimpressive feat (as it is forced to agree
there).
-
Due Wednesday, March 6:
- Read
12.7, 12.8, 12.9
-
Homework: Note: the notation for this homework is a bit annoying. For example,
imagine we have a function \(f:\mathbb{R}^3\to\mathbb{R}\) and \(x, y, z:\mathbb{R}^2
\to \mathbb{R}\), so we have \(A(u,v) = f(x(u,v), y(u,v), z(u,v))\). If we
want to figure out how this compound function changes with \(u\), I prefer to
write \(\frac{\partial A}{\partial u}\); however, the book will often overload
the notation and write \(\frac{\partial f}{\partial u}\). I think this greatly
increases the chance of making an error, and
strongly
suggest introducing another function name.
Page 960: #2:
Find \(dw/dt\) both by using the chain rule and by expressing \(w\) explicitly
as a function of \(t\) before differentiating, with \(w = \frac{1}{u^2 +
v^2}\), \(u = \cos(2t)\), \(v = \sin(2t)\).
Page 960: #5:
Find \(\partial w/\partial s\) and \(\partial w/\partial t\) with \(w = \ln(x^2
+ y^2 + z^2)\), \(x = s - t\), \(y = s + t\), \(z = 2\sqrt{st}\).
Page 960: #8:
Find \(\partial w/\partial s\) and \(\partial w/\partial t\) with \(w = yz +
zx + xy\), \(x = s^2 - t^2\), \(y = s^2 + t^2\), \(z = s^2t^2\).
Page 960: #34:
A rectangular box has a square base. Find the rate at which its volume and
surface area are changing if its base is increasing at 2 cm/min and its height
is decreasing at 3cm/min at the instant when each dimension is 1 meter.
Page 960: #41:
Suppose that \(w = f(u)\) and that \(u = x + y\). Show that \(\partial
w/\partial x = \partial w/\partial y\).
Due Friday, March 8:
- Read
12.8, 12.9
- Hand in Quiz 2.
- Homework: Due Friday March 8:
Page 971:
Question 3: Find the gradient \(\nabla f\) at \(P\) where \(f(x,y) =
\exp(-x^2-y^2)\) and \(P\) is (0,0).
Page 971: Question 10:
Find the gradient \(\nabla f\) at \(P\) where \(f(x,y,z) = (2x-3y+5z)^5\) and
\(P\) is (-5,1,3).
Page 971: Question 11:
Find the directional derivative of \(f(x,y) = x^2+2xy+3y^2\) at \(P(2,1)\) in
the direction \(\overrightarrow{v} = (1,1)\). In other words, compute \((D_{\overrightarrow{u}}f)(P)\)
where \(\overrightarrow{u} = \overrightarrow{v}/|\overrightarrow{v}|\).
Page 971: Question 19:
Find the directional derivative of \(f(x,y,z) = \exp(xyz)\) at \(P(4,0,-3)\)
in the direction \(\overrightarrow{v} = (0,1,-1)\) (which is \(\textbf{j}-\textbf{k}\)).
In other words, compute \((D_{\overrightarrow{u}}f)(P)\) where \(\overrightarrow{u}
= \overrightarrow{v}/|\overrightarrow{v}|\).
Page 971: Question 21:
Find the maximum directional derivative of \(f(x,y) = 2x^2+3xy+4y^2\) at
\(P(1,1)\) and the direction in which it occurs.
Due Monday, March 11:
- Reading: Read 13.1.
- Homework: Question 1: Use Newton's
Method to find a rational number that estimates the square-root of 5 correctly
to at least 4 decimal places. Question 2: Let \(w(r,s,t) =
f(u(r,s,t), v(r,s,t))\) with \(f(u,v) = u^2 + v^2, u(r,s,t) = t \cos(rs)\) and
\(v(r,s,t) = t \sin(rs)\). Find the partial derivatives of \(w\) with respect
to \(r\), \(s\) and \(t\) both by direct substitution (which is very nice
here!) and by the chain rule. Question 3: Write \((1/2, \sqrt{3}/2)\)
in polar coordinates. Question 4: Find the tangent plane to \(z = f(x,y)\)
with \(f(x,y) = x^2 y + \sqrt{x+y}\) at \((1,3)\), and approximate the
function at \((.9,1.2)\). General comments: These problems are all done the
same way. Let's say we have functions of three variables, \(x,y,z\). Find the
function to maximize \(f\), the constraint function \(g\), and then solve \(\nabla
f(x,y,z) = \lambda \nabla g(x,y,z)\) and \(g(x,y,z) = c\). Explicitly, solve
\(\frac{\partial f}{\partial x}(x,y,z) = \lambda \frac{\partial g}{\partial
x}(x,y,z)$, $\frac{\partial f}{\partial y}(x,y,z) = \lambda \frac{\partial
g}{\partial y}(x,y,z)\), \(\frac{\partial f}{\partial z}(x,y,z) = \lambda \frac{\partial
g}{\partial z}(x,y,z)\), and \(g(x,y,z) = c\). For example, if we want to
maximize \(xy^2z^3\) subject to \(x+y+z = 4\), then \(f(x,y,z) = xy^2z^3\) and
\(g(x,y,z) = x+y+z = 4\). The hardest part is the algebra to solve the system
of equations. Remember to be on the lookout for dividing by zero. That is
never allowed, and thus you need to deal with those cases separately.
Specifically, if the quantity you want to divide by can be zero, you have to
consider as a separate case what happens when it is zero, and as another case
what happens when it is not zero. Page 981: Question 1: Find the
maximum and minimum values, if any, of \(f(x,y)=2x+y\) subject to the
constraint \(x^2+y^2=1\). Page 981: Question 14: Find the maximum and
minimum values, if any, of \(f(x,y,z)=x^2+y^2+z^2\) subject to the constraint
\(x^4+y^4+z^4=3\).
Due Wednesday, March 13:
- Read 13.1, 13.2.
- For additional reading on some of the background and related material, see
the following links. If you're interested in a math major, I strongly urge you
to read these.
(we'll get to the Taylor series part later in the
semester).
Proofs by Induction
(as well as other items, including the notes above!).
Due at the start of class on Friday, March 15:
first problem on the midterm.
You may start whenever, but once you start it is closed book, no
calculators, ....
Homework: Due Wednesday March 13:
Page 981: Question 19:
Find the point on the line \(3x+4y=100\) that is closest to the origin. Use
Lagrange multipliers to minimize the SQUARE of the distance.
Page 981: Question 35:
Find the point or points of the surface \(z=xy+5\) closest to the origin.
Page 981: Question 51:
Find the point on the parabola \(y= (x-1)^2\) that is closest to the origin.
Note: after some algebra you'll get that \(x\) satisfies \(2(x-1)^3+x=0\)
(depending on how you do the algebra it may look slightly different). You may
use a calculator, computer program, ... to numerically approximate the
solution.
Due Monday, April 1: Exam: Friday, March 15 in class
- Read
13.2, 13.3, 13.4.
- Due at the start of class on Friday, March 15:
first problem on the midterm.
You may start whenever, but once you start it is closed book, no
calculators, ....
- Homework: Due Monday, April 1:
Page 1004: Question 15:
Evaluate \(\int^3_0 \int^3_0 (xy+7x+y) dx dy\).
Page 1004: Question 24:
Evaluate \(\int^1_0 \int^1_0 e^{x+y} dx dy\).
Page 1004: Question 25:
Evaluate \(\int^\pi _0 \int^\pi _0 (xy+\sin x) dx dy\).
Page 1005: Question 37:
Use Riemann sums to show, without calculating the value of the integral, that
\(0\leq \int^\pi_0 \int^\pi_0 \sin \sqrt{xy}dxdy\leq \pi^2\).
Extra credit:
Let \(G(x) = \int_{t = 0}^{x^3} g(t) dt\). Find a nice formula for G'(x) in
terms of the functions in this problem.
Due Wednesday, April 3:
Due Friday, April
5:
Remember quiz 4 is due at the start
of Friday's class
- Read 13.7: Just know the statements of cylindrical and spherical change of
variables
- Video of the week:
Coin
Sorting.
- Play with Mathematica (or go online to
http://www.wolframalpha.com/).
- Here are
some more problems (with solutions) in setting up double integrals.
- Homework: Due Friday, April 5: Page 1011: #13: Evaluate the iterated
integral \(\int_0^3 \int_0^y \sqrt{y^2 + 16}\ dx\ dy. \). Page 1011: #25:
Sketch the region of integration for the integral \(\int_{-2}^2 \int_{x^2}^4
x^2y\ dy\ dx. \) Reverse the order of integration and evaluate the integral.
Page 1011: #30: Sketch the region of integration for the integral \(\int_{0}^1
\int_{y}^1 \exp(-x^2)\ dx\ dy. \). Reverse the order of integration and
evaluate the integral. Additional Problem: Give an example of a region in the
plane that is neither horizontally simple nor vertically simple. Page 1018:
#13: Find the volume of the solid that lies below the surface \(z=f(x,y)= y+e^x\)
and above the region in the \(xy\)-plane bounded by the given curves: \(x=0\),
\(x=1\), \(y=0\), \(y=2\). Page 1018: #42: Find the volume of the solid
bounded by the two paraboloids \(z=x^2+2y^2\) and \(z=12-2x^2-y^2\). Page
1026; #13: Evaluate the given integral by first converting to polar
coordinates: \(\int_0^1 \int_0^{\sqrt{1-y^2}} \frac{1}{1+x^2+y^2} dx dy.\)
Due Monday, April
8: Exam will be Wednesday, April 17th.
Know the definition of Jacobian determinants. Read the statement of the Change of
Variables formula. We will not deal with
this theorem in its full generality, but I want you to at least be aware of
its statement. We will concentrate on several special cases: polar
coordinates, cylindrical coordinates, and spherical
coordinates. If you want more information, click here for my handout
on the
Change of Variable formula.
Homework: Due Monday, April 8: Find \(\int_{y = 0}^1 \int_{x=-y}^y x^9 y^8
dx dy\). Page 1026: Question 4: Evaluate \(\int_{-\pi/4}^{\pi/4}
\int_0^{2\cos2\theta} r drd\theta\). Additional Question 1: Evaluate
\(\int_0^1\int_{-y}^y \sin(xy) \cdot \exp(x^2y^2) dxdy\). Hint: in what way is
this similar to the first problem on this homework assignment? Additional
Question 2: Let \(f(x,y,z) = \cos(xy + z^2)\). Find D\(f(x,y,z)\). Additional
Question 3: Find the maximum value of \(f(x,y) = xy\) given that \(g(x,y) =
x^2 + 4 y^2 = 1\).
Due Wednesday, April
10:
Exam will be Wednesday, April 17th.
- Read
multivariable calculus (Cain and Herod)
and my lecture notes.
- Read: 13.9.
Know the definition of Jacobian determinants. Read the statement of the Change of
Variables formula. We will not deal with
this theorem in its full generality, but I want you to at least be aware of
its statement. We will concentrate on several special cases: polar
coordinates, cylindrical coordinates, and spherical
coordinates. If you want more information, click here for my handout
on the
Change of Variable formula.
- Due Wednesday, April 10: Page 1056: #37a: Use spherical coordinates to
evaluate the integral \(I \ = \ \int \int \int_B \exp(-\rho^3)\ dV, \) where
\(B\) is the solid ball of radius \(a\) centered at the origin. Page
1056: #37b. Let \(a \rightarrow \infty\) in the result of part (a) to show
that \(\int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}
\exp(-(x^2 + y^2 + z^2)^{3/2})\ dx\ dy\ dz = \frac{4}{3}\pi. \)
Due Friday, April
12: Exam will be Wednesday, April 17th.
- Read
multivariable calculus (Cain and Herod)
and my lecture notes.
- Homework: Due Friday April 12: Page 1071: Solve for \(x\) and \(y\)
in terms of \(u\) and \(v\), and compute the Jacobian \(\partial(x,y)/\partial(u,v)\)
with \(u = x - 2y, v = 3x + y\). Page 1071: #3: Solve for \(x\) and \(y\) in
terms of \(u\) and \(v\), and compute the Jacobian \(\partial(x,y)/\partial(u,v)\)
with \(u = xy, v = y/x\).
Due
Monday, April
15: Exam will be Wednesday, April 17th.
Slides for
Monday's class.
Homework: Due Monday April 15: Problem 1. Give an example of a sequence
\(\{a_n\}_{n = 1}^{\infty}\) that diverges. Problem 2. Give an example of a
sequence of distinct terms \(a_n\) such that the sequence \(\{a_n\}_{n = 1}^\infty\)
converges. Problem 3. Give an example of a sequence of distinct terms \(a_n\)
such that \(|a_n| < 2013\) and the sequence \(\{a_n\}_{n = 1}^\infty\) does
not converge. Problem 10-4 (Cain-Herod): Find the limit of the sequence \(a_n
= 3/n^2\), or explain why it does not converge. Problem 10-5 (Cain-Herod):
Find the limit of the sequence \(a_n = \frac{3n^2+2n-7}{n^2}\).
NO HW DUE Friday, April
17th.
Due
Monday, April
22:
No written HW
Due
Wednesday, April
24:
For Taylor series, see my handout here
(essentially just pages 2 and 3).
Homework: Dues Wednesday April 24: (1) Cain-Herod: Find the limit of the
series \(\sum_{n=1}^\infty \frac{1}{3^n}\). (2) Cain-Herod: Find a value of
\(n\) that will insure that \(1+1/2+1/3+\cdots+1/n > 10^6\). Prove your value
works. (3) Cain-Herod: Question 14: Determine if the series \(\sum_{k=0}^\infty
\frac{1}{2e^k+k}\) converges or diverges. (4) Cain-Herod: Question 15:
Determine if the series \(\sum_{k=0}^\infty \frac{1}{2k+1}\) converges or
diverges. (5) Let \(f(x)=\cos x\), and compute the first eight derivatives of
\(f(x)\) at \(x=0\), and determine the \(n\)-th derivative.
Due
Friday, April
26:
-
For Taylor series, see my handout here
(essentially just pages 2 and 3).
- Homework: Dues Friday April 26: (1) Cain-Herod 10-18: Is the series
\(\left(\sum_{k=0}^n\frac{10^k}{k!}\right)\) convergent or divergent? (2)
Cain-Herod 10-21: Is the following series convergent or divergent? \(\sum_{k=1}^n
\frac{3^k}{5^k(k^4+k+1)}\). (3) Let \(a_n = \frac{1}{(n \ln n)}\) (one divided
by \(n\) times the natural log of \(n\)). Prove that this series diverges. \emph{Hint:
what is the derivative of the natural log of \(x\)? Use \(u\)-substitution.}
(4) Let \(a_n = \frac{1}{ (n\ln^2 n)}\) (one divided by n times the square of
the natural log of \(n\)). Prove that this series diverges. \emph{Hint: use
the same method as the previous problem. (5) Give an example of a sequence or
series that you have seen in another class, in something you've read, in
something you've observed in the world, ....
Due
Monday, April 29:
-
For Taylor series, see my handout here
(essentially just pages 2 and 3).
-
Cain-Herod: Question 20: Does the series \(\sum_{n=1}^\infty
\frac{3^{2k+1}}{10^k}\) converge or diverge? Additional Question 1: Compute
the first five terms of the Taylor series expansion of \(\ln (1-x)\) (the
natural logarithm of x) about \(x = 0\), and conjecture the answer for the
full Taylor series. Additional Question 2: Compute the first five terms of the
Taylor series expansion of \(\ln (1+x)\) (the natural logarithm of x) about
\(x = 0\), and conjecture the answer for the full Taylor series. Additional
Question 3: Give an example of a sequence or series you like.
Due Friday, May 3:
-
For Taylor series, see my handout here
(essentially just pages 2 and 3).
- Question 1: Find the second order Taylor series expansion of \(\cos(xy)\)
about \((0,0)\). Question 2: Find the second order Taylor Series expansion of
\(\cos(\sqrt{x+y})\) about \((0,0)\). Question 3: Find the second order Taylor
series expansion of \(\cos(x^3 y^4)\) about \((0,0)\). Extra Credit 1: Give a
product of infinitely many distinct, positive terms such that the product
converges to a number \(c\) with \(0 < c < \infty\). Extra Credit 2: Let \(\{a_n\}_{n=1}^\infty\)
be a sequence of positive numbers such that \(\sum_{n = 1}^\infty 1/a_n\)
converges. Let \(B_n = 1/n \sum_{k = 1}^n a_k\). Prove that \(\sum_{n = 1}^\infty
1/B_n\) converges.
The following are almost surely the assignments,
but the dates will change as these are from 2011.
Suggested
Problems and Extra Credit Problems for Math 105: The suggested problems are not
to be turned in, but are for your own personal edification or for additional
practice, though of course I and the TAs are happy to chat about these (or any)
problems. If you submit an extra credit problem, please clearly mark that it is
an extra credit problem.
- Introduction: THREE Extra Credit Problems:
(1) Let N be a
large integer. How should we divide N into positive integers ai such
that the product of the ai is
as large as possible. Redo the problem when N and the ai need
not be integers. (2) Without using any computer,
calculator or computing by brute force, determine which is larger: eπ or
πe. (In other words, find out which is larger without actually
determining the values of eπ or
πe). If you're
interested in formulas for π,
see also my paper A
probabilistic proof of Wallis' formula for π, which
appeared in the American
Mathematical Monthly (there are
a lot of good articles in this magazine, many of which are accessible to
freshmen). (3) Prove that the product of the slopes of
two perpendicular lines in the plane that are not parallel to the coordinate
axes is -1. What is the generalization of this to lines in three-dimensional
space? What is the analogue of the product of the slopes of the line equaling
-1?
- Section 11.1: Page 823: Is #38 true for all points (i.e., if you take any
three points in the plane)?
- Section 11.2: Page 833: #59, #61.
- Section 11.3: Page 842: #7, #17a.
- Section 11.4: Page 849: #25, #54, #58, #60.
- Section 11.8: Page 893: #33, #53. Extra Credit: #55.
- Section 12.2: Page 908: #41, #43, #45.
- Section 12.3: Page 917: #41, #51, #55.
- Section 12.4: Page 981: #55, #57, #58, #68.
- Section 12.5: Page 940: #10, #17, #46.
- Section 12.6:
Extra Credit:
Let f(x) = exp(-1/x2) if |x| > 0 and 0 if x = 0.
Prove that f(n)(0) = 0 (i.e., that all the derivatives at the
origin are zero). Show this implies the Taylor series approximation to f(x) is
the function which is identically zero. As f(x) = 0 only for x=0, this means
the Taylor series (which converges for all x) only agrees with the function at
x=0, a very unimpressive feat (as it is forced to agree there).
- Section 12.7: Page 960: #38, #53.
- Section 12.8: Page 971: #29, #40, #41, #60.
- Section 12.9: Page 981: #36, #37, #47, #49, #62 (important).
- Section 13.1: Page 1004: #33.
- Section 13.2: Page 1011: #41, #44, #49.
- Section 13.3: Page 1018: #29.
- Section 13.4: Page 1026: #7, #34.
- Section 13.7: Page 1056: #47, #48. (Extra credit for solving both of
these.)
- Section 13.9: Page 1070: #10, #28, #29.
-
From
multivariable calculus (Cain and Herod): Exercise 1 (page 10.3).
Extra
credit:
Find a series where the ratio test provides no information on whether or not
it converges but the root test says whether or not it converges or diverges.
-
Problems
leading up to Green's Theorem TBD.