Below is a tentative list of homework problems for Math 150. 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.
HOMEWORK: Homework problems for 2022 listed
below; suggested problems collected together at the end. Note dates MAY change
- Due at Class 02: Email me what you want to get out of the course, what
majors you are considering, and comments on Michael Uslan's graduation speech:
https://mycommunitysource.com/top-stories/video-michael-uslan-advice-for-college-graduates/#:~:text=Uslan%20is%20the%20executive%20producer%20of%20the%20Batman,many%20doors%20will%20be%20slammed%20in%20their%20faces.
Or try https://tinyurl.com/468hj5v9
- Due at Class 04:
Homework: You do not need to turn in, but make sure you can do all the
differentiation and integration problems from the calculus problem review
sheet, and make sure you are comfortable with the material from Calculus I and
II listed:
https://web.williams.edu/Mathematics/sjmiller/public_html/140Sp22/handouts/105CalcReviewProblems.pdf
(note the solution to the problems are later in the link).
- Due at Class 05:
- Read: Section 11.1, 11.2.
-
Video of the week:
Fibonacci numbers
(click
here for more on the Fibonacci numbers).
- HW #1 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).
-
Think
about the following: The sum of two numbers is 8 and their product is 15.
What is the sum of their reciprocals? How would you solve this?
-
Extra
Credit (due at Class 08, separate sheet): 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). Hand this in to me, NOT to the graders.
-
Due at Class 06:
-
Read: 11.2, 11.3,
11.4. NOTE: we will skip sections 11.5, 11.6, 11.7
-
HW
#2 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 at Class 07:
- Read:
11.4, 11.8.
NOTE: we will skip sections 11.5, 11.6, 11.7
-
HW #3 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. This extra credit should be written right after this
problem, or as part of this problem.) 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
\(\left(\begin{array}{cc}1 & 2 \\
3 & 4 \end{array}\right)\); 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 \(\left(\begin{array}{ccc}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{array}\right)\); 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 \(1, 2,
\dots, n^2\) filled as above, and prove your claim. This extra credit should
be turned in on a separate sheet of paper.) Question 4: Find
the area of the parallelogram with vertices (0,0), (2,4), (1,6), (3,10).
FOR FUN -- DO NOT SUBMIT:
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).
-
BONUS LECTURE
-
Due at Class 08:
-
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 #4 problems:
Section 11.3: Page 842: #1, #5, #11, #12. Section 11.4: Page 849: #1,
#2, #3, #22.
-
Due at Class 09:
- Video: Watch:
http://www.youtube.com/watch?v=71_8kHSAE4w
(February 21, 2014: Limits, Defn Partial Derivatives)
-
Read: 12.3,
12.4, 12.5.
-
HW #5: Section 11.8: Page 893: #1, #26.
Section 12.2: Page 908: #2, #4, #5, #27, #32. (Extra Credit: Section 11.8:
#55.)
-
EXAM IN THE NEAR FUTURE
Due at Class 10:
- Video: Watch:
http://www.youtube.com/watch?v=S6wiYRCiQhs
(March 3, 2014: Derivatives)
-
Read 12.4,
12.5, 12.6, 12.7.
-
HW #6:
Section 12.3: Page 917:
#1, #8, #10, #24, #38, #54. Section 12.4: Page 928: #1, #4, #5.
Due at Class 11:
Read 12.4,
12.5, 12.6, 12.7. Read
my
notes on the Method of Least Squares and Sections 12.6, 12.7.
Video of the week:
Flatland trailer. (The
full movie is available here for class purposes only.) There's also
projections of 4-dimensional cubes in our 3-dimensional space.
HW #7: Section 12.4: Page 928: #21, #25, #33, #36, #63 (is this surprising?).
Due at Class 12:
Due at Class 13:
- Due at Class 14:
http://www.youtube.com/watch?v=Q1TQtH6POyI (March 19, 2014: Fundamental
Theorem of Calculus in a Day). You should be working on the take-home exam,
but you can also be working on the HW due at the start of Class 16. That
homework involves the linear approximation lecture.
-
Due at Class 15:
-
Read 12.7, 12.8
- Video: Watch http://youtu.be/cgIoKYr11sY
(March 12, 2014: Chain Rule)
-
HW #10: (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 at Class 16:
-
Read 12.8, 12.9
- Video: Watch:
http://youtu.be/Da0cO905Aj8
(March 10, 2014: Linear Approximation) (originally
assigned for Class 13)
-
Video 3/14/22: No video, going over exam
-
NOTE: MOVING
TO LATER: Pythagorean Formula:
https://youtu.be/EH6PUS2OwUY (slides here)
-
HW #11:
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 at Class 17:
-
Video 3/16/22:
https://youtu.be/grtkHEIdlkU
(slides
here)
- HW#12:
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 at Class 18:
Homework #13: DUE AT THE START
OF CLASS 19: 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 at Class 19:
Optional:
Supplemental lecture:
Fractals: From Khan to
Frozen
(March 20, 2014: Mathematics of Fractals and applications to films)
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!).
-
Homework #13: DUE AT THE START OF CLASS
19: 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 at Class 20:
Sixteenth
day lecture: http://youtu.be/G9d9lcYevnM
(April 9, 2014: Iterated integrals, changing order)
Seventeenth
day lecture:
http://youtu.be/JP78q_ri-4o
(April 11, 2014: Polar Change of Variables, Circles and Spheres)
Homework #14:
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 at Class 21:
Lecture on Monte Carlo Integration, Simple Regions, Spherical
Coordinates:
https://youtu.be/ygskSmmshKg (slides
here)
-
Read 13.3, 13.4.
We will not cover 13.5 or 13.6 (though we will of course discuss triple
integrals).
-
Video of the week:
Coin
Sorting. (We talked about this when doing the proof of the Fundamental
Theorem of Calculus -- you can integrate by adding as you go, or by grouping
by value).
-
Eighteenth
day lecture: http://youtu.be/Nz7ahXOMTus
(April 14, 2014: Monte Carlo Integration, Change of Variables for Ellipses)
-
OPTIONAL:
http://youtu.be/MN80UyR2Rh8
(April 21, 2014: Gamma Function and Calc I/II/III review)
-
Play with Mathematica (or go online to
http://www.wolframalpha.com/).
-
Handout
with correctly worked example from Monday's class on vertically/horizontally
simple region
-
Here are
some more problems (with solutions) in setting up double integrals.
-
THERE ARE TWO
HOMEWORK ASSIGNMENTS DUE TODAY. TO ASSIST THE GRADERS HAND THEM IN AS
SEPARATE ASSIGNMENTS, I.E., TWO DIFFERENT SETS, SO EACH GRADER CAN TAKE
ONE.
-
Homework #15:
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.
-
Homework #16: Page 1011: Question 4: Evaluate
\(\int_0^2 \int_{y/2}^1 (x+y) dxdy\). Page 1012: Question 11: Evaluate
\(\int_0^1 \int_0^{x^3} \exp(y/x)dydx\). Additional Problem: Let \(f(x)=x^3-4x^2+
\cos(2x^3)+ \sin(x+1701)\). Find a finite \(B\) such that \(|f'(x)| \leq B\)
for all \(x\) in \([2,3]\).
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.
Due at Class 22:
Baseball Lecture
- Read
13.7: Just know the statements of cylindrical and spherical change of
variables (did baseball lecture instead)
- Nineteenth
Lecture:
http://youtu.be/tZ9lcdk6hLOUt (April
23, 2014: Review Class: Perimeter Ellipse, Word Problems, Viewing Algebra)
-
Optional lecture:
http://youtu.be/t3Pt4E1BeUTwt
(April 16, 2014: Cylindrical and Spherical Coordinates, Newton's Shell
Theorem)
- 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 #17: 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: \(\int_0^1 \int_0^{\sqrt{1-y^2}} \frac{1}{1+x^2+y^2} dx dy.\).
Find \(\int_{y = 0}^1 \int_{x=-y}^y x^9 y^8
dx dy\). Additional Question 1: 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 2: Evaluate
\(\int_0^1\int_{-y}^y \sin(xy) \cdot \exp(x^2y^2) dxdy\). Hint: in what way is
this similar to an earlier problem on this homework assignment? Additional
Question 3: Let \(f(x,y,z) = \cos(xy + z^2)\). Find D\(f(x,y,z)\). Additional
Question 4: Find the maximum value of \(f(x,y) = xy\) given that \(g(x,y) =
x^2 + 4 y^2 = 1\).
Due at Class 23:
Spherical Coordinates, Basketball, Taylor Series: 4-13-22:
https://youtu.be/qUP-giFb_f8https://youtu.be/qUP-giFb_f8
(slides
here)
- Twenty-third
day lecture: http://youtu.be/aigdKmu-5ow (April
28, 2014: Geometric and Harmonic Series, Memoryless Processes)
-
ADDITIONAL LECTURE:
https://youtu.be/grs3Pt4E1BeUTw
Cylindrical and Spherical Coordinates, Newton's Shell Theorem-
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 at Class 24:
Sequences/Series, Harmonic Series, p-Series, Comparison Test, Streaming
video:
https://youtu.be/NxpZlV3M2Zs
(slides
here)
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.
Homework: #18: 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 at Class 25:
Alternating Series, Integral Test:
https://youtu.be/0_3_xDg1i2k
(slides
here)
Read
multivariable calculus (Cain and Herod)
and my lecture notes.
Homework #19: THIS ASSIGNMENT IS ENTIRELY EXTRA CREDIT! IT INVOLVES YOU WATCHING THE VIDEO
AND DOING THESE PROBLEMS. IT IS OPTIONAL.
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 at Class 26:
No In-Person Class (due to Midterm)
Due at Class 27:
Ratio and Root Test 4-22-22
https://youtu.be/o2ATZBoVxII
(slides
here)
-
Read
multivariable calculus (Cain and Herod)
and my lecture notes.
- Twenty-seventh
day lecture: http://youtu.be/ujJbUpCab6M
(May 7, 2014: Root Test, Integral Test)
- Homework #20: 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 \(a_n = \frac{3n^2+2n-7}{n^2}\).
Due at Class 28:
Taylor Series Trick, Multivariable Taylor SEries:
https://youtu.be/P5t7eLYRFTI
(slides
here)
Due at Class 29:
- Twenty-ninth
day lecture:
http://youtu.be/4OcxtpxuSJw (May 14, 2014:
Special Series, Alternating Series, Pi
formulas, Birthday Problem: Not doing 2018)
- Read
multivariable calculus (Cain and Herod)
and my lecture notes. (You
should have already done this).
-
For Taylor series, see my handout here
(essentially just pages 2 and 3).
- Homework #21: (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 at Class 30: Library Trip
Due at Class 31: Difference Equations
-
HW: (1) Calculate, to at least 40 decimal places, 100/9801. Do
you notice a pattern? Do you think it will continue forever - why or why not?
(2) Calculate, to at least 40 decimal places, 1000/998999. Do you notice a
pattern? Do you think it will continue forever - why or why not?
Class 32: Differential Equations and Trafalgar
-
HW (1) Solve the difference equation a(n+1) = 7a(n) - 12a(n-1)
with initial conditions a(0) = 3 and a(1) = 10. (2) Consider the whale problem
from class, but now assume that on every two pairs of 1 year old whales give
birth to one new pair of whales, and every four pairs of 2 year old whales
give birth to one new pair. Prove or disprove: eventually the whales dies out.
HOMEWORK: Homework problems for 2021 listed below;
suggested problems collected together at the end. Note dates MAY change
(original dates are from 2018)
- Due Monday, Feb 22 :
Handout from first day
of class is here
Videos: 2/7/20: Sec 1:
https://youtu.be/gfeXKy3fk1M
- 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. I am trying to get a
rotated version posted on glow.
-
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).
-
Read
Uslan's graduation speech, and email me something about it that resonates
with you:
http://www.graduationwisdom.com/speeches/0018-uslan.htm (do not include
this with the HW you hand in to the TAs).
-
Think
about the following: The sum of two numbers is 8 and their product is 15.
What is the sum of their reciprocals? How would you solve this?
-
Extra
Credit (due 2/12/20): 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). Hand this in to me, NOT to the graders.
-
Due Friday, February 26:
-
Read: 11.2, 11.3,
11.4. NOTE: we will skip sections 11.5, 11.6, 11.7
-
HW
#2 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 Monday, March 1:
- Read:
11.4, 11.8.
NOTE: we will skip sections 11.5, 11.6, 11.7
-
HW #3 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. This extra credit should be written right after this
problem, or as part of this problem.) 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
\(\left(\begin{array}{cc}1 & 2 \\
3 & 4 \end{array}\right)\); 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 \(\left(\begin{array}{ccc}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{array}\right)\); 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 \(1, 2,
\dots, n^2\) filled as above, and prove your claim. This extra credit should
be turned in on a separate sheet of paper.) Question 4: Find
the area of the parallelogram with vertices (0,0), (2,4), (1,6), (3,10).
FOR FUN -- DO NOT SUBMIT:
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).
-
BONUS LECTURE
-
Due Wednesday, March 3:
-
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 #4 problems:
Section 11.3: Page 842: #1, #5, #11, #12. Section 11.4: Page 849: #1,
#2, #3, #22.
-
Due Friday, March 5:
- Video: Watch:
http://www.youtube.com/watch?v=71_8kHSAE4w
(February 21, 2014: Limits, Defn Partial Derivatives)
-
Read: 12.3,
12.4, 12.5.
-
HW #5: 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, March 8: EXAM - IN
CLASS
Due Wednesday, March 10:
- Video: Watch:
http://www.youtube.com/watch?v=S6wiYRCiQhs
(March 3, 2014: Derivatives)
-
Read 12.4,
12.5, 12.6, 12.7.
-
HW #6:
Section 12.3: Page 917:
#1, #8, #10, #24, #38, #54. Section 12.4: Page 928: #1, #4, #5.
Due Friday, Mar 12:
Read 12.4,
12.5, 12.6, 12.7. Read
my
notes on the Method of Least Squares and Sections 12.6, 12.7.
Video of the week:
Flatland trailer. (The
full movie is available here for class purposes only.) There's also
projections of 4-dimensional cubes in our 3-dimensional space.
HW #7:
Due Friday, Mar 12: Section 12.4: Page 928: #21, #25, #33, #36, #63 (is this surprising?).
Due Monday, Mar 15:
Due Wednesday, March 17:
-
Due Friday March 19:
-
Read 12.7, 12.8
- Video: Watch http://youtu.be/cgIoKYr11sY
(March 12, 2014: Chain Rule)
-
HW #10:
Due Wednesday March 24: (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 Monday, March 22:
-
Read 12.8, 12.9
- Video: Watch:
http://youtu.be/Da0cO905Aj8
(March 10, 2014: Linear Approximation)
-
NOTE: MOVING
TO LATER: Pythagorean Formula:
https://youtu.be/EH6PUS2OwUY (slides here)
-
HW #11:
Due Wednesday March 24:
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
Wednesday March 24:
- HW#12: Due Friday March 26:
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 Friday, March 26:
-
Reading: Read 13.1, 13.2
-
Thirteenth
day lecture: http://youtu.be/pgwC2vOwRuE
(March 17, 2014: Lagrange
Multipliers)
-
Homework #13: Due Mon Mar 29: 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 Monday,
March 29:
DO NOT WATCH: Fourteenth
day lecture:
http://www.youtube.com/watch?v=Q1TQtH6POyI (March 19, 2014: Fundamental
Theorem of Calculus in a Day)
Optional:
Supplemental lecture:
Fractals: From Khan to
Frozen
(March 20, 2014: Mathematics of Fractals and applications to films)
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!).
Homework
#14: Due
Wed Mar 31:
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
Wednesday, March 31:
Due Friday, April 2:
- Read
13.2, 13.3, 13.4.
-
Fifteenth
day lecture: http://youtu.be/N8nFFWG_6J4
(April 7, 2014: Integration in two variables)
- Homework #15: Due Monday, April 5:
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
Monday, April 5:
-
Read 13.3, 13.4.
We will not cover 13.5 or 13.6 (though we will of course discuss triple
integrals).
-
Sixteenth
day lecture: http://youtu.be/G9d9lcYevnM
(April 9, 2014: Iterated integrals, changing order)
-
Video of the week:
Coin
Sorting. (We talked about this when doing the proof of the Fundamental
Theorem of Calculus -- you can integrate by adding as you go, or by grouping
by value).
-
Play with Mathematica (or go online to
http://www.wolframalpha.com/).
-
Handout
with correctly worked example from Monday's class on vertically/horizontally
simple region
-
Here are
some more problems (with solutions) in setting up double integrals.
-
Homework #16: Due Wednesday, April 7: Page 1011: Question 4: Evaluate
\(\int_0^2 \int_{y/2}^1 (x+y) dxdy\). Page 1012: Question 11: Evaluate
\(\int_0^1 \int_0^{x^3} \exp(y/x)dydx\). Additional Problem: Let \(f(x)=x^3-4x^2+
\cos(2x^3)+ \sin(x+1701)\). Find a finite \(B\) such that \(|f'(x)| \leq B\)
for all \(x\) in \([2,3]\).
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.
Due
Wednesday, April 7:
- Read
13.7: Just know the statements of cylindrical and spherical change of
variables
- Seventeenth
day lecture:
http://youtu.be/JP78q_ri-4o
(April 11, 2014: Polar Change of Variables, Circles and Spheres)
- 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 #17: Due Friday, April 9: 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: \(\int_0^1 \int_0^{\sqrt{1-y^2}} \frac{1}{1+x^2+y^2} dx dy.\).
Find \(\int_{y = 0}^1 \int_{x=-y}^y x^9 y^8
dx dy\). Additional Question 1: 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 2: Evaluate
\(\int_0^1\int_{-y}^y \sin(xy) \cdot \exp(x^2y^2) dxdy\). Hint: in what way is
this similar to an earlier problem on this homework assignment? Additional
Question 3: Let \(f(x,y,z) = \cos(xy + z^2)\). Find D\(f(x,y,z)\). Additional
Question 4: Find the maximum value of \(f(x,y) = xy\) given that \(g(x,y) =
x^2 + 4 y^2 = 1\).
Due
Friday, April 9:
- Baseball lecture
- Eighteenth
day lecture: http://youtu.be/Nz7ahXOMTus
(April 14, 2014: Monte Carlo Integration, Change of Variables for Ellipses)
- Twenty-first
day lecture:
OPTIONAL:
http://youtu.be/MN80UyR2Rh8
(April 21, 2014: Gamma Function and Calc I/II/III review)
- 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
Monday, April
12:
https://www.youtube.com/watch?v=qVmHBnisZW4
https://www.youtube.com/watch?v=Z9lcdk6hLOU
ead: 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.
Homework: #18: Due Wednesday, April
14: 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
Wednesday, April 14:
- Twenty-third
day lecture: http://youtu.be/aigdKmu-5ow (April
28, 2014: Geometric and Harmonic Series, Memoryless Processes)
-
Read
multivariable calculus (Cain and Herod)
and my lecture notes.
-
Homework
#19: Due Friday April
16:
THIS ASSIGNMENT IS ENTIRELY EXTRA CREDIT! IT INVOLVES YOU WATCHING THE VIDEO
AND DOING THESE PROBLEMS. IT IS OPTIONAL.
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
Friday, April 16:
Due
Monday, April 19:
-
NO CLASS DUE TO MIDTERM 3 - Exam due at the start of Friday's class
-
Twenty-sixth lecture: Watch before start of
your section on Monday April 19th:
https://youtu.be/jFgCKfUTOQ8
-
Read
multivariable calculus (Cain and Herod)
and my lecture notes.
- Homework #20: Due Friday April 23: 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 \(a_n = \frac{3n^2+2n-7}{n^2}\).
Wednesday April 21: Mental Health Day (aka Spring Break)
Friday April 23:
Review Day (class
optional, consider it an extended mental break day)
Due Monday, April
26:
Due
Wednesday,
April 28:
For Taylor series, see my handout here
(essentially just pages 2 and 3).
Homework #21: Dues Friday, April 30: (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
30:
-
Twenty-seventh
day lecture:
http://youtu.be/yr01SLw9t4c
(May 12, 2014: Taylor Series)
-
For Taylor series, see my handout here
(essentially just pages 2 and 3).
- Homework #22: Dues Monday May 3: (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 converges. \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, May 3:
Review Class: These videos will also be the review videos for Friday May 12
Review: Change of Variables, Odd/Even Functions, Bounding Functions,
Comparison Test: http://youtu.be/8EZQKzoWxEo
Review:
Mostly series test (one series and we apply all four tests!):
http://youtu.be/HANb4mVLOOc
Due
Wednesday, May 5: not assigned 2018
Due
Friday, May 7: NO CLASS - MENTAL HEALTH DAY Due
Monday, May 10: Class is optional.
Due
Wednesday, May 12:
Due
Friday, May 14:
REVIEW CLASS
- Homework: Due at the start of class on Monday May 15: Take-home part of
the final
- Here are some videos you can watch to help review.
- Review:
Spherical Integral, Changing
Orders, Series:
https://youtu.be/Jgb3z3E58Zw
Review: Change of Variables, Odd/Even Functions, Bounding Functions,
Comparison Test: http://youtu.be/8EZQKzoWxEo
Review:
Mostly series test (one series and we apply all four tests!):
http://youtu.be/HANb4mVLOOc
Due
Monday, May 17: Part 1 of the Final (in class)
Due Wednesday, May 19: Part 2 of the Final (in class)
DATES FROM HERE ON HAVE NOT BEEN UPDATED.
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.