Tuesday, December 5: Polar Coordinates
In class worksheet.
Here is a worksheet with more space, in case you are downloading the pdf to your tablet.
Here are the scanned
solutions.
Monday, December 4: Arc Length and Speed
One of the WeBWorK problems feature Cornu’s Spiral. Don't forget the Fundamental Theorem of Calculus for this one:
it's easy to compute \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) using the FToC!
Here’s a figure of Cornu’s spiral traced out with \(-10 \leq t \leq 10\),
and here is an animation of the spiral traced out with \(0 \leq t \leq T\).
The worksheet today is on paper since it has a printed figure,
but here is a pdf.
Here are the scanned
solutions.
Friday, December 1: More about Parametric Curves:
eliminating the parameter, and tangent lines to parametric curves
Here are a few more Desmos parametric plots.
Cycloid (desmos) (like set 22, problem 5).
A blue wheel of radius 1 is rolling and a point at distance a from the axel traces out a parametric
curve. If \(a = 1\), this curve is a cycloid.
Here is the Desmos graph of the tangent line to the curve
\(x = 5 \cos(t), y = 3 \sin(t)\) at time \(t = \pi/3\).
In class worksheet.
Here are the scanned
solutions.
Wednesday, November 29:
Parametric Equations for curves in the plane.
You can plot parametric curves using the Desmos Graphing Calculator
x = f(t), y = g(t). You
can change the functions f and g. You can save the new functions if you have a Desmos account. The link on
this webpage will always have the original Lissajou figure.
In class worksheet.
Here are the scanned
solutions.
Tuesday, November28 24: MT 3
Monday, November 27: Review for MT 3
Wednesday, November 22: (The day before Thanksgiving!)
Here are some youtube videos suggested by a student:
Taylor Series and Maclaurin Series
Taylor Polynomials and Maclaurin Polynomials With Approximations. Note: The Organic Chemistry Tutor used the formula of last resort, \(\displaystyle c_n = \frac {f^{(n)}(a)}{n!}\).
A simpler method is to write \(\ln(x) = \ln(1 + (x-1) )\) and substitute
\(x-1\) for \(t\) in the known series
\(\ln(1+t) = t - t^2/2 + t^3/3 - t^4/4 + \ldots\).
Today we will mostly go over WeBWorK problems.
We already did something like problems 1 and 2 on the worksheet
(like WeBWorK problem 8).
So we will focus on problems 3, 4 and 5
on the worksheet (like WeBWorK problems 7 and 9).
In class worksheet.
Here are the scanned
solutions.
Tuesday, November 21:
Applications of Taylor Series,
including Taylor Polynomials.
The partial sums of a Taylor series are called Taylor polynomials.
Here is a Mathematica demonstration of some Taylor polynomials for
\(\frac 1{1-x}\) \( = 1+ x + x^2 + x^3 + \cdots \).
The series convegences iff \(-1< x < 1\).
Here is a Mathematica demonstration of some Taylor polynomials for
\( \sin(x)\)
\(= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \) .
The series
convergence at all \(x\).
Here is an animated animated gif of some Taylor Polynomials for
\(\frac {1}{1+x^2}\)
\( = 1 - x^2 + x^4 - x^6 + x^8 - \cdots\).
The series converges iff \(-1< x < 1\).
In class worksheet.
Here are the scanned
solutions.
Monday, November 20: Taylor Series, with Paul's Notes
and Sudipto's Notes.
The Taylor series of \(f\), centered at \(a\), is \(\displaystyle f(x) = \sum_{n=0}^\infty c_n (x-a)^n\),
where \(\displaystyle c_n = \frac {f^{(n)}(a)}{n!}\). That is,
\(\displaystyle f(x) = \sum_{n=0}^\infty \frac {f^{(n)}(a)}{n!} (x-a)^n\).
The equality holds if \(f\) is a
‘nice’ (analytic) function and the P.S. converges.
In class worksheet.
Here are the scanned
solutions.
Friday, November 17: Power Series Repersentations of More Functions
Also: The partial sums of a Taylor series are called Taylor polynomials.
Here is a Mathematica demonstration of some Taylor polynomials for
\(\frac 1{1-x}\) \( = 1+ x + x^2 + x^3 + \cdots \).
The series convegences iff \(-1< x < 1\).
Here is an animated animated gif of some Taylor Polynomials for
\(\frac {1}{1+x^2}\)
\( = 1 - x^2 + x^4 - x^6 + x^8 - \cdots\).
The series converges iff \(-1< x < 1\).
In class worksheet.
Bonus question: Find the sum of the series \( 1-\frac 1 3 + \frac 1 5 - \frac 1 7 + \cdots \) by recognizing this as \(f(1)\) for some function \(f\) of which you know the power series.
Here are the scanned
solutions.
Wednesday, November 15: Power Series of Functions
The link here is to Paul's notes.
A power series representing a function is also called a Taylor series.
Most of the Taylor series we consider have their
center at 0. These are sometimes called Maclaurin series.
In class worksheet.
Here are the scanned
solutions.
Tuesday, November 14: The Radius of Convergence and Interval of Convergence of a Power Series
Every power series has a radius of convergence which can usually be determined using
the Ratio Test.
In class worksheet.
Here are the scanned
solutions.
Monday, November 13: Power Series.
A power series has a variable \(x\), and is written as
\(\displaystyle \sum_{n=0}^\infty a_n (x-c)^n\) or \(\displaystyle \sum_{n=0}^\infty c_n (x-a)^n\).
(Sorry about the two different notations!)
Every power series centered at \(c\) has a radius of convergence \(R\) such that
The P.S. converges if \(|x-c| < R\), and
The P.S. diverges if \(|x-c| > R\).
(If \(R\) is a positive number, then anything can happen if \(|x-c| = R\). Furthermore, \(R = 0\) and \(R = \infty\) are possible.)
Sometimes the power series is a geometric series, and it's easy to determine the interval of convergence.
In class worksheet.
Here are the scanned
solutions.
Wednesday, November 8: Review of convergence tests
Handouts: Series Convergence and Divergence Tests,
and Flow Chart for determining if a series converges.
In class worksheet.
Here are the scanned
solutions.
Tuesday, November 7: The Ratio Test
The Ratio Test:
If \(\displaystyle \lim_{n\to\infty} \left | \frac{a_{n+1}}{a_n} \right | < 1 \),
including \(\displaystyle \lim_{n\to\infty} \left | \frac{a_{n+1}}{a_n} \right | = 0 \), then
\(\sum a_n\) converges absolutely (and thus \(\sum a_n\) converges).
If \(\displaystyle \lim_{n\to\infty} \left | \frac{a_{n+1}}{a_n} \right | = 1 \), then the ratio test is inconclusive. (This is very common!)
If \(\displaystyle \lim_{n\to\infty} \left | \frac{a_{n+1}}{a_n} \right | > 1 \), including \(\displaystyle \lim_{n\to\infty} \left | \frac{a_{n+1}}{a_n} \right | = \infty \), then \(\sum a_n\) diverges.
In class worksheet.
This worksheet will not be collected. Here is the solution, without the reasons.
You will explain the reason in a paper assignment.
Paper Assignment due Wednesday, Nov. 8 at the beginning of class, worth 5 class points:
For series A and C in the worksheet, write up a careful proof, using
the guidance of the solutions to quiz 5.
(Series B is already in that quiz 5 solution.)
Here are the solutions showing the proper Application of Convergence Tests.
This assignment is worth 5 extra credit points.
Monday, November 6: Absolute and Conditional Convergence
The Alternating Series Test: If \(a_n >0\) and \(a_{n+1}>a_n\) for all integers \(n \geq 1\), and \(\displaystyle \lim_{n\to \infty} a_n = 0\), then \(\sum \pm (-1)^n a_n\) converges.
The Definition of Absolute and Conditional Convergence
The Absolute Convergence Test
Let \(C = \) “The series \( \sum a_n\) converges.”
Let \(D = \) “The series \( \sum a_n\) diverges.”
Let \(A = \) “The series \( \sum a_n\) converges absolutely.” (That is, \(\sum |a_n|\) converges.)
Let \(CC = \) “The series \( \sum a_n\) converges conditionally.”
(That is, \(\sum a_n\) converges but \(\sum |a_n|\) diverges.)
Then we have the following implications:
\(C \iff \text{not} (D)\)
\(CC \iff (C \ \text{and} \ \text{not}(A))\)
\(A \implies C \quad \) This is the Absolute Convergence Test.
Here is another link to the Mathematica notebook that plots the sequence of terms and the
sequence of partial sums.
In class worksheet.
Here are the scanned
solutions.
Wednesday, November 1, and Friday, November 3: The Integral Test, and the Comparison Test, and the Limit Comparison Test
The Integral Test:
If \(a_n = f(n)\), for all \(n \geq 1\) and \(f(x)\) is decreasing for all real numbers \(x \geq 1\), and \(\displaystyle \lim_{x \to \infty} f(x) = 0\), then
\( \displaystyle \sum_{n=1}^\infty a_n\) converges if and only if
\(\displaystyle \int_1^\infty f(x) dx\) converges.
That is, they both converge or they both diverge.
The Comparison Test:
Assume \(0 \leq a_n \leq b_n\), for all \(n \geq 1\).
If \( \displaystyle \sum_{n=1}^\infty a_n = \infty\), then
\( \displaystyle \sum_{n=1}^\infty b_n = \infty\).
That is, if the smaller series diverges, then the larger series diverges.
If \( \displaystyle \sum_{n=1}^\infty b_n < \infty\), then
\( \displaystyle \sum_{n=1}^\infty a_n < \infty\).
That is, if the larger series converges, then the smaller series converges.
Warning! About half the time the comparison test is inconclusive. That is why the limit comparison test is helpful:
The Limit Comparison Test:
Assume \(a_n > 0\) and \(b_n > 0\), for \(n \geq 1\), and \(\displaystyle 0 < \lim_{n\to \infty} \frac{a_n}{b_n} < \infty\). Then
\( \displaystyle \sum_{n=1}^\infty a_n < \infty\) if and only if
\( \displaystyle \sum_{n=1}^\infty b_n < \infty\). That is, they both converge or they both diverge.
In all of these tests, \(n = 1\) can be replace by \(n = 2\) or \(n = 10\), etc.
In class worksheet.
Here are the scanned
solutions.
Tuesday, October 31: The Test for Divergence, the Integral Test, and the Comparison Test
The Test for Divergence: If \(\displaystyle \lim_{n \to \infty} a_n = L \neq 0\),
or \(\displaystyle\lim_{n \to \infty} a_n\) does not exist, then \(\sum a_n\) diverges.
This was in the previous section of Sudipto's notes.
After the test for divergence,
the focus of this set is on series with \(a_i > 0\).
In class worksheet.
Here are the scanned
solutions.
Monday, October 30: More about Series
Examples where we can get a closed form expression for \(s_n = a_1 + a_2 + \cdots + a_n\).
(1) Geometric series.
(2) Telescoping series.
In class worksheet.
Here are the scanned
solutions.
Friday, October 27: Series (that is, Infinite Series (plural))
\(\displaystyle s_n = \sum_{i=1}^n a_i\), is the \(n\)th partial sum of
\(\displaystyle \sum_{n=1}^\infty a_n\). The series \(\displaystyle \sum_{n=1}^\infty a_n\) converges if and only if the sequence of partial sums,
\(\{s_n\}\), converges.
\(\displaystyle \sum_{n=1}^\infty a_n := \lim_{n \to \infty} \sum_{i=1}^n a_i\)
SeriesGraph.nb Mathematica notebook. You can get Mathematica on your own machine for free. Google NAU Mathamatica.
In class worksheet.
Here are the scanned
solutions.
Wednesday, October 25: More about Sequences
Paul's notes on
Sequences
and More Sequences.
Definitions:
The sequence \(\{a_n\}_{n = 1}^\infty\) converges to \(L\) if
\(\displaystyle \lim_{n \to \infty}a_n = L\). Note that \(L\) must be a number, and
\( \infty\) is not a number.
The sequence \(\{a_n\}_{n = 1}^\infty\) diverges if
\(\displaystyle \lim_{n \to \infty} a_n\) does not exist.
The sequence \(\{a_n\}_{n = 1}^\infty\) diverges to \(\infty\) if
\(\displaystyle \lim_{n \to \infty} a_n = \infty\). We also say the sequence increases without bound.
The sequence \(\{a_n\}_{n = 1}^\infty\) diverges to \(-\infty\) if
\(\displaystyle \lim_{n \to \infty} a_n = -\infty\). We also say the sequence decreases without bound.
The number ''1'' in each of these definitions can be replaced by any integer. For example,
\(\displaystyle \left \{\frac{n}{n+1} \right \}_{n = 0}^\infty\) and
\(\displaystyle \left \{\frac{n}{n+1} \right \}_{n = 1}^\infty\) both converge to 1
since \(\displaystyle \lim_{n \to \infty} \frac{n}{n+1} = 1\). Thus we say that
the sequence \(\displaystyle \left \{\frac{n}{n+1} \right \}\) converges to 1.
In class worksheet.
Here are the scanned
solutions.
Tuesday, October 24:
Sequences
(Infinite Sequences)
In class worksheet.
Here are the scanned
solutions.
Wednesday, October 18: More Modeling with First Order ODEs
Tank problems, and Newton's Law of Cooling.
Multiple ways to represent exponential functions: \(e^{-kt} = a^{t/h}\)
In class worksheet on Newon's law of cooling.
Here are the scanned
solutions.
Tuesday, October 17: Modeling with First Order ODEs
Here are Paul's notes, which are are better for this set. Sudipto doesn't talk much about setting up the ODE.
We will only do applications in WeBWorK set 14 with ODEs of two forms:
(1)
\(\displaystyle \frac{dy}{dt} = -k y\), with the general solution \(y(t) = C e^{-kt}\) (exponential decay), or
(2)
\(\displaystyle\frac{dy}{dt} = -k (y - A)\), with the general solution \(y(t) = A + C e^{-kt}\) (exponential decay to \(y = A\)).
In both cases, we assume that \(k\) is a positive constant.
The constant \(A\) is important since \(y(t) = A\) is a constant solution. The arbitrary constant in the general solution is \(C\).
Be aware that a common application which is not in WeBWorK set 14 is
(3) \(\displaystyle\frac{dy}{dt} = k y\), with the general solution \(y(t) = C e^{kt}\) (exponential growth).
In class worksheet.
Here are the
solutions.
Monday, October 16: The interval of existence of a solution
In class worksheet.
Here are the scanned
solutions.
Friday October 13: Separation of Variables, a technique to solve some 1st order ODEs.
In class worksheet, which is printed out today.
Here are the scanned
solutions.
Wednesday October 11: Introduction to ODEs (Ordinary Differential Equations), with a focus on
Euler's method
Here is the slope field for
\(\frac{dy}{dx} = x^2 - y^2\) using Daryll Nester's Slope Field App.
Click Initial Points, and plot the approximate solution with \(y(0) = 1\).
The default method is Runge-Kutta 4 (RK4), with a step size of \(h = 0.1\).
I can see some kinks in the approximate curve, so change the step size to \(h = 0.01\) and replot the approximate solution.
This gives a very accurate numerical approximation to the solution to the Initial Value Problem.
The simplest numerical approximation to an IVP is Euler's method. Click on that link again to get a new copy of the slope field. Change the method to Euler, and the step size to \(h = 0.5\). First we will draw the approximate solution on the whiteboard, then let the app draw it. Then reduce the step size \(h\) and see how the approximate solution gets better and better. In the limit \(h \to 0\) this gives the exact solution.
Then we will open up google sheets and get the approximate solution with the spreadsheet. Then you can do your own problem 9 on the WeBWorK.
Here is a video from MAT 239 about
Euler's Method
and how to do it on a spreadsheet.
Tuesday 10-10: Introduction to ODEs (Ordinary Differential Equations)
Darryl Nester's Slope Field Application.
In class worksheet. Here are the scanned
solutions.
Monday, October 9: Word done pumping fluids, Hydrostatic Pressure, and Hydrostatic Force
The Webwork set 11 was delayed until October 9.
In class worksheet. Here are the scanned
solutions.
Friday, October 6: Start of Hydrostatic Pressure and Hydrostatic Force.
Solutions to the quiz on differentiation and integration.
Wednesday, October 4:
Two Physics Applications of the definite integral,
Work
and
hydrostatic pressure.
In class worksheet. Here are the scanned
solutions.
Tuesday, October 3:
I have a lot of supplementary videos for this section, since
last semester the class was canceled by snow days and I went virtual.
My video,
computing the
Arc Length of y = f(x), on YouTube.
The area of the surface obtained when the curve \(y = f(x)\) on the interval \(a \leq x \leq b\) is
rotated about the \(x\)-axis is \(\displaystyle A = \int_a^b 2 \pi f(x) \sqrt{1 + (f'(x))^2} dx\).
Area of a Surface of Revolution on YouTube.
I made a Mathematica notebook, surfaceArea.nb (also available as surfaceArea.pdf) that makes
a graphic of the surface of revolution in the previous video. You can see my
YouTube video
describing that notebook,
and the famous Garbriel's horn
shape that has finite volume and infinite surface area.
Both the worksheet and one of the webwork problems involve hyperbolic trig functions.
Here is a YouTube video introducing Hyperbolic Sine and Cosine.
This adds a few new differentiation formulas you need to memorize. We also had
a new trig identity you needed to memorize to do some integrals. So here is
the MAT 137 version of the
Differentiation Shortcuts
you need to know.
In class worksheet.
Here are the scanned
solutions.
Monday October 2:
Average Value of a Function on an interval, and
Arc Length.
The average value of a function \(f\) on the interval \(a \leq x \leq b\) is \(\displaystyle f_{ave} = \frac{1}{b-a}\int_a^b f(x) dx\)
The arc length of the curve \(y = f(x)\) on the interval \(a \leq x \leq b\) is \(\displaystyle L = \int_a^b \sqrt{1 + (f'(x))^2} dx\).
Note: You will not be given these formulas on the exam. You need to memorize them, or derive them for yourself as needed.
Here is a pdf of the worksheet which was handed out on paper today,
and here are the scanned solutions.
Friday September 28:
More about the Volume of a Solid of Revolution using Cylindrical Shells
Quiz about applications of the integral: Computing areas and volumes.
Here are the scanned
solutions to Friday's quiz.
Wednesday, September 27:
Volume of a Solid of Revolution using Cylindrical Shells
A great resource for this and all calculus is from the organic chemistry tutor on youtube.
Here is his video on the shell method.
The "official" textbook has a section with both the slicing method and the shell method:
Whitman Calculus, Section 9.3
The explanations in this textbook are similar to what I did in class.
In class worksheet.
Here are the scanned
solutions.
Tuesday, September 26: Computing volume of solids of revolution using Pancakes and Washers
Here is a picture of the solid in Problem 1 of Set 8:
pdf and mathematica notebook. Note that an extra credit opportunity is included!
In class worksheet.
Here are the scanned
solutions.
Monday, September 25: Computing Volume of
solids by Parallel Slices. (Set 8, Volume by Slices)
\(\displaystyle V = \int_a^b A(x)dx \approx \sum_{i = 1}^n A(x_i) \Delta x\), or \(\displaystyle V = \int_a^b A(y)dx \approx \sum_{i = 1}^n A(y_i) \Delta y\).
This is covered in two sections of Mollik's notes:
Volume of a Solid, and Volume of a Solid of Revolution.
In class worksheet. Here are the scanned
solutions.
Friday, September 22:
More about Area Between Curves.
In class worksheet. Here are the scanned
solutions.
Wednesday, September 20:
Area Between Curves.
In class worksheet. Here are the scanned
solutions.
This worksheet is similar to Problem 3 on the WeBWorK.
Tuesday, September 19 Midterm 1 = Exam 1, in class.
Monday, September 18: Review for exam
Practice exam is on Canvas, with solutions.
Practice on Integration Techniques.
In class worksheet. Here are the scanned
solutions.
Friday, September 15: More about Improper Integrals, including the Comparison Test.
Desmos graph to help you memorize when \(\displaystyle \int_0^1 \frac 1 {x^p} dx\) and \(\displaystyle \int_1^\infty \frac 1 {x^p} dx\) converge.
If \(p > 1\) then \(\int_0^1 \ldots\) diverges and \(\int_1^\infty \ldots\) converges.
If \(p = 1\) then \(\int_0^1 \ldots\) and \(\int_1^\infty \ldots\) both diverge.
If \(p < 1\) then \(\int_0^1 \ldots\) converges and \(\int_1^\infty \ldots\) diverges.
In class worksheet. Here are the scanned
solutions.
Wednesday, September 13: Improper Integrals
Here are some examples of improper integrals, and how they are evaluated:
\(\displaystyle \int_1^\infty \frac{1}{x^p}\, dx = \lim_{b \to \infty} \int_1^b \frac{1}{x^p}\, dx\). The integral is improper since the interval \([1, \infty ) \) is infinitely long.
\(\displaystyle \int_0^1 \frac{1}{x^p}\, dx = \lim_{a \to 0^+} \int_a^1 \frac{1}{x^p}\, dx\). The integral is improper since \( \frac{1}{x^p} \) “blows up” at 0.
In class example: Evaluate \(\displaystyle \int_1^\infty \frac 1 {x^3}dx\), or show that the improper integral diverges. (Desmos graph of
area to compute, with the
answer.)
Here is today's worksheet on improper integrals.
Here are the solutions.
Tuesday, September 12: Numerical Integration
To approximate \(\int_a^b f(x) dx\),
divide the interval \([a, b]\) into \(n\) intervals of length \(\Delta x = \frac{b-a}n\). Define \(x_i = a + i \Delta x\) and \(\bar x_i = a + (i-1/2) \Delta x\).
Also define \( y_i = f(x_i)\) and \( \bar y_i = f(\bar x_i)\).
\(T_n = \frac{ \Delta x }{2} (y_0 + 2 y_1 + 2 y_2 + \cdots + 2 y_{n-1} + y_n)\) is the Trapezoid rule approximation with \(n\) subintervals.
\(M_n = \Delta x (\bar y_1 + \bar y_2 + \cdots + \bar y_n)\) is the Midpoint rule approximation with \(n\) subintervals.
\(S_n = \frac{ \Delta x }{3} (y_0 + 4 y_1 + 2 y_2 + 4 y_3 + \cdots + 2 y_{n-2} + 4 y_{n-1} + y_n)\) is Simpson's rule approximation with \(n\) subintervals. Warning: \(n\) must be even.
Monday, September 11: Partial Fractions
How to integrate some rational functions (ratios of polynomials), for example
\(\int \frac{1}{1-x^2} dx\).
Partial Fractions at Sudipto Mollik's notes. We will us the fact that
\( \displaystyle \int \frac{1}{ax + b}\, dx = \frac 1 a \ln|ax + b| + C\) for all constants \(a \neq 0\) and \(b\).
In class worksheet. Here are the scanned
solutions.
Friday, September 8: More about Set 4
In the 10:20 class I did a calculation of one of parts of Set 4, Problem 8.
Here is a photo of the whiteboard after I corrected the calculation.
We also had a quiz. Here are the scanned solutions.
Wednesday, September 6: More about Set 4
More about how to do integrals of some trig functions, for example \(\int \sin^3(x)(\cos(x))^{2} dx\).
Trig substitution for doing some integrals with square roots, for example \(\int \frac{1}{\sqrt{25+x^2}} dx\).
In class worksheet. Here are the scanned
solutions.
Tuesday, September 5:
More about Parts, and start Trig integrals.
Trigonometric Integrals, and
Trigonometric Substitiutions
The names are similar, but these are two techniques. The links are to Dr. Mallik's notes.
In class worksheet.
Friday, September 1:
More Integration by Parts, prep for the quiz, and the quiz.
In class worksheet on which technique
of integration to use. The scanned solutions are here
Here are the scanned solutions to Friday's quiz. (To appear.)
Wednesday, August 30:
Integration by Parts
This new (for most of us) material is covered in Sudipto's notes.
In class worksheet on Integration by Parts,
with the scanned solutions.
Tuesday, August 29:
Paul's Notes on Linear Approximations. This should help with the last problem in WeBWorK set 1.
The local linearization of \(f\) at \(x = a\) is \(L(x) = f(a) + f'(a)\cdot(x-a)\).
The tangent line approximation is \(f(x) \approx L(x)\) if \(x \approx a\).
Review of three techniques of integration:
(1) By inspection, from a derivative formula.
(2) Algebraic manipulation, and then by inspection.
(3)
\(u\)-substitution.
Today's
practice problems on Integration.
Here are the solutions.
Here is a Desmos Graph illustrating
problem 2.
Monday, August 28, 2023:
The Big picture of Calculus
Differentiation Shortcuts
Required knowledge from Precalculus.
Review problems on Differentiation and Integration.
Here's that page
with solutions.
If this schedule disagrees with what I announced in class, please tell me or send e-mail to Jim.Swift@nau.edu.