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♦ Course Videos:

Class 01: https://youtu.be/cS65Uxhaur4 (slides here)

Class 02: Extending P y t h a g o r a s: https://youtu.be/iPvL0XLtXMw (slides here)

Class 03: Probability and Mathematical Modeling I: https://youtu.be/QaCL95AharE (slides online here). Will not do, but the second lecture is here: https://youtu.be/caFcPaQp9GU (slides online here)

Class 04: Review of Calculus I and II: https://youtu.be/rzzyAgQ1KCE (slides here)

Class 04: Coding Handouts:

Go to https://web.williams.edu/Mathematics/sjmiller/public_html/math/handouts/latex.htm for Mathematica / LaTeX

Video: Introductory lecture to using Mathematica (YouTube) and sample notebook for version 6: MathematicaIntroVer6.nb

Practice coding problems and solutions: BasicCoding.nb BasicCoding.pdf

Coding lecture (from Probability: Start at 28:26): https://youtu.be/2agUkFQJtnU?t=1704

Die Problem from Probability (can watch up to 26:13, coding starts at 13:01): https://www.youtube.com/watch?v=ZhPypMaQWhc.

Here is the problem: Have a fair die, roll once, whatever number you roll you get that many dollars. How much would you pay to play the game? What if you have two die, you roll the first and can either keep that number OR you can roll the second die but you must then take that -- how much would you pay to play?

Aces up problem and coding: video: https://youtu.be/sSgjBysixdQ; code file here, pdf here)

Coding general tic-tac-toe: https://youtu.be/zE0zc2hACg4 (code here, code pdf here)

Class 05: Definition and properties of vectors, through dot product)

Part 1: http://youtu.be/fghStY0cqIg Part 2: http://youtu.be/AA2UEYbnkRw
Video from 2/14/22: Vectors, Lines: https://youtu.be/T9VmzAkSbC8 (slides here)

Class 06: http://youtu.be/aIZmP640Sn4 (February 12, 2014: Dot Products, Cross Products, Determinants, Areas). Lecture 2/16/22: https://youtu.be/f8JkOmTuuyY (slides here)
Class 07: http://youtu.be/ft2zViDFXcY (February 17, 2014: Lines)
Class 08: http://youtu.be/np_Icg33Lug (February 19, 2014: Coordinate systems, level sets)

https://youtu.be/gz64jWipVSU

(slides here)

Class 08: BONUS: Programming:
A mathematica template is available here: MathematicaIntroVer6.nb

Mathematica Video Intro (YouTube) A video of an introduction to Mathematica lecture is available here (file).

Section 1: https://youtu.be/WfLd6cZlW7w Section 2: https://youtu.be/RvhO6pp_cvw

Program from today's lecture: BasicCommands2.nb BasicCommands2.pdf
Class 09: http://www.youtube.com/watch?v=71_8kHSAE4w (February 21, 2014: Limits, Defn Partial Derivatives). Video from 2/25/22: https://youtu.be/xolV2TnWvlI (slides here)

Class 10: http://www.youtube.com/watch?v=S6wiYRCiQhs (March 3, 2014: Derivatives). Video from 2/28/22: https://youtu.be/epJbnWCOqCA (slides here)

2/26/20: Mathematica Notebook Section 2: https://youtu.be/v01gK35Eo1Q
3/5/21: Bonus lecture before exam: Section 1 (recording error). Section 2: https://youtu.be/X3GUGb9ZXuo (Level Sets, Spherical/Cylindrical Coordinates)
3/10/21: Bonus lecture: Section 1: https://youtu.be/B12mhUgZrGw (Partial Derivatives) Section 2: https://youtu.be/c9jcVbOVZeQ (Partial Derivatives)

BONUS LECTURE: Programming: Sec 1: https://youtu.be/2xmH6w7xdyo Sec 2: https://youtu.be/2xmH6w7xdyo

Class 11: http://www.youtube.com/watch?v=L1WUWmjvnE0 (March 5, 2014: Optimization)

. Video 3/2/22: https://youtu.be/J24iLtvJCFE (slides here)

BONUS LECTURE: More on Optimization

3/4/20: Section 1: https://youtu.be/QSgieLP8QDY Section 2: https://youtu.be/X5lIDlZuR3w
3/15/21: Section 2: https://youtu.be/CFb9luNvZWk (Chain Rule, Constraints, Linearization)

Class 12: (March 7, 2014: Method of Least Squares)

3/9/20: Section 2: Method of Least Squares: https://youtu.be/I2Z47ypMtBI (my notes on the Method of Least Squares) ( my slides on the method of least squares )
3/17/21: Section 2: Method of Least Squares: https://youtu.be/CXx4FxawgNQ

Class 13: http://youtu.be/Da0cO905Aj8 (March 10, 2014: Linear Approximation): Video 3/7/22: https://youtu.be/6uQFRz2KCkA (dimension to fractals to Newton's method) (slides here)

3/11/20: Sec 1: https://youtu.be/Q8WuIoT06dI Sec 2: https://youtu.be/ap-JM19by2A

Class 14: http://www.youtube.com/watch?v=Q1TQtH6POyI (March 19, 2014: Fundamental Theorem of Calculus in a Day): NO CLASS IN PERSON!!!

3/9/18: Sec 1: https://youtu.be/0bkHRFEwIe0 Sec 2: https://youtu.be/BbAil7J0UGg
3/29/21: Sec 2: https://youtu.be/_SpbvPXyA9w

Due at Class 15:

Read 12.7, 12.8

Video: Watch http://youtu.be/cgIoKYr11sY (March 12, 2014: Chain Rule)

. Video 3/11/22: https://youtu.be/HFQpLfyha5s (slides here)

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:

Read 12.9, 3.1
Twelfth day lecture: Part I: http://youtu.be/FbwGZfkf9P8 Part II: http://youtu.be/RbhK3a308sg (March 14, 2014: Directional Derivatives, Exp Fn, Trig in a Day)
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:

Reading: Read 13.1, 13.2
Thirteenth day lecture: http://youtu.be/pgwC2vOwRuE (March 17, 2014: Lagrange Multipliers)
NO WRITTEN HW DUE. Watch additional lecture: Fourteenth day lecture: http://www.youtube.com/watch?v=Q1TQtH6POyI (March 19, 2014: Fundamental Theorem of Calculus in a Day)
Video from 3/18/22: https://youtu.be/MzVRbIEf1to (slides here)
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:

Read 13.1, 13.2.

Fifteen

th day lecture: http://youtu.be/N8nFFWG_6J4

(April 7, 2014: Integration in two variables)

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.

Intermediate and Mean Value Theorems and Taylor Series (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:

Read 13.2, 13.3, 13.4. Lecture 4/6/22: Interchanging Order, Polar Integration: https://youtu.be/1OwwNrg8x3U (slides here)
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

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)

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)
- Twenty-fifth lecture: https://youtu.be/fa4X1AcGFOA (May 2, 2014: Comparison Test, Implications of Limits of Terms: II)
- 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)
- Twenty-sixth lecture: https://youtu.be/jFgCKfUTOQ8 (Comparison Test, Implications of Limits of Terms: III)
- Optional lecture: http://youtu.be/u3BbXey5dVY (May 5, 2014: Ratio Test, Patton Movie Clip)
- Read multivariable calculus (Cain and Herod) and my lecture notes.
- Homework: no written HW due.
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.
- 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)

Twenty-eight day lecture: http://youtu.be/yr01SLw9t4c (May 12, 2014: Taylor Series: Should have already watched)
5/4/18: Sec 1: https://youtu.be/ultVUCHrl1c Sec 2: https://youtu.be/ss-uBM3lY2g
5/7/18: Sec 1: https://youtu.be/Mepc9GWlQB0
5/9/18: M&M game: Lecture from Probability: M&M Game: https://youtu.be/DFobthridWI (slides) (paper)
5/5/21: Taylor Series and the Birthday Problem: Sec 1: https://youtu.be/FdFDc7sz15k Sec 2: https://youtu.be/ZvsZyGzQYhs
5/11/18: Review: Sec 1: Spherical Integral, Lagrange Multipliers, Level Sets: https://youtu.be/Jgb3z3E58Zw; Sec 2: Spherical Integral, Changing Orders, Series: https://youtu.be/Jgb3z3E58Zw
4/30/21: Sec 1: https://youtu.be/Jq2j4FP87m4 Sec 2: https://youtu.be/Ga1SnvKxSws
Bonus Review Lecture: Twenty-eighth day lecture: http://youtu.be/UfL8o_IetYQ (May 12, 2014: Review by Partitioning: Not doing 2018)
Bonus clip on Tangent Lines: http://youtu.be/lEJ06epMLEQ

Due at Class 29: Circles on Circles, Multivariable Taylor, Taylor Convergence: 5-27-22: https://youtu.be/cTPKck5msBI (slides here)

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: 5/2/22: https://youtu.be/0f2OJ6AMvx0 (slides here)

Watch: Double Plus Ungood: https://www.youtube.com/watch?v=Esa2TYwDmwA&t=309s

For Taylor series, see my handout here (essentially just pages 2 and 3).

Homework #22: (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, ....

Video on logarithms: https://youtu.be/-SsbkPaB6j8

Here are some videos you can watch to help review.

Review: Spherical Integral, Lagrange Multipliers, Level Sets: https://youtu.be/Jgb3z3E58Zw

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

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: 5-4-22: https://youtu.be/C3E4J2Q1OW4 (slides here)

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.

Class 33: Green's Theorem in a Day: 5-6-22: https://youtu.be/luld2zsO9mk (slides here)

HW: Watch by the start of class the video from 2013: https://youtu.be/Iq-Og1GAtOQ (2013)
Videos from later year: 5/12/21: Sec 1: https://youtu.be/aQbPrQ82K-Y Sec 2: https://youtu.be/MV3jxoscvh0

Class 34: Application: Mathematical Modeling I: German Tank Problem: 5-9-22; https://youtu.be/J2CJRwhjM-M (slides pdf)

Class 35: Application: Mathematical Modeling II: German Tank Problem: 5-11-22: https://youtu.be/DbVmGKLO_FE (slides pdf)

Class 36: M&M game: Lecture from Probability: M&M Game: 5-13-22 https://youtu.be/H5HOIMAD9VU (slides) (paper)

General Videos:

Review of Calculus I and II: part I part II (right click to download)
Calculus Continuing Education Lectures: Wellesley High School
- Part 1: October 13, 2017
  - Review of differentiation, exponential function, need for proofs, Russell's paradox, inverse funcctions: https://youtu.be/8jmrccy5_Xc
  - Inverse trig functions, trigonometry in a minute, trig differentiation, birthday problem: https://youtu.be/jckjVogsSKs
  - Big No's in math, 0 * oo = -1, Babylonian multiplication, Horner's algorithm, Newton's method, Divide and conquer: https://youtu.be/piUHHHCW3lw
  - Stacking dominoes, harmonic series, integral test, egg drop problem: https://youtu.be/75AtD14hlA4
- Part 2: December 15, 2017
  - Sequences and series: Convergence, Geometric series, Term by term differentiation, Differentiating Identities, IVT, MVT, Fund Thm Calculus: https://youtu.be/sQhZdfFL_Zk
  - First order Taylor theorem, Taylor with remainder, Streaming Video, Fibonacci Numbers: https://youtu.be/H5BdJGfOkRc
How to switch orders of integration (by Cam)
Tangent lines
Introduction to LaTex
Introduction to Mathematica

Lectures: PLAYLIST: https://www.youtube.com/playlist?list=PLcWNuL61dIBjye9hIa8T8Yd0R8WJfr8av

First day lecture: http://youtu.be/Sabcuhsxekg (February 7, 2014: Introduction, Pythagoras)
- 2/7/18: handout: Handout from first day of class is here Section 1 Lecture: https://youtu.be/gfeXKy3fk1M
- 2/19/21: Videos: Section 1: https://youtu.be/nMC-h5oIQY0 Section 2: https://youtu.be/y7bxQL-yQFA
Second day lecture: (February 10, 2014: Definition and properties of vectors, through dot product)
- Part 1: http://youtu.be/fghStY0cqIg Part 2: http://youtu.be/AA2UEYbnkRw (NO PHYSICAL CLASS)
- 2/22/21: Section 1: https://youtu.be/ujJxXuDf-GE Section 2: https://youtu.be/Qdzn_JurhLo (Notation, Vectors, Sniffing out Product Rule)
Third day lecture: http://youtu.be/aIZmP640Sn4 (February 12, 2014: Dot Products, Cross Products, Determinants, Areas)
- 2/12/20: Video from Wednesday's lecture: Sec 1: https://youtu.be/JQAR1xRrGGc Sec 2: https://youtu.be/GcwiBh_OOq0
- 2/24/21: Sec 1: https://youtu.be/7xfYlkp2Abg Sec 2: https://youtu.be/S-82n-lpml8
Fourth day lecture: http://youtu.be/ft2zViDFXcY (February 17, 2014: Lines)
- Slides: Slides from lecture on 2/17/2014 (lines and art).
- 2/17/20: Section 1: https://youtu.be/PZn78-AaY3o Section 2: https://youtu.be/tbVEjeKff-Q
- 2/26/21: Section 1: https://youtu.be/p8d7EdMMcic Section 2: https://youtu.be/qYQExySyBmE
Fifth day lecture: http://youtu.be/np_Icg33Lug (February 19, 2014: Coordinate systems, level sets)

2/19/20: Section 1: https://youtu.be/EXGqCEje2Ow
3/1/21: Section 1: https://youtu.be/4mw-X4aAsfs Section 2: https://youtu.be/GycirhBAW_Q

BONUS LECTURE: MATHEMATICA:

Sixth day lecture: http://www.youtube.com/watch?v=71_8kHSAE4w (February 21, 2014: Limits, Defn Partial Derivatives)

2/24/20: Section 1: https://youtu.be/vnhtigmUfHM
3/3/21: Section 1: https://youtu.be/cXibnH-DT6Q Section 2: https://youtu.be/a6leyhrOMOk

Seventh day lecture: http://www.youtube.com/watch?v=S6wiYRCiQhs (March 3, 2014: Derivatives)

2/26/20: Mathematica Notebook Section 2: https://youtu.be/v01gK35Eo1Q
3/5/21: Bonus lecture before exam: Section 1 (recording error). Section 2: https://youtu.be/X3GUGb9ZXuo (Level Sets, Spherical/Cylindrical Coordinates)
3/10/21: Bonus lecture: Section 1: https://youtu.be/B12mhUgZrGw (Partial Derivatives) Section 2: https://youtu.be/c9jcVbOVZeQ (Partial Derivatives)

BONUS LECTURE: Programming: Sec 1: https://youtu.be/2xmH6w7xdyo Sec 2: https://youtu.be/2xmH6w7xdyo

Eighth day lecture: http://www.youtube.com/watch?v=L1WUWmjvnE0 (March 5, 2014: Optimization)

3/2/20: Section 1: https://youtu.be/YmjNs5onVAY Section 2: https://youtu.be/oZvyU50bUp8
3/12/21: Section 1: https://youtu.be/f8uhNfWv4Tg Section 2: https://youtu.be/1oWBqNOB8F8 (Max/Min, Partial Derivatives)

BONUS LECTURE: More on Optimization

3/4/20: Section 1: https://youtu.be/QSgieLP8QDY Section 2: https://youtu.be/X5lIDlZuR3w
3/15/21: Section 2: https://youtu.be/CFb9luNvZWk (Chain Rule, Constraints, Linearization)

Ninth day lecture: (March 7, 2014: Method of Least Squares)

3/9/20: Section 2: Method of Least Squares: https://youtu.be/I2Z47ypMtBI (my notes on the Method of Least Squares) ( my slides on the method of least squares )
3/17/21: Section 2: Method of Least Squares: https://youtu.be/CXx4FxawgNQ

Tenth day lecture: http://youtu.be/Da0cO905Aj8 (March 10, 2014: Linear Approximation)

3/11/20: Sec 1: https://youtu.be/Q8WuIoT06dI Sec 2: https://youtu.be/ap-JM19by2A

ADJUSTED LECTURES UP TO HERE -- HERENOW

Eleventh day lecture: http://youtu.be/cgIoKYr11sY (March 12, 2014: Chain Rule)

3/13/18: Sec 1: https://youtu.be/g1ZO1k3gzdA Sec 2: https://youtu.be/SSCik_ejvaw
3/11/20: Video: Section 1: https://youtu.be/AZPdiWwfap8
3/19/21: Section 1: https://youtu.be/GAKcz2gGWkU Section 2: https://youtu.be/FrhJStuf2MA

Twelfth day lecture: Part I: http://youtu.be/FbwGZfkf9P8 Part II: http://youtu.be/RbhK3a308sg (March 14, 2014: Directional Derivatives, Exp Fn, Trig in a Day)

3/5/18: Sec 1: https://youtu.be/Ici6M_UJvwk Sec 2: https://youtu.be/4Nopzrgq6Z8
3/24/21: Sec 1: https://youtu.be/aYJwRbMSIjI Sec 2: https://youtu.be/07XZaOiciy4 Directional Derivatives, L'Hopital's Rule, Trig and Pi

Thirteenth day lecture: http://youtu.be/pgwC2vOwRuE (March 17, 2014: Lagrange Multipliers)
- 3/7/18: Sec 2: https://youtu.be/8YOXUPcvP-U
- 3/26/21: Sec 1: https://youtu.be/TE4KweETMeg (Bad Algebra!) https://youtu.be/wLapkwqZBbM (friendlier algebra)
Fourteenth day lecture: http://www.youtube.com/watch?v=Q1TQtH6POyI (March 19, 2014: Fundamental Theorem of Calculus in a Day)
- 3/9/18: Sec 1: https://youtu.be/0bkHRFEwIe0 Sec 2: https://youtu.be/BbAil7J0UGg
- 3/29/21: Sec 2: https://youtu.be/_SpbvPXyA9w
Supplemental lecture: Fractals: From Khan to Frozen (March 20, 2014: Mathematics of Fractals and applications to films)
Fifteenth day lecture: http://youtu.be/N8nFFWG_6J4 (April 7, 2014: Integration in two variables)
- 3/12/18: Sec 1: https://youtu.be/2mLTZRtSAq0 Sec 2: https://youtu.be/U1PEPt1If4Q
Sixteenth day lecture: http://youtu.be/G9d9lcYevnM (April 9, 2014: Iterated integrals, changing order)
- 3/14/18: Sec 1: https://youtu.be/SIej3SZ4p7M Sec 2: https://youtu.be/M6pvAWfNlgU
- 4/5/21: Sec 1: https://youtu.be/JOGfHPScpDQ Sec 2: https://youtu.be/qU-fXr-80-g (issues recording, much missing, see section 1)
Seventeenth day lecture: http://youtu.be/JP78q_ri-4o (April 11, 2014: Polar Change of Variables, Circles and Spheres)
- 4/4/18: Sec 1: https://youtu.be/UBnjwxaTkXg Sec 2: https://youtu.be/IsXFx7ikzIs
- 4/7/21: Sec 1: https://youtu.be/KnSdtuUcTns Sec 2: https://youtu.be/3cJbsyXlWuI
Eighteenth day lecture: http://youtu.be/Nz7ahXOMTus (April 14, 2014: Monte Carlo Integration, Change of Variables for Ellipses)
- No class, traveling
Nineteenth day lecture: http://youtu.be/t3Pt4E1BeUTwt (April 16, 2014: Cylindrical and Spherical Coordinates, Newton's Shell Theorem)
- 4/9/18: Sec 1: https://youtu.be/qVmHBnisZW4 Sec 2: recording issue
- 4/12/21: Sec 1: Recording Issue Sec 2: https://youtu.be/nkBJTSqiwso (spherical integration a la gravity, length of curves)
Extra lecture: Probability and Spheres: Sec 1: https://youtu.be/1PwQevlIoW8 Sec 2: https://youtu.be/0TeX4c2E43A
Twentieth day lecture: http://youtu.be/gFDly_6qOn4 (April 18, 2014: Sabermetrics and Multivariable Calculus - Lecture from 2013)
Twenty-first day lecture: http://youtu.be/MN80UyR2Rh8 (April 21, 2014: Gamma Function and Calc I/II/III review)
Twenty-second day lecture: http://youtu.be/tZ9lcdk6hLOUt (April 23, 2014: Review Class: Perimeter Ellipse, Word Problems, Viewing Algebra): Did baseball lecture instead
Twenty-third day lecture: http://youtu.be/aigdKmu-5ow (April 28, 2014: Geometric and Harmonic Series, Memoryless Processes)
Twenty-fourth day lecture: http://youtu.be/6bf9fjwMs2o (May 2, 2014: Comparison Test, Implications of Limits of Terms)
- 4/18/18: Sec 1: https://youtu.be/C5FpOCX3eR4 Sec 2: https://youtu.be/XnAIGqDrtaQ
- Bonus lecture on sequences and series: http://youtu.be/QZ1_5SmxUo8
- 4/16/21: Sec 1: https://youtu.be/LlKirVntRhQ Sec 2: https://youtu.be/KHuqURbVa80 Comparison Test, L'Hopital, Special Sequences
Lecture from 4/20/2018:
- 4/20/18: Sec 1: https://youtu.be/TJWuCwp1tUo Sec 2: https://youtu.be/tRHtA6U-YsQ
Twenty-fifth lecture: https://youtu.be/fa4X1AcGFOA (May 4, 2014: Comparison Test, Implications of Limits of Terms: II) QZ1_5SmxUo8
- 4/20/18: Sec 1: https://youtu.be/TSvffHyFHgM Sec 2: https://youtu.be/tHDtnF8cf4k Review: https://youtu.be/8EZQKzoWxEo
Twenty-fifth day lecture: http://youtu.be/u3BbXey5dVY (May 5, 2014: Ratio Test, Patton Movie Clip)
- 4/26/21: Sec 1: https://youtu.be/jOCKI4dpmZI Sec 2: https://youtu.be/X-AGHD-Vhh8
Twenty-sixth day lecture: http://youtu.be/ujJbUpCab6M (May 7, 2014: Root Test, Integral Test)
- 5/2/18: Sec 1: https://youtu.be/gaC6R01aTjs (Sec 2 was outside)
- 4/28/21: Sec 1: https://youtu.be/oEMSWOdYnPs Sec 2: https://youtu.be/KrDijs-6OwE
No class on May 9, watched streaming video lecture from last year: https://www.youtube.com/watch?v=WjnHNHPhJtU
Twenty-seventh day lecture: http://youtu.be/yr01SLw9t4c (May 12, 2014: Taylor Series: Should have already watched)
- 5/4/18: Sec 1: https://youtu.be/ultVUCHrl1c Sec 2: https://youtu.be/ss-uBM3lY2g
- 5/7/18: Sec 1: https://youtu.be/Mepc9GWlQB0
- 5/9/18: M&M game: Lecture from Probability: M&M Game: https://youtu.be/DFobthridWI (slides) (paper)
- 5/5/21: Taylor Series and the Birthday Problem: Sec 1: https://youtu.be/FdFDc7sz15k Sec 2: https://youtu.be/ZvsZyGzQYhs
- 5/11/18: Review: Sec 1: Spherical Integral, Lagrange Multipliers, Level Sets: https://youtu.be/Jgb3z3E58Zw; Sec 2: Spherical Integral, Changing Orders, Series: https://youtu.be/Jgb3z3E58Zw
- 4/30/21: Sec 1: https://youtu.be/Jq2j4FP87m4 Sec 2: https://youtu.be/Ga1SnvKxSws
Twenty-eighth day lecture: http://youtu.be/UfL8o_IetYQ (May 12, 2014: Review by Partitioning: Not doing 2018)
- Bonus clip on Tangent Lines: http://youtu.be/lEJ06epMLEQ
Twenty-ninth day lecture: http://youtu.be/4OcxtpxuSJw (May 14, 2014: Special Series, Alternating Series, Pi formulas, Birthday Problem: Not doing 2018)
Review sessions:
- Day 0: Video on logarithms: https://youtu.be/-SsbkPaB6j8 (slides here)
- Day 1: Change of Variables, Odd/Even Functions, Bounding Functions, Comparison Test: http://youtu.be/8EZQKzoWxEo
- Day 2: Mostly series test (one series and we apply all four tests!): http://youtu.be/HANb4mVLOOc
- 5/14/21: Level sets, volume between surfaces, interchanging orders of integration, Taylor series: Sec 1: https://youtu.be/_o4-uqfDFmI
- 5/14/21: Taylor series (similar to section 1), Integral Test: Sec 2: https://youtu.be/_mvKJzvZjV0

♦ Course Videos 2018:

General Videos:

Review of Calculus I and II: part I part II (right click to download)

How to switch orders of integration (by Cam)

Tangent lines

Introduction to LaTex

Introduction to Mathematica

Lectures: PLAYLIST: https://www.youtube.com/playlist?list=PLcWNuL61dIBjye9hIa8T8Yd0R8WJfr8av

First day lecture: http://youtu.be/Sabcuhsxekg (February 7, 2014: Introduction, Pythagoras)
- 2/2/18: Sec 1: https://youtu.be/h8UiShlawOs Sec 2: https://youtu.be/8At4WaOLZkU
Second day lecture: (February 10, 2014: Definition and properties of vectors, through dot product)
- Part 1: http://youtu.be/fghStY0cqIg Part 2: http://youtu.be/AA2UEYbnkRw
- 2/5/18: Sec 1: https://youtu.be/WsStSGvorgE Sec 2: https://youtu.be/qNn9bs7ETWQ
Third day lecture: http://youtu.be/aIZmP640Sn4 (February 12, 2014: Dot Products, Cross Products, Determinants, Areas)
- 2/7/18: Sec 1: https://youtu.be/toKszypxiKE Sec 2: https://youtu.be/N8_I5Ao6B24
Fourth day lecture: http://youtu.be/ft2zViDFXcY (February 17, 2014: Lines)
- Slides: Slides from lecture on 2/17/2014 (lines and art).
- 2/9/18: Sec 1: https://youtu.be/0WH7jdUKizQ Sec 2: https://youtu.be/B3a_Bn1mmFs
Fifth day lecture: http://youtu.be/np_Icg33Lug (February 19, 2014: Coordinate systems, level sets)

2/12/18: Sec 1: https://youtu.be/r5SGAMFiVdE Sec 2: https://youtu.be/IkEGfXhbhhc

Sixth day lecture: http://www.youtube.com/watch?v=71_8kHSAE4w (February 21, 2014: Limits, Defn Partial Derivatives)
- 2/14/18: Sec 1: https://youtu.be/aKPWpMW0lYM Sec 2: https://youtu.be/SxF7Eg0RY7g
Seventh day lecture: http://www.youtube.com/watch?v=S6wiYRCiQhs (March 3, 2014: Derivatives)
- 2/19/18: Sec 1: https://youtu.be/BxqloF6FNLg Sec 2: https://youtu.be/vA071GquwGo Mathematica Notebook
Eighth day lecture: http://www.youtube.com/watch?v=L1WUWmjvnE0 (March 5, 2014: Optimization)

2/21/19: Sec 1: https://youtu.be/hS1XXbXTdNk Sec 2: https://youtu.be/u4R6NRIEWwA

BONUS LECTURE: Programming: Sec 1: https://youtu.be/2xmH6w7xdyo Sec 2: https://youtu.be/2xmH6w7xdyo

Ninth day lecture: (March 7, 2014: Method of Least Squares)

2/26/18: Sec 1 and Sec 2: https://youtu.be/Wnth5d0oT1I

Tenth day lecture: http://youtu.be/Da0cO905Aj8 (March 10, 2014: Linear Approximation)
- 2/28/18: Sec 1: https://youtu.be/Q8WuIoT06dI Sec 2: https://youtu.be/ap-JM19by2A
Eleventh day lecture: http://youtu.be/cgIoKYr11sY (March 12, 2014: Chain Rule)
- 3/2/18: Sec 1: https://youtu.be/g1ZO1k3gzdA Sec 2: https://youtu.be/SSCik_ejvaw
Twelfth day lecture: Part I: http://youtu.be/FbwGZfkf9P8 Part II: http://youtu.be/RbhK3a308sg (March 14, 2014: Directional Derivatives, Exp Fn, Trig in a Day)

3/5/18: Sec 1: https://youtu.be/Ici6M_UJvwk Sec 2: https://youtu.be/4Nopzrgq6Z8

Thirteen

th day lecture: http://youtu.be/pgwC2vOwRuE (March 17, 2014: Lagrange Multipliers)

3/7/18: Sec 2: https://youtu.be/8YOXUPcvP-U

Fourteenth day lecture: http://www.youtube.com/watch?v=Q1TQtH6POyI (March 19, 2014: Fundamental Theorem of Calculus in a Day)
- 3/9/18: Sec 1: https://youtu.be/0bkHRFEwIe0 Sec 2: https://youtu.be/BbAil7J0UGg
Supplemental lecture: Fractals: From Khan to Frozen (March 20, 2014: Mathematics of Fractals and applications to films)
Fifteenth day lecture: http://youtu.be/N8nFFWG_6J4 (April 7, 2014: Integration in two variables)
- 3/12/18: Sec 1: https://youtu.be/2mLTZRtSAq0 Sec 2: https://youtu.be/U1PEPt1If4Q
Sixteenth day lecture: http://youtu.be/G9d9lcYevnM (April 9, 2014: Iterated integrals, changing order)
- 3/14/18: Sec 1: https://youtu.be/SIej3SZ4p7M Sec 2: https://youtu.be/M6pvAWfNlgU
Seventeenth day lecture: http://youtu.be/JP78q_ri-4o (April 11, 2014: Polar Change of Variables, Circles and Spheres)
- 4/4/18: Sec 1: https://youtu.be/UBnjwxaTkXg Sec 2: https://youtu.be/IsXFx7ikzIs
Eighteenth day lecture: http://youtu.be/Nz7ahXOMTus (April 14, 2014: Monte Carlo Integration, Change of Variables for Ellipses)
- No class, traveling
Nineteenth day lecture: http://youtu.be/t3Pt4E1BeUTwt (April 16, 2014: Cylindrical and Spherical Coordinates, Newton's Shell Theorem)
- 4/9/18: Sec 1: https://youtu.be/qVmHBnisZW4 Sec 2: recording issue
Extra lecture: Probability and Spheres: Sec 1: https://youtu.be/1PwQevlIoW8 Sec 2: https://youtu.be/0TeX4c2E43A
Twentieth day lecture: http://youtu.be/gFDly_6qOn4 (April 18, 2014: Sabermetrics and Multivariable Calculus - Lecture from 2013)
Twenty-first day lecture: http://youtu.be/MN80UyR2Rh8 (April 21, 2014: Gamma Function and Calc I/II/III review)
Twenty-second day lecture: http://youtu.be/tZ9lcdk6hLOUt (April 23, 2014: Review Class: Perimeter Ellipse, Word Problems, Viewing Algebra)
Twenty-third day lecture: http://youtu.be/aigdKmu-5ow (April 28, 2014: Geometric and Harmonic Series, Memoryless Processes)
- 4/16/18: Sec 1: https://youtu.be/pVwg8eg-Qi4 Sec 2: https://youtu.be/ZplES7XRkjg
Twenty-fourth day lecture: http://youtu.be/6bf9fjwMs2o (May 2, 2014: Comparison Test, Implications of Limits of Terms)
- 4/18/18: Sec 1: https://youtu.be/C5FpOCX3eR4 Sec 2: https://youtu.be/XnAIGqDrtaQ
- Bonus lecture on sequences and series: http://youtu.be/QZ1_5SmxUo8
Lecture from 4/20/2018:
- 4/20/18: Sec 1: https://youtu.be/TJWuCwp1tUo Sec 2: https://youtu.be/tRHtA6U-YsQ
Twenty-fifth lecture: https://youtu.be/fa4X1AcGFOA (May 4, 2014: Comparison Test, Implications of Limits of Terms: II)
- 4/20/18: Sec 1: https://youtu.be/TSvffHyFHgM Sec 2: https://youtu.be/tHDtnF8cf4k Review: https://youtu.be/8EZQKzoWxEo
Twenty-fifth day lecture: http://youtu.be/u3BbXey5dVY (May 5, 2014: Ratio Test, Patton Movie Clip)
Twenty-sixth day lecture: http://youtu.be/ujJbUpCab6M (May 7, 2014: Root Test, Integral Test)
No class on May 9, watched streaming video lecture from last year: https://www.youtube.com/watch?v=WjnHNHPhJtU
5/2/18: Sec 1: https://youtu.be/gaC6R01aTjs (Sec 2 was outside)
Twenty-seventh day lecture: http://youtu.be/yr01SLw9t4c (May 12, 2014: Taylor Series: Should have already watched)
- 5/4/18: Sec 1: https://youtu.be/ultVUCHrl1c Sec 2: https://youtu.be/ss-uBM3lY2g
- 5/7/18: Sec 1: https://youtu.be/Mepc9GWlQB0
- 5/9/18: M&M game: Lecture from Probability: M&M Game: https://youtu.be/DFobthridWI (slides) (paper)
- 5/11/18: Review: Sec 1: Spherical Integral, Lagrange Multipliers, Level Sets: https://youtu.be/Jgb3z3E58Zw; Sec 2: Spherical Integral, Changing Orders, Series: https://youtu.be/Jgb3z3E58Zw
Twenty-eighth day lecture: http://youtu.be/UfL8o_IetYQ (May 12, 2014: Review by Partitioning: Not doing 2018)
- 5/10/21: Review of Convergence Tests for Series: Sec 1: https://youtu.be/lep1YxcXrfQ (has physics length contraction at end) Sec 2: https://youtu.be/nG6A-2APq18
- Bonus clip on Tangent Lines: http://youtu.be/lEJ06epMLEQ
Twenty-ninth day lecture: http://youtu.be/4OcxtpxuSJw (May 14, 2014: Special Series, Alternating Series, Pi formulas, Birthday Problem: Not doing 2018)
Twenty-ninth day lecture: Green's Theorem in a Day: https://youtu.be/Iq-Og1GAtOQ (2013)
- 5/12/21: Sec 1: https://youtu.be/aQbPrQ82K-Y Sec 2: https://youtu.be/MV3jxoscvh0
Review sessions:
- Day 1: Change of Variables, Odd/Even Functions, Bounding Functions, Comparison Test: http://youtu.be/8EZQKzoWxEo
- Day 2: Mostly series test (one series and we apply all four tests!): http://youtu.be/HANb4mVLOOc

Videos of the week:

Week 1: Fibonacci numbers (click here for more on the Fibonacci numbers).
Week 2: Light cycle scene from Tron.
Week 3: 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.
Week 4: The Battle of Trafalgar: Animated battle clip Facts on the battle Best animated re-enactment I could find (For more on this see the Additional Comments.)
Week 5: Mandelbrot zoom: video 1, video 2. Here's a cubic fractal zoom. (other copies: Videos: Mandelbrot set Newton Fractal (cubic) )
- Mathematica notebook for Newton's Method (pdf version here)
Week 6: Trick shots in pool (directions and constraints!).
- Week 6: Audio of the semester: I'm my own Grandpa; graphical version here (don't view this until you try to graph it!). Actually, this is a much better illustration of the relationships.
Week 7: coin sorting (this leads to Lebesgue's Measure Theory).
Week 8: Pacman (actually Ms. Pacman) You can play Pacman online here.
Week 9: Stacking blocks
Week 10: Conway's Game of Life: Gosper's glider gun, and a breeder leaving Gosper glider guns in its wake.
Week 11: The video of the week is a TED lecture (these are great talks, and worth hearing). We'll do Malcolm Gladwell on Spaghetti Sauce (but only a snippet starting around 5 minutes).

♦ Interesting news articles involving math (see also the course disclaimer about not suing me!)

Election fraud:

♦ Interesting videos

TED Lectures: http://www.ted.com/talks

Did You Know (2009 version)

Yes, Prime Minister: Manipulation Polls (YouTube video)

♦ Course disclaimer

I may occasionally say things such as `Probability is one of the most useful courses you can take' or 'If you know probability, stats and a programming language then you'll always be able to find employment'. I really should write `you should always be able to find employment', as nothing is certain. Thus, please consider yourself warned and while you may savor the thought of suing me and/or Williams College, be advised against this! I'm saying this because of the recent lawsuit of a graduate who was upset that she didn't have a job, and sued her school!