♦ Course Videos: 2023
Class 01: Introduction: https://youtu.be/tEkxqje0_A8 (slides here)
Class 02: Review of Calc I: https://youtu.be/C73M7A-KN54
♦ Course Videos: 2022
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
Class 10: http://www.youtube.com/watch?v=S6wiYRCiQhs (March 3, 2014: Derivatives). Video from 2/28/22: https://youtu.be/epJbnWCOqCA (slides here)
Due at Class 16:
Due at Class 17:
Due at Class 18:
Reading: Read 13.1, 13.2
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.
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.
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:
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)
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.
Read multivariable calculus (Cain and Herod) and my lecture notes.
Read multivariable calculus (Cain and Herod) and my lecture notes.
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 \(u = xy, v = y/x\).
Twenty-eight day lecture: http://youtu.be/yr01SLw9t4c (May 12, 2014: Taylor Series: Should have already watched)
Due at Class 29: Circles on Circles, Multivariable Taylor, Taylor Convergence: 5-27-22: https://youtu.be/cTPKck5msBI (slides here)
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
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.
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)
2/19/21: Videos: Section 1: https://youtu.be/nMC-h5oIQY0 Section 2: https://youtu.be/y7bxQL-yQFA
2/22/21: Section 1: https://youtu.be/ujJxXuDf-GE Section 2: https://youtu.be/Qdzn_JurhLo (Notation, Vectors, Sniffing out Product Rule)
2/24/21: Sec 1: https://youtu.be/7xfYlkp2Abg Sec 2: https://youtu.be/S-82n-lpml8
2/26/21: Section 1: https://youtu.be/p8d7EdMMcic Section 2: https://youtu.be/qYQExySyBmE
BONUS LECTURE: MATHEMATICA:
Sixth
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
3/26/21: Sec 1: https://youtu.be/TE4KweETMeg (Bad Algebra!) https://youtu.be/wLapkwqZBbM (friendlier algebra)
ADDITIONAL LECTURE: https://youtu.be/grs3Pt4E1BeUTw
4/16/21: Sec 1: https://youtu.be/LlKirVntRhQ Sec 2: https://youtu.be/KHuqURbVa80 Comparison Test, L'Hopital, Special Sequences
Twenty-fifth lecture: https://youtu.be/fa4X1AcGFOA (May 4, 2014: Comparison Test, Implications of Limits of Terms: II) QZ1_5SmxUo8
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)
5/14/21: 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:
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
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
5/12/21: Sec 1: https://youtu.be/aQbPrQ82K-Y Sec 2: https://youtu.be/MV3jxoscvh0
♦ Interesting news articles involving math (see also the course disclaimer about not suing me!)
♦ Interesting videos
♦ Course disclaimer