Class 03: From Pascal to Calculus https://youtu.be/v3j0oiTrj7w (slides here)
Class 04: Review of Calculus I: https://youtu.be/58TLpnIthF8 (slides here)
Class 04: Coding Handouts:
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Go to https://web.williams.edu/Mathematics/sjmiller/public_html/math/handouts/latex.htm for Mathematica / LaTeX
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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: Fundamental Theorems, Simpson's Rule: watch https://www.youtube.com/watch?v=Q1TQtH6POyI (50 minutes, proofs): Video from 2/14/22: https://youtu.be/osKvsXoCuIM (code here, slides here)
Class 10: Review day / application day: no HW due, lecture topics chosen by class. Video 2/28/22: https://youtu.be/9W-qVOSl7h0 (slides here)
Class 13: Review day / application day: no HW due, Birthday problem. Video: 3/7/22: https://youtu.be/Vhk_25kBy8U (slides here)
Class 15: Lecture: Partial Fractions II: https://youtu.be/fmdWptrelmE (part II) slides: pdf Video 3/11/22: https://youtu.be/Mpe9w4TlH14 (slides here)
Class 16: No HW Due: Partial Fractions III and u-Substitution: https://youtu.be/Kw3K6EElo8Y (slides here)
Class 18: No HW DUE: Watch the following videos: Lecture 3/18/21 https://youtu.be/XRCay8-LrWM (prod rule for finite sums, intro sec, geom series formula): slides: (slides here)
- Introduction to Sequences and Series: https://youtu.be/vYArtl-KBr4 slides: pdf
- Sequences and Series II: Part 1: https://youtu.be/VRpfWQ-URLA and Part II: https://youtu.be/BTcaX8UCWfo slides: pdf
- Sequences and Series III: Differentiating under the summation sign, generating functions, guessing integrals: https://youtu.be/_ZRJersOjSY slides: pdf
Class 19: Review day / application day: no HW due. Lecture 4/4/22: Drowning Swimmer, Sequences/Series: https://youtu.be/SuJ7ly2RVQE (slides here) Watch the following videos:
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Integral Test: https://youtu.be/WD6bFf0hW9o slides: pdf
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Review of Series Convergence, Root Test: https://youtu.be/KZDwcP0WAuw slides: pdf
Class 20: Sequences and Series; read handout https://people.math.gatech.edu/~cain/notes/cal10.pdf Lecture 4/6/22: https://youtu.be/OAY3ycA_Yq0 (slides here)
- Homework due: 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 https://people.math.gatech.edu/~cain/notes/cal10.pdf ): Find the limit of the sequence \(a_n = \frac{3n^2+2n-7}{n^2}\).
Class 21: Topic Sequences and Series: Lecture 4/8/22 (Ratio/Root tests): https://youtu.be/37HF1t4czAU (slides here)
- Homework: (1) Cain-Herod https://people.math.gatech.edu/~cain/notes/cal10.pdf : 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.
Class 22: Topic Baseball Lecture: Watch before class: http://youtu.be/tZ9lcdk6hLOUt
- Homework: (1) Cain-Herod https://people.math.gatech.edu/~cain/notes/cal10.pdf 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, ....
Class 23: Taylor Series: Basketball Problem, Taylor Series: https://youtu.be/5J9g8_9gbmQ (slides here)
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No HW due in class
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Watch the following videos before class:
- Taylor Series Introduction, Trig Identities: https://youtu.be/MqAV2OhPoD0 slides: pdf
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Taylor Series II: https://youtu.be/yc_ihgQw7GI slides: pdf
Class 24: Taylor Series: Convergence of Taylor Series, Streaming Information: https://youtu.be/mnWFqigY4Kk (slides here)
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No HW due in class
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Watch the following video before class: https://www.youtube.com/watch?v=t3Pt4E1BeUT
Class 25: Differential Equations: Newton's Laws, Kinematics of Projectiles: https://youtu.be/DVZjUZTA9bk (slides here)
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HW: Due at the start of class
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Watch: Lecture 16: Differential Equations I: https://youtu.be/YUbMk3MvLM0 slides: pdf
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Watch: Lecture 17: Differential Equations II: Integrating Factors: https://youtu.be/RHUeFV__Bak slides: pdf
Class 26: Differential Equations: No class in person (due to takehome)
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HW: Work on HW due at the start of Class 27
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Watch: Differential Equations III: Slope Fields, Euler's Method: https://youtu.be/6PGFIThLhmo slides: pdf
Class 27: Farmer Brown (and Bob) Problem, Simplify and Symmetry, Volumes of Revolution: https://youtu.be/21seJ_Lh5Ys (slides here)
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HW: (1) Solve y'(x) = x^2. (2) Solve dy/dx = y. (3) Solve d^2 y / dx^2 = y. (4) Solve d^2 y / dx^2 = -y. (5) Solve dy/dx = x + xy. (6) A player hits a fastball 5 ft above the ground. If the ball leaves his bat at 100mph, what angle should he hit it so that the horizontal distance is maximized, and what is that maximum distance? Note: You can use a computer to estimate / approximate some quantities. (6) Imagine the Earth has a uniform density of 1 kg/meter^3. If there is a massive explosion and all the Earth more than half the radius from the center shoots off into space, what is the ratio of the new acceleration due to gravity at the reduced Earth's surface relative to the original acceleration at the surface of the full Earth?
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Watch: Volumes of Revolution, Polar Areas: https://youtu.be/SXZJIIDIV_s slides: pdf
- HW: (1) Find the volume of a sphere of radius R (without loss of generality you can argue you can do a sphere of radius 1 and multiply by R^3). (2) Find the volume of region formed by rotating y = sin(x) about the x axis, with x ranging from 0 to 2pi. (3) Find the volume of the region formed by rotating the region between y = sin(x) and y = 0 about the y-axis for x ranging from 0 to pi. You may use a program such as Mathematica to evaluate the integral.
Class 30: Library Trip (no written HW)
Class 31: Difference Equations: 5/2/22: https://youtu.be/r8xr1gcwZb4 (slides here)
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Watch: Double Plus Ungood: https://www.youtube.com/watch?v=Esa2TYwDmwA&t=309s
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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/0v4oDfDXwjM (slides here)
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Watch: https://youtu.be/jFgCKfUTOQ8 (Differential Equations)
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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: Review Day: Watch Thinking About Formulas: 5-6-22: https://youtu.be/JlQOL2GkgVQ (slides here)
Class 34: Application: Mathematical Modeling I: German Tank Problem: 5-9-22: https://youtu.be/_ULiGrhEDiQ (slides pdf)
Class 35: Application: Mathematical Modeling II: German Tank Problem: 5-11-22: https://youtu.be/ipVkQwQ6Z8I (slides pdf)
Class 36: M&M game: Lecture from Probability: M&M Game: 5-13-22: https://youtu.be/l_MRNRKrDUI (slides) (paper)