Below is a tentative list of homework problems for Math 140. In general, HW is due at the start of each class, and we will typically cover on the order of one section a day. All problems are worth 10 points; I will drop whatever assignment helps your HW average the most. Homework solutions are available here.
HOMEWORK: Homework problems listed below; suggested problems collected together at the end. Note dates MAY change
Class 14: Takehome midterm - no class
Class 15: (no HW due) Partial Fractions II: https://youtu.be/fmdWptrelmE (part II) slides: pdf
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)
Class 19: Review day / application day: no HW due. Watch the following videos:
Integral Test: https://youtu.be/WD6bFf0hW9o slides: pdf
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)
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)
No HW due in class
Watch the following videos before class:
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)
No HW due in class
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)
HW: Due at the start of class:
Watch: Lecture 16: Differential Equations I: https://youtu.be/YUbMk3MvLM0 slides: pdf
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)
HW: None, work on problems due at Class 27.
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)
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?
Watch: Volumes of Revolution, Polar Areas: https://youtu.be/SXZJIIDIV_s slides: pdf
Class 30: Library Trip (no written HW)
Class 31: Difference Equations: 5/2/22: https://youtu.be/r8xr1gcwZb4 (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: 5/4/22: https://youtu.be/0v4oDfDXwjM (slides here)
Watch: https://youtu.be/jFgCKfUTOQ8 (Differential Equations)
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: (slides pdf)
Class 35: Application: Mathematical Modeling II: German Tank Problem: (slides pdf)
Class 36: Review Day