assignments
There will be regular homework assignments. The assignments contribute 40% of your grade. There will be three quizzes; expect the first two after we complete Chapters 7 and 9 and the third at the end of the semseter. Your best two of the three quizzes contribute 10% each to your grade. A three hour final exam contributes 40% to your grade.
Attendance, class participation and prompt submission of homework are expected. Your performance in these areas will influence your final result by up to one letter grade.
due Fri Jan 27
Assignment 1
- Strang: 3.2.42, 3.8.20, 4.2.26, 6.2.44.
due Wed Feb 8
Assignment 2
Evaluate the following integrals.
- \( I_1 = \int \sin^2x \cos^3 x \ dx \)
- \( I_2 = \int \sin^2x \cos^4 x \ dx \)
- \( I_3 = \int \sin x \sin 2x \sin 3x \ dx \)
- \( I_4 = \int \frac{x}{x^4+5x^2+4} \ dx \)
- \( I_5 = \int \frac{x \ln x}{\sqrt{x^2 - 1}} \ dx \)
- \( I_6 = \int e^{3x} \cos{5x} \ dx \)
- \( I_7 = \int \frac{x}{\sqrt{1-x^2}} \ dx\)
- \( I_8 = \int_0^{\infty} \frac{1}{x^2 + 5x + 6} \ dx \)
due Sat Feb 18
Assignment 3
- From Strang: 8.1.12, 8.1.22, 8.1.32, 8.2.4, 8.2.12, 8.3.4.
- Valentine's Day Special: Find a function that when revolved around the y-axis gives something that's roughly heart-shaped. Set-up the integral that gives its volume. Solve if you are feeling ambitious.
due Mon Feb 27
Assignment 4
- From Strang: 6.7.26, 6.7.36, 9.1.18, 9.2.14, 9.3.10, 9.3.28.
Quiz 1
due Wed Mar 29
Assignment 5
due Mon Apr 10
Assignment 6
These are mostly from Stewart; I'm typing them out here because of the differing numbers between editions (the numbers correspond to the 4th ed.).
- 9.2.10. Sketch a direction field for the differential equation \( y' = xy + y^2\). Use it to sketch three solution curves.
- 9.3.18. Solve the equation \(e^{-y}y' + \cos{x} = 0\) and graph several members of the family of solutions. How does the solution curve change as the constant of integration varies?
- 9.5.14. Another model for a population growth function is given by the Gompertz function, which is a solution to the differential equation \( \frac{dP}{dt} = c \ln(\frac{K}{P})P\) where \(c\) is a constant and \(K\) is the carrying capacity.
- Solve this differential equation.
- Compute \(\lim_{t \rightarrow \infty} P(t)\).
- Graph the Gompertz growth function for \( K = 1000\), \( P(0) = 100\) and \( c=0.05\). How does it compare to the logistic growth function?
- The logistic growth function grows fastest when \( P = K/2 \) (bonus points: prove this). Use the Gompertz differential equation to show that the Gompertz function grows fastest when \( P = K/e \).
- 9.6.18. Solve the initial value problem: \( (1+x^2)y' + 2xy = 3 \sqrt{x} \) where \(y(0)=2\).
- Zombies. Design a system of differential equations similar in spirit to the Lotka-Volterra equations that model the zombie apocalypse. (You might, for example, have three populations: healthy humans, infected humans that are not yet zombies, and zombies. How do these populations interact?)
Quiz 2
due Mon Apr 24
Assignment 7
Numbers from 4th Edition Stewart:
- From 11.7. Test the following series for divergence or convergence.
- \( \sum_{n=1}^{\infty} \frac{3^n n^2}{n!} \)
- \( \sum_{n=1}^{\infty} \frac{1}{n^2 + n} \)
- \( \sum_{n=1}^{\infty} \frac{(-1)^n n}{(n+1)(n+2)} \)
- 11.10.6 Find the Maclaurin series for \( \ln(1+ x) \).
- 11.10.38 Evaluate \( \int \frac{\sin{x}}{x} \ dx \) as an infinite series.
- Problems Plus 5. Construct the snowflake curve by starting with a equilateral triangle with sides of length 1 and proceeding as we did in class.
- Let \( s_n \), \(l_n\) and \(p_n\) represent the number of sides, the length of a side and the total length of the curve after step \( n \). Find formulas for \( s_n \), \(l_n\) and \(p_n\).
- Find \( \lim_{n \rightarrow \infty} p_n \)
- Sum an infinite series to find the area enclosed by the snowflake curve.
- Go to the wikipedia page on approximating \( \pi \). Choose one of the nasty-looking infinite-sum approximation formulas and use the first few terms to approximate \( \pi \). How accurate is the result?
due Wed May 3
Assignment 8
- Investigate the bifurcation diagram on p.353 of HSD (or, better, find one online that you can zoom in on). Describe as many low-period windows as you can. Is there a pattern to how they are related?
- Find the smallest mini-Mandelbrot you can in the Mandelbrot set. What is the approximate scaling factor compared to the original? Same question, but with the stipulation that the mini-Mandelbrot does not lie on the real line.
- Find the coordinates for points in various lobes of the Mandelbrot set. What is their behaviour as the function is iterated?
Quiz 3
Final Exam
Final Grade
