# Math Insight

### Overview of: Discrete dynamical systems exam

The discrete dynamical systems module exam for Math 1241 is based on part 2 from the Math 1241 thread, with the knowledge from part 1 also assumed.

#### Material for the exam

The following highlights what is and what is not good potential material for this exam.

1. Functions, variables, and parameters (Functions from Math 1241 thread).

1. Be comfortable with variables, parameters, and functions. Manipulating them is a key basic skill for this course.
2. Of particular importance is being able to work with parameters that don't have a specific numerical value.
3. Exam problems are likely to be more involved than a simple calculation from this section, but the ability to do such calculations is assumed.
2. Exponentials and logarithms (Exponentials and logarithms from Math 1241 thread).

You should be able to manipulate exponential functions and logarithms.

3. Discrete-time dynamical systems (Discrete dynamical system from Math 1241 thread).

Discrete-time dynamical systems will be the core this exam.

Key points include:

1. Calculating the first few points of a trajectory from an initial condition.
2. Solving the system if it is linear without a constant term. In this case, one obtains exponential growth or decay.
3. Estimating a solution with cobwebbing using the graph of the updating function.
4. Finding equilibria analytically and graphically.
5. Determining the stability of an equilibrium via cobwebbing or by observing trajectories near the equilibrium.
6. Determining the doubling time or the half-life of a linear dynamical system, or the time it takes to obtain a certain value.
7. Analyzing the influence of parameters on the behavior of the system.
8. Setting up a dynamical system from a verbal description of a system. This process includes
1. carefully defining the notation used.
2. possibly introducing parameters to describe quantities whose numerical value is unknown or whose value may change between experiments.
3. possibly estimating parameters of the system from data.
The dynamical systems you would be asked to set up will be linear (possibly with a constant term) unless any nonlinear component is clearly specified. They would be along the lines of the bacteria growth or penicillin clearance examples. (Sorry, no rabbit control challenges on the exam.)

#### Study aids

1. Sample problems

Review questions: Elementary discrete dynamical system problems and Review questions: Elementary discrete dynamical system biology problems contain problems similar to those of the exam. Generate multiple versions of these problems to see a larger array of the possible problems.

2. Quizzes

Online quiz: Quiz 1, Online quiz: Quiz 2, and Online quiz: Quiz 3 contain material that will be on the exam. Even after the due date of quizzes, you can still generate more attempts and check out how well you can answer those questions.

3. Problem Sets

The problem sets from the Math 1241 thread are also good review.

#### Exam rules

1. Permitted items:
1. The exams will be open book, notes, and the internet. You may research the questions on the internet.
2. You are allowed to have a calculator in the exam. A graphing calculator is OK.
3. You may not communicate with another person about the exam, except for the professor or one of the teaching assistants.
1. Show your work, in a reasonably neat and coherent way, in the space provided. All answers must be justified by valid mathematical reasoning. To receive full credit on a problem, you must show enough work so that your solution can be followed by someone without a calculator.
2. Mysterious or unsupported answers will not receive full credit. Your work should be mathematically correct and carefully and legibly written.
3. A correct answer, unsupported by calculations, explanation, or algebraic work will receive no credit; an incorrect answer supported by substantially correct calculations and explanations will receive partial credit.
4. Full credit will be given only for work that is presented neatly and logically; work scattered all over the page without a clear ordering will receive from little to no credit.
3. You will have your registered 50-minute class time to take the exam. The exam will be posted on Canvas 10 minutes before the class period, giving you time to print the exam, if you desire or otherwise prepare for the exam. You will have a 10 minute grace period afterwards to scan your work and post it on Gradescope.
4. You can write your answers on blank (lined is OK) paper, on a printout of the exam PDF, or electronically using your computer. The only requirement is that your answers must eventually be in a PDF file to submit to Gradescope.

#### Exam times

You can take the exam up to two times. You can take it once on Thursday, October 1, and once on Thursday, October 8.

If you take the exam twice, your exam score will be the maximum of your scores from the two attempts.

Exam posting times and submission deadlines based on your section are as follows.

• Sections 11 and 12 (11:15 - 12:05 sections)
Exam posted on Canvas: 11:05 AM
Exam due: 12:05 PM
End of grace period for submission: 12:15 PM
• Sections 13 and 14 (12:20 - 1:10 sections)
Exam posted on Canvas: 12:10 PM
Exam due: 1:10 PM
End of grace period for submission: 1:20 PM
• Sections 15 and 16 (1:25 - 2:15 sections)
Exam posted on Canvas: 1:15 PM
Exam due: 2:15 PM
End of grace period for submission: 2:25 PM

Important note: Check Canvas for the time we consider your exam due (and add a ten minute grace period). Gradescope does not have the ability to modify due dates based on your section, so it will show 2:25 PM as the deadline for everyone. We will manually check the submission time on Gradescope to determine if you submitted the exam on time.

#### Points and due date summary

Total points: 200
Assigned: Oct. 1, 2020, 11:15 a.m.
Due: Oct. 8, 2020, 2:15 p.m.