# 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. You must bring your University ID card.
2. You are allowed 50 minutes to take the exam.
3. You are allowed to have one-half of one letter (8.5in $\times$ 11in) sized sheet of paper of handwritten notes. Double-sided is OK.
4. You are allowed to have a calculator in the exam. A graphing calculator is OK.
5. No textbook or electronic equipment (other than calculator) allowed.

You can also check out the rules that will be printed on the exam cover page.

#### Exam times

You can take the exam up to two times. You can take it once on Thursday, September 26, and once on Thursday, October 3. Each day, you can choose one of the following times (even if it is different from the time that you are officially registered):

• 11:15 AM - 12:05 PM
• 12:20 PM - 1:10 PM
• 1:25 PM - 2:15 PM

Exam location: Tate Hall, room 20

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

#### Points and due date summary

Total points: 200
Assigned: Sept. 26, 2019, 11:15 a.m.
Due: Oct. 3, 2019, 2:15 p.m.