### Overview of: Final exam

The final exam for Math 1241 is based on parts 2-6 from the Math 1241 thread. It is a cumulative exam, meaning it covers all the materials from the previous four modules, plus the new material on autonomous differential equations.

#### Material for the exam

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

See discrete dynamical system exam for a description of material and study aids for this section.

Differentiation (The derivative from Math 1241 thread).

See derivative exam for a description of material and study aids for this section.

Applications of differentiation (Applications of differentiation from Math 1241 thread).

See applications of differentiation exam for a description of material and study aids for this section.

Pure-time differential equations (Pure-time differential equations from Math 1241 thread).

See integration exam for a description of material and study aids for this section.

Autonomous differential equations (Autonomous differential equations from Math 1241 thread).

Solving linear equations (Introduction and solution to linear equations from Math 1241 thread).

- We can solve a linear autonomous differential equation such $\diff{x}{t}=ax$. The general solution is an exponential: $x(t)=Ce^{at}$.
- If $a \lt 0$, we have exponential decay. If $a \gt 0$, we have exponential growth.
- Given an initial condition $x(0)=x_0$, we can change the general solution with arbitrary constant $C$ to the specific solution $x(t)=x_0e^{at}$.

Solving a differential equation graphically (Graphical methods from Math 1241 thread).

- For the autonomous equation $\diff{u}{t} = g(u)$, determine where $u(t)$ is increasing, decreasing, or constant from a plot of $g(u)$. Use this information to sketch the solution $u(t)$. (Here we use $u$ for the state variable, but of course, the actual letter will be different for different problems.)
- One can summarize where $u(t)$ is increasing, decreasing, or constant using a phase-line diagram, with points for equilibria and arrows (a vector field) showing the direction of movement.
- Be able to go between a plot of $g$ versus $u$ (where $u$ is the horizontal axis) and plot of $u$ versus $t$ (where $u$ is the vertical axis). In the phase line, $u$ is also the horizontal axis.

Stability of equilibria (Stability of equilibria from Math 1241 thread).

- For the autonomous equation $\diff{u}{t} = g(u)$, the equilibria are the points where the rate of change is zero, i.e., the values of $u$ where $g(u)=0$.
- One can determine the equilibria analytically (solve $g(u)=0$) or graphically (find points where plot of $g(u)$ crosses the $u$-axis).
- From the graph of $g(u)$, or the vector field on the phase-line, determine the stability of equilibria.
- Use the stability theorem to determine stability of the equilibria without looking at the graph of $g(u)$. The stability theorem says for an equilibrium $u_e$ (with $g(u_e)=0$), the equilibrium is stable if $g'(u_e) \lt 0$ and unstable if $g'(u_e) \gt 0$.

Numerical solution (Numerical solution from Math 1241 thread).

- We can approximate the solution to an autonomous differential equation $\diff{v}{t} = f(v)$ using the Forward Euler algorithm.
- To use Forward Euler, divide time into interval of length $\Delta t$.
- If we know the value of the state variable $v$ at time $t = a$, we can take one Forward Euler step to estimate the value of $v$ at time $t=a+\Delta t$.
- Plug in the known value $v(a)$ into the right hand side of the differential equation, getting the number $f(v(a))$, which is the slope $\diff{v}{t}$ at time $t=a$.
- Assume that from $t=a$ to $t=a+\Delta t$, the slope doesn't change. In other words, approximate $v(t)$ by its linear approximation at $a$: $v(t) \approx L_a(t) = v(a) + f(v(a))(t-a)$.
- Calculate the value of $v$ at $t=a+\Delta t$ using the linear approximation.
- The resulting formula is that $v(a+\Delta t) \approx L_a(a+\Delta t) = v(a)+f(v(a))\Delta t.$

- Repeatedly apply Forward Euler steps to march forward in time. At each time step, calculate a new slope, march forward for a time step of $\Delta t$ with that slope (i.e., add $\Delta t$ times that slope to your previous value), and obtain an estimate for $v$ at the next time step.

Bifurcations (Bifurcations from Math 1241 thread).

- If a dynamical system depends on a parameter $\beta$, which we might write as $\diff{w}{t} = f(w,\beta)$, we can look for qualitative changes in the behavior of the dynamical system as we change $\beta$.
- If the stability or number of equilibria changes as we change $\beta$, we say we have a bifurcation.
- A bifurcation point is the value of $\beta$ where such a change occurs.
- Given a sequence of graphs of $f(w,\beta)$ or a sequence of phase lines of the dynamical system for different values of $\beta$, sketch a bifurcation diagram, which is a graph of equilibria on a plot of $w$ versus $\beta$.
- To draw a bifurcation diagram, draw the equilibria as curves, showing how the value of the equilibrium changes with $\beta$. Indicate which equilibria are stable and which are unstable. (Usually, we draw stable equilibria with solid lines and unstable equilibria with dashed lines.)
- Be able to interpret a bifurcation diagram. Given a bifurcation diagram, determine the phase line for different parameter values.

#### Study aids

Review problems

The majority of the questions that may appear in the final exam are available so that you can practice them. In both these problems and on the actual exam, the set of problems as well as values of numbers, variables, parameters, and other quantities are selected randomly. You will want to generate multiple versions of the problems to see the larger array of problems. Given the random nature, we cannot guarantee that you will actually see all the problems that will appear on the exam. But the more problems you work on, the greater the chance you will work on problems that will show up on the test.

The format of the exam questions may differ from the practice problems in that the exam problems may have been modified to remove any answer blanks and the instructions specific to entering answers in the right format for computer grading.

For all problems on this exam, you will be expected to

**show your work**, and the grading will be based on the work that you show.Free form questions

See the problems listed in the first four exams (1 2 3 4). In addition, Review questions: Single autonomous differential equation problems contains short answer problems that reflect the format of questions as they would appear on the exam.

Computer-scored problems

See the problems listed in first four exams (1 2 3 4). In addition, Review questions: Simple linear differential equations practice, Review questions: Bifurcation of single differential equation practice, Online quiz: Quiz 9, Online quiz: Quiz 10 and Online quiz: Quiz 11 contains problems where the computer will the score your answer for you, so you can see how well you did. When these questions appear on the exam, they may be modified to the short answer format.

Problem sets

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

#### Review session

We will hold a review session on the last day of class, Wednesday, December 16.

#### Final exam rules

- Permitted items:
- The exams will be open book, notes, and the internet. You may research the questions on the internet.
- You are allowed to have a calculator in the exam. A graphing calculator is OK.
- You may not communicate with another person about the exam, except for the professor or one of the teaching assistants.

- Expectations for answers:
**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.- Mysterious or unsupported answers will not receive full credit. Your work should be mathematically correct and carefully and legibly written.
- 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.
- 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.

- You will have 180 minutes to take the exam (see below). The exam will be posted on Canvas 10 minutes ahead of time, giving you time to print the exam, if you desire or otherwise prepare for the exam. You will have a 15 minute grace period afterwards to scan your work and post it on Gradescope.
- You can write your answers on blank (lined or graph 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.

#### Final exam time

Date: Thursday, December 17

Time: noon - 3:00 PM

Exam posted on Canvas: 11:50 AM

Exam due: 3:00 PM

End of grace period for submission: 3:15 PM

Provisions are available for students who have two Math common final exams at the same time. See instructor if you have any conflict with the final exam time.