# Math Insight

### Overview of: Applications of differentiation exam

The applications of differentiation module exam for Math 1241 is based on part 4 from the Math 1241 thread.

#### Material for the exam

1. Linear approximation (Linear approximation from Math 1241 thread).

1. Be able to compute linear approximations of a function around a point, or, equivalently, the equation for the tangent line.
2. The linear approximation should match the function and its derivative at the point.
3. The linear approximation is the basis behind determining the stability of equilibria (both for discrete dynamical systems from this module and for continuous dynamical systems in the next module).
2. Stability of equilibria in discrete dynamical systems (Stability of equilibria from Math 1241 thread).

1. To calculate the stability of an equilibrium, put the system in function iteration form: $x_{t+1}=f(x_{t})$. Then, an equilibrium $x_n =E$ is stable if $|f'(E)| < 1$ and is unstable if $|f'(E)| >1$.
2. Be able to calculate equilibria and stability of dynamical systems that include parameters. This is where the derivative method for stability shines, as we cannot determine stability graphically if we don't have numbers for all the parameters. Be able able to determine the values of a parameter that make an equilibrium stable.
3. Logistic growth (Logistic growth from Math 1241 thread).

1. The discrete logistic equation captures how growth slows down when a population begins to exhaust its environment.
2. It is primarily an example that we can use for determining equilibria and stability.
3. We're primarily concerned with the case where the upper positive equilibrium is stable. In this case, the upper equilibrium is the carry capacity, and small population sizes will grow to this carrying capacity.
4. If both equilibria are unstable, the discrete logistic equation can have complicated behavior, but we won't worry about this complicated behavior on the exam.
4. The derivative and graphing (The derivative and graphing from Math 1241 thread).

1. Be able to identify critical points.
2. Determine where the function is increasing and decreasing by checking the sign of the derivative in the intervals determined by the critical points.
3. Sketch a graph of the function consistent with this information from the derivative.
5. Minimization and maximization (Minimization and maximization from Math 1241 thread).

1. Be able to identify global minima and maxima (global extrema) by checking the critical points and the endpoints.
2. Be able to identify local minima and maxima (local extrema) by checking the sign of the derivative on either side of the critical points.

#### Study aids

1. Review problems

All questions that may appear in the differentiation 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.

1. Free form questions

Review questions: Elementary discrete dynamical systems problems, part 2, Review questions: Elementary discrete dynamical systems biology problems, part 2, and Review questions: Critical points, maximization and minimization problems contain short answer problems that reflect the format of questions as they would appear on the exam.

2. Computer-scored problems

Review questions: Linear approximation practice, Review questions: Critical points, maximization, and minimization practice, Review questions: Discrete dynamical system stability practice, and , and contain problems where the computer will the score your answer for you, so you can see how well you did. With the exception of the quizzes, these computer generated scores do not count toward your grade. When these questions appear on the exam, they may be modified to the short answer format.

2. Problem sets

The from the problem sets 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.
2. Expectations for answers:
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 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.

#### Exam times

You can take the exam up to two times. You can take it once on Thursday, November 5, and once on Thursday, November 12.

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: Nov. 5, 2020, 10:15 a.m.
Due: Nov. 12, 2020, 2:15 p.m.
Time limit: 50 minutes