Portland Community College | Portland, Oregon

Upon completion of the course the students should be able to:

• Application – Analyze real world scenarios to: recognize when ordinary differential equations, sequences, and series are appropriate, formulate and model these scenarios (using technology, if appropriate) in order to find solutions using multiple approaches, judge if the results are reasonable, and then interpret these results.

• Concept – Recognize the underlying mathematical concepts of ordinary differential equations, sequences, and series.

• Computation – Use ordinary differential equations, sequences, and series with correct mathematical terminology, notation, and symbolic processes.

• Communication – Communicate mathematical applications, concepts, computations, and results with classmates and colleagues in the fields of science, technology, engineering, and mathematics.

Enjoy a life enriched by exposure to one of humankind's great achievements.

1. Demonstrate an understanding of the concepts of vectors and series and their application to real world problems in:

   • at least two proctored exams, one of which is a comprehensive final that is worth at least 25% of the overall grade

   • proctored exams should be worth at least 50% of the overall grade

   • and at least one of the following:

     • Take-home examinations

     • Graded homework problems

     • Quizzes

2. Consistently demonstrate proper notation, documentation, and use of language throughout all assessments and assignments. For proper documentation standards see Addendum.

3. Demonstrate an ability to work and communicate with colleagues, on the topics of vectors and series, in at least two of the following:

   • A team project with a written report and/or in-class presentation

   • Participation in discussions

   • In-class group activities

• Students will learn to solve first- and second-order differential equations.

• Students will explore numerical methods of approximation.

• Students will learn to determine the convergence status of a given series.

• Students will learn estimation techniques for convergent series.

• Students will learn to model functions with Taylor series and use Taylor Series to solve application problems.

• Classroom activities will include lecture/discussion and group work.

• Students will communicate their results in oral and written form.

• Students will apply concepts to real world problems.

• The use of technology will be demonstrated and encouraged by the instructor where appropriate. Technology will be used (at least) when estimating convergent series.

1. Introduction to Differential Equations

   The goal is to solve differential equations using graphical, numerical and analytic methods.

   1. Define first and second order differential equations.

   2. Verify that a function is a solution of a given differential equation.

   3. Discuss families of solutions for first and second order differential equations.

   4. Use slope fields to solve a differential equation graphically.

   5. Use the separation of variables technique.

   6. Cover at least one of the following applications of differential equations in similar depth to the required topics:

      1. Solutions to exponential growth and decay differential equation problems

      2. Newton’s Law of Heating and Cooling

      3. The logistic equation

2. Approximation Strategies

   The goal is to use numerical methods of approximation.

   1. Linear Approximations

   2. Cover at least one of the following approximation strategies in similar depth to the required topics:

      1. Euler’s Method

      2. Newton’s Method

3. Sequences and Series

   The goal is to study convergent/divergent sequences and series, and to approximate functions with simpler functions.

   1. Study convergent/divergent infinite sequences.

      1. Define conditions for convergent/divergent infinite sequences (including geometric, arithmetic, algebraic and recursive sequences).

      2. Work with convergent/divergent infinite sequences geometrically, numerically, and symbolically.

   2. Study convergent/divergent infinite series.

      1. Define conditions for convergent/divergent series.

      2. Demonstrate strategies for testing series for convergence/divergence.

         1. Apply the Test for Divergence, Telescoping Series, Integral Test, Comparison Test, Limit Comparison Test, Ratio Test, and Alternating Series Test for various convergent/divergent infinite series.

         2. Estimate the sum of convergent infinite series using partial sums or estimation techniques such as comparing to an integral.

         3. Estimate the error in using partial sums to approximate the sum of a convergent alternating series. 

      3. Find sums of convergent series when possible (e.g. geometric series and telescoping series).

   3. Work with power series.

      1. Find a new series by using substitution (e.g. by comparing \(\frac{1}{1+2x}\) to the series representation of \(\frac{1}{1-x}\).

      2. Integrate or differentiate power series.

      3. Find a new power series representations by using differentiation or integration (e.g. find the series representation of \(\ln(1+x)\) by using an appropriate integral).

      4. Find the interval and radius of convergence for a power series.

   4. Exposure to using the binomial series to expand \((1 + x)^p\).

   5. Use Taylor series to approximate functions locally.

      1. Work with Taylor polynomials of degree \(n\) to approximate \(f(x)\) centered at \(x = c\).

      2. Study particular Taylor and Maclaurin series and their intervals of convergence, including \(e^x\), \(\sin(x)\), and \(\cos(x)\).

      3. Calculate the error in using Taylor polynomials to approximate \(f(x)\) by using Taylors Inequality (or the Alternating Series Estimation Theorem where appropriate). 

   6. Exposure to series solutions for differential equations 

4. Optional Topics
   1. The following topics may be of interest to the instructor or the student and can be covered as time allows. 

      1. Fourier series

      2. The root test.

      3. Proof by Induction

All work in this course will be evaluated for your ability to meet the following writing objectives as well as for mathematical content.

1. Every solution must be written in such a way that the question that was asked is clear simply by reading the submitted solution.
2. Any table or graph that appears in the original problem must also appear somewhere in your solution.
3. All graphs that appear in your solution must contain axis names and scales. All graphs must be accompanied by a figure number and caption. When the graph is referenced in your written work, the reference must be by figure number. Additionally, graphs for applied problems must have units on each axis and the explicit meaning of each axis must be self-apparent either by the axis names or by the figure caption.
4. All tables that appear in your solution must have well defined column headings as well as an assigned table number accompanied by a brief caption (description). When the table is referenced in your written work, the reference must be by table number.
5. A brief introduction to the problem is almost always appropriate.
6. In applied problems, all variables and constants must be defined.
7. If you used the graph or table feature of your calculator in the problem solving process, you must include the graph or table in your written solution.
8. If you used some other non-trivial feature of your calculator (e.g., SOLVER), you must state this in your solution.
9. All (relevant) information given in the problem must be stated somewhere in your solution.
10. A sentence that orients the reader to the purpose of the mathematics should usually precede symbol pushing.
11. Your conclusion shall not be encased in a box, but rather stated at the end of your solution in complete sentence form.
12. Remember to line up your equal signs.
13. If work is word-processed all mathematical symbols must be generated with a math equation editor.

Emphasis should be placed on using technology such as Desmos and GeoGebra appropriately; such as when computing approximations, graphing curves, or visualizing or checking answers.  Technology should not be used as a substitute for meeting the outcomes and skills for the course that are expected to be done by hand.