Portland Community College | Portland, Oregon

This is the third course of four courses in the Calculus sequence.

Upon completion of this course the learner should be able to do the following in the outside world:

• Analyze real world scenarios to recognize when elementary differential equations, vectors, or series are appropriate, formulate problems about the scenarios, creatively model these scenarios (using technology, if appropriate) in order to solve the problems using multiple approaches, judge if the results are reasonable, and then interpret and clearly communicate the results.
• Appreciate elementary differential equation, vector, and series concepts that are encountered in the real world, understand and be able to communicate the underlying mathematics involved to help another person gain insight into the situation.
• Work with elementary differential equations, vectors, and series in various situations and use correct mathematical terminology, notation, and symbolic processes in order to engage in work, study, and conversation on topics involving vectors and series with colleagues in the field of mathematics, science or engineering.
• Enjoy a life enriched by exposure to Calculus.

1. Demonstrate an understanding of the concepts of vectors, series, and differential equations 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 60% of the overall grade;
   • at least one of the exams should require the use of technology;
   • 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

1. Context Specific Skills

   • 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.
   • Students will learn to model and solve several types of applications using vectors.
   • Students will learn to recognize a differential equation, draw and read a slope field, and algebraic solve simple differential equations.

2. Learning Process Skills

   • 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 calculators and/or computers 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 and discuss families of solutions.
   2. Verify that a function is a solution of a given differential equation.
   3. Use slope fields to solve a differential equation graphically.
   4. Use the Separation of variables technique.
2. 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.
      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. Demonstrate use of the Integral test, Comparison test, Root test, Ratio test, and Alternating Series test for various convergent/divergent infinite series.
         2. Demonstrate estimating the sum of convergent infinite series.
   3. Study techniques for evaluating infinite series.
      1. Find a new series by using substitution.
      2. Find a new series by using integration.
   4. Use the binomial series to expand \((1 + x)^p\).
   5. Work with sums of finite/infinite geometric series.
   6. Study 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. Study the error in Taylor approximations.
         1. Use the Mean Value Theorem and a graphical approach.
         2. Use a symbolic to bound the error using the residual.
3. Vectors

   The goal is to use vectors, in the plane and \(3\)-space, to represent quantities that have direction as well as magnitude.

   1. Define a vector.
      1. Represent a vector graphically.
      2. Represent a vector using component notation.
      3. Represent a vector in terms of its unit vectors \(\hat{\imath}\), \(\hat{\jmath}\), \(\hat{k}\).
      4. Define the magnitude and direction of a vector in terms of the above representations.
   2. Define and apply operations of vectors to perform addition, subtraction, scalar multiplication graphically and symbolically.
   3. Define and use the dot product geometrically and symbolically.
      1. Define work using vector notation.
      2. Define parallel and orthogonal vectors.
      3. Calculate projections of vectors.
      4. Write parametric equations and symmetric equations for a line.
   4. Define and use the cross product geometrically and symbolically.
      1. Write the equation of a plane.
      2. Calculate the area of a parallelogram.
4. Extended Topics

   At least one of the following topics must also be included in the course.

   1. Euler's method
   2. Solutions to growth and decay differential equation problems.
   3. Newton's Law of Heating and Cooling.
   4. The logistic equation.
   5. Predator-prey systems.
   6. Laplace transforms.
   7. Power series solutions to differential equations.
   8. Fourier series.

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.