Portland Community College | Portland, Oregon

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

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

• Analyze real world scenarios to recognize when series, vectors, and geometry of space 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.
• Recognize  series,  vectors, and geometry of space 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  series,  vectors, and geometry of space 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.

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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 visualize and manipulate two-variable functions presented in graphical, numeric, and symbolic form.

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. 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(s), 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. Use 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). 
2. Vectors

   The goal is to use vectors, in \(\mathbb{R}^2\) and \(\mathbb{R}^3\), 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, and 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. Define and use the cross product geometrically and symbolically.
      1. Calculate the area of a parallelogram.
3. Lines and Planes​
   1. Write parametric equations and symmetric equations for a line.
   2. Write the equation for a plane given a variety of conditions such as:
      1. Three points on the plane
      2. A point and a line contained within the plane.
      3. Two lines contained within the plane.
4. Two-variable Functions
   1. Three-dimensional graphs of two-variable functions.
      1. Plot points on a three-dimensional axes system.
      2. Sketch planes on a three-dimensional axes system.
      3. Sketch quadric surfaces on a three-dimensional axes system using a CAS.
      4. Match equations of quadric surfaces to their graphs.
   2. Two-variable functions presented in symbolic form.
      1. Evaluate a two-variable function given its formula.
      2. Find the domain of a two-variable function given its formula.
   3. Two-variable functions presented in tabular form.
      1. Visualize and describe a three dimensional function presented in tabular form.
5. Spherical and Cylindrical coordinates.
   1. Plot points in cylindrical and spherical coordinates.
   2. Convert back and forth between rectangular form, cylindrical coordinate form, and spherical coordinate form.
6. Optional Topics 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

This course is an essential, required pre-requisite for MTH 254. The topics related to vectors and \(\mathbb{R}^3\) should be given adequate time for coverage as these topics will only be briefly reviewed in MTH 254. Students who took MTH 253 at another institution that does not cover topics of \(\mathbb{R}^3\) should be provided with resources to review those topics. 

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.