Calculus II

Course Prerequisites:

Successful completion of Saxon or Shormann Calculus (or pre-approved equivalent.) Please note that this course is by invitation or interview only.

Course Description:

Single-variable calculus and analytic geometry: Advanced techniques of integration, polar equations, parametric equations, introduction to differential equations, infinite sequences and series, convergence, power series, and Taylor polynomials.

Course Objectives:

1. To look at the quadrivium's mathematical arts of arithmetic and geometry from a biblical and classical approach.
2. To study the language of mathematics with a goal of improving fluency and to recognize the integral relationship between the Christian Faith and mathematics.
3. To build upon Calculus I knowledge; to learn, review, and practice intellectual skills of learning in the fundamentals of mathematics.
4. To contemplate the truth, beauty, and goodness via journaling and actively participating in discussions of mathematics.
5. To research and extol the virtues of a historical mathematician via a paper and speech.

6. To gain understanding and wisdom in the arts of mathematics using Cicero's Five Canons of Rhetoric to participate in various methods of discussions.

   Student Learning Outcomes:

Upon completion of this course, student should be able to:

1. Interpret a volume of revolution of a function’s graph around a given axis as a (Riemann) sum of disks or cylindrical shells, convert to definite integral form and compute its value.
2. Express the length of a curve as a (Riemann) sum of linear segments, convert to definite integral form and compute its value.
3. Express the surface area of revolution of a function’s graph around a given axis as a (Riemann) sum of rings, convert to definite integral form and compute its value.
4. Integrate products of trigonometric functions.
5. Decompose a rational integrand using partial fractions.
6. Interpret the concept of a series as the sum of a sequence and use the sequence of partial sums to determine convergence of a series.
7. Decide whether and to what value an infinite geometric series converges.
8. Distinguish between absolute and conditional convergence of series and be aware of the consequences of reordering terms in conditionally converging series.
9. Perform the ratio and root test to determine convergence of infinite series.
10. Determine the Taylor series of the nth order and determine an upper bound on its remainder.
11. Manipulate Taylor series by substitution and (anti-) differentiation to obtain expansions for other functions.
12. Devise parametric representations for conic sections and other relations.

Required Resources:

• TBD

Optional Resources:

• Kline, Morris (Calculus: An Intuitive and Physical Approach, ISBN-13:‎ 978-0486404530)
• Nickel, James (Mathematics: Is God Silent?, ISBN-13: 978-1879998223)
• Stokes, Mitch, PhD (Calculus for Everyone, ISBN-13 978-1944482541)