9. Calculus II

Course Description:

Calculus 2 is a continuation of Calculus 1, covering approximately the equivalent of the AP Calculus BC course, together with topics in second-order differential equations and multivariable calculus. Topics are taught at a higher level of logical rigor than Calculus 1. The key new concept introduced in this course is Taylor series and polynomials, giving a systematic way of approximating functions by polynomials (possibly of 'infinite' degree). In addition, many applications outside of mathematics are considered, especially to physics.

The textbook is:

Stewart: Calculus, Early Transcendentals 6th edition (Covering Chapters 7-13, 17)

The teacher is:

Peter Mannisto

Detailed Course Topic List:

0. Review of Integration basics (1.5 weeks)

• • 1. Integrals as areas, limits of Riemann sums, and antiderivatives

    2. Fundamental Theorem of Calculus

    3. u-substitution

• 1. Further integration techniques and applications (4 weeks)

     1. Integration by parts

     2. Integration by trigonometric substitution

     3. Integration by partial fractions

     4. Improper integrals

     5. Applications: probability distributions, surface area (if time)

  2. Infinite series (10 weeks)

• • 1. Infinite sequences

    2. Recursive sequences and induction

    3. Infinite series; the geometric series

    4. Absolute vs. conditional convergence of series; tests for convergence

    5. Power series: radius and integral of convergence; techniques for computing

    6. Taylor and Maclaurin series; definition and techniques for evaluation

    7. Applications of Taylor series: numerical approximation, physics applications

• 3. Differential equations (7 weeks)

• • 1. Review of first-order differential equations: slope fields, separation of variables

    2. Linear first-order DEs and integrating factors

    3. Second-order differential equations: characteristic equations

    4. Method of undetermined coefficients

    5. Applications to physics: springs, RLC circuits and resonance

    6. Power series solutions to differential equations (if time)

• 4. Multivariable Calculus (10 weeks)

• • 1. Parametric equations

    2. Polar equations

    3. 3-dimensional coordinate systems

    4. Vectors: dot product and cross product

    5. Lines and planes in 3 dimensions

    6. Arc length and curvature in 3 dimensions

    7. Motion in space: velocity and acceleration vectors, relation to curvature

    8. Physics application: Kepler's laws