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

    1. 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)

    1. 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