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:
Detailed Course Topic List:
0. Review of Integration basics (1.5 weeks)
Integrals as areas, limits of Riemann sums, and antiderivatives
Fundamental Theorem of Calculus
u-substitution
Further integration techniques and applications (4 weeks)
Integration by parts
Integration by trigonometric substitution
Integration by partial fractions
Improper integrals
Applications: probability distributions, surface area (if time)
Infinite series (10 weeks)
Infinite sequences
Recursive sequences and induction
Infinite series; the geometric series
Absolute vs. conditional convergence of series; tests for convergence
Power series: radius and integral of convergence; techniques for computing
Taylor and Maclaurin series; definition and techniques for evaluation
Applications of Taylor series: numerical approximation, physics applications
Differential equations (7 weeks)
Review of first-order differential equations: slope fields, separation of variables
Linear first-order DEs and integrating factors
Second-order differential equations: characteristic equations
Method of undetermined coefficients
Applications to physics: springs, RLC circuits and resonance
Power series solutions to differential equations (if time)
Multivariable Calculus (10 weeks)
Parametric equations
Polar equations
3-dimensional coordinate systems
Vectors: dot product and cross product
Lines and planes in 3 dimensions
Arc length and curvature in 3 dimensions
Motion in space: velocity and acceleration vectors, relation to curvature
Physics application: Kepler's laws