For your convenience, I also provide here a rough sketch of topics and corresponding lectures as described on the UCSD math department website where they list a recommended 20B syllabus. This is not a rigid schedule and the class will likely deviate from it.
- Lec. 1. Sec. 5.2-5.5: Review of the Fundamental Theorem of Calculus
- Lec. 2. Sec. 5.6: Total change as the integral of a rate
- Lec. 3. Sec. 5.7: The substitution algorithm for integrals
- Lec. 4. Sec. 6.1–6.2: Areas between curves, computing volumes using the “slicing” technique (using pieces of known cross-sectional area), average values of a function. Skip the discussions of flow rate in Section 6.2.
- Lec. 5. Sec. 6.3: Computing the volume of solids of revolution as a special case of Sec. 6.2.
- Lec. 6. Sec. 7.1: Integration by parts
- Lec. 7–8. Sec. 11.3-11.4: Polar coordinates; areas in polar coordinates. Skip the discussion of arc length at the end of Section 11.4. The use of the cosine double-angle identity to integrate functions involving sine squared and cosine squared will have to be introduced here.
- Lec. 9–10. Supp. 1–2: Complex numbers and complex exponentials: Discuss de Moivres theorem, complex roots, and Eulers formula. Discuss how the trigonometric functions are related to the complex exponential and how trigonometric identities may be derived using the complex exponential.
- Lec. 11. Sec. 7.2, 7.4, Supp. 3: Trigonometric integrals, integrals of hyperbolic functions: Illustrate that trigonometric integrals can be done using complex exponentials, with the goal of giving the students experience with complex exponentials (which is particularly useful to engineering students). Integrals of hyperbolic functions use real exponentials and can be done quickly. Integrals of inverse hyperbolic functions should be skipped.
- Lec. 12. Sec. 7.3: Trigonometric substitution (If behind schedule, this topic may be omitted.)
- Lec. 13–14. Supp. 4–5, Sec. 7.5: The fundamental theorem of algebra; partial fractions and integration of rational functions using partial fractions. The discussion in Section 4 of the Supplement introduces students to the complex viewpoint of partial fraction expansions (PFE). Keep in mind that if a PFE is obtained using complex numbers, no complex numbers can appear in denominators when computing antiderivatives.
- Lec. 15–16. Sec. 7.7: Improper integrals: Take care to emphasize that improper integrals are limits; this will be reinforced during the discussion of sequences and series. The comparison test will also show up again during the discussion of sequences and series.
- Lec. 17. Sec. 10.1: Sequences: limits, convergence, and divergence. Emphasize that a sequence is a function of the positive integers. Refer students to Section 2.3 of Rogawski for the corresponding properties of limits for functions of a real variable, and Section 2.7 for definition of limit at infinity: there is not time to formally list the properties of limits in class. The discussion of bounded and monotonic sequences may be omitted.
- Lec. 18. Sec. 10.2: Series: Use the geometric series and harmonic (or other) series to illustrate a convergent versus a divergent series. Illustrate with further examples.
- Lec. 19. Sec. 10.3: Series with positive terms: the integral and comparison tests. Emphasize that these are just two variations of the idea of comparison.
- Lec. 20. Sec. 10.4: Absolute convergence; the ratio and root tests; conditional convergence and the Leibnitz test for alternating series.
- Lec. 21. Sec. 10.5: Emphasize that the ratio and root tests are based on comparison with an appropriate geometric series, but avoid the technical proofs.
- Lec. 22–23. Sec. 10.6: Power series: Emphasize that power series are important because they are so much like polynomials. Note that the radius of convergence of a power series can usually be checked using the ratio test (or root test).
- Lec. 24–25. Sec. 10.7: Taylor series: Emphasize that the process for finding coefficients of a Taylor series for a function comes from treating the function as a polynomial. State the remainder theorem, but avoid discussing the proof.
- Lec. 26. Sec. 9.1–9.2: Solving differential equations; exponential models: Illustrate the method of separation of variables. Exponential models are a source of applications. (May be omitted.)
- Lec. 27. Sec. 9.4: The logistic equation (May be omitted.)