Every lesson in the Calculus II slide course, in full text: 27 decks, 1819 slides.
3.1 Integration by PartsReversing the product rule: the parts formula as a trade, choosing u and dv with LIATE, repeated application for higher powers, the circular case solved by algebra, definite integrals, and knowing when substitution is the better tool.
3.2 Trigonometric IntegralsProducts of powers of sine and cosine sorted by the parity of the exponents, the odd-power substitution, half-angle identities for all-even powers, secant and tangent products, product-to-sum identities, and reduction formulas.
3.3 Trigonometric SubstitutionThe three radical forms and the substitution each calls for, why the Pythagorean identity clears the root, the reference triangle for converting back, completing the square to reach a standard form, definite integrals with converted limits, and the hyperbolic alternative.
3.4 Partial FractionsBreaking a rational function into simpler fractions: long division first, the forms contributed by distinct linear, repeated linear and irreducible quadratic factors, finding the coefficients by substituting roots or comparing them, and integrating the pieces.
3.5 Other Strategies for IntegrationChoosing among the techniques in order of cost, rewriting before reaching for a method, using a table of integrals by matching forms, computer algebra systems and why their answers differ from yours, and recognising a non-elementary integral.
3.6 Numerical IntegrationThe midpoint and trapezoidal rules and which over- or underestimates, Simpson's rule and its weights, absolute and relative error, the error-bound formulas, and choosing the number of subintervals to meet a tolerance.
3.7 Improper IntegralsIntegrals over unbounded intervals and with unbounded integrands, defined as limits of ordinary integrals, the p-integrals and their opposite boundaries at the two ends, splitting when several points misbehave, and the comparison theorem.
4.1 Basics of Differential EquationsEquations whose unknown is a function, the order of an equation, what counts as a solution, general and particular solutions, initial-value problems and how many conditions each order needs, and verifying a proposed solution by substitution.
4.2 Direction Fields and Numerical MethodsDrawing the field of slopes a differential equation prescribes, reading solution curves and equilibria off it, judging stability without solving, and approximating a solution numerically with Euler's method.
4.3 Separable EquationsRecognising when the two variables can be pulled apart, separating and integrating both sides, why only one arbitrary constant survives, implicit against explicit solutions, the equilibrium solutions lost by dividing, and applications.
4.4 The Logistic EquationCarrying capacity and the factor that switches growth off, the direction field and its two equilibria, solving by separation and partial fractions, the S-shaped solution and its inflection at half capacity, and fitting the model to data.
4.5 First-order Linear EquationsStandard form, the integrating factor and the condition defining it, the product-rule collapse that makes the left side integrable, solving initial-value problems, and applications to mixing tanks and circuits.
5.1 SequencesSequences as functions on the integers, finding a general term, convergence and divergence, the limit laws and the theorem borrowing limits from continuous functions, the squeeze theorem, and monotone bounded sequences.
5.2 Infinite SeriesThe sequence of partial sums as the definition of an infinite sum, geometric series and their closed form, telescoping series, the algebraic properties of series, and the harmonic series as the standard warning.
5.3 The Divergence and Integral TestsThe divergence test as the contrapositive of the necessary condition, the integral test comparing a series against an improper integral, the p-series family and the location of its boundary, and estimating a sum by bounding its remainder.
5.4 Comparison TestsThe direct comparison test and which of its four arrangements are informative, the limit comparison test and why a finite positive limit forces agreement, and the practical skill of reading a term's dominant behaviour to choose a comparison.
5.5 Alternating SeriesThe alternating series test and the nesting of its partial sums, the remainder bound that comes free with it, absolute against conditional convergence, and the rearrangement theorem that gives the distinction its meaning.
5.6 Ratio and Root TestsThe ratio test and the root test, both built by comparison against a geometric series, why both are silent at a limit of exactly one, how a term's shape picks between them, and the strategy assembling every test in the chapter.
6.1 Power Series and FunctionsWhat a power series is, why its convergence set is always an interval centred on the series, computing the radius with the ratio test, checking the endpoints separately, and reading the geometric series backwards as a function's representation.
6.2 Properties of Power SeriesCombining power series by addition, scaling and substitution, multiplying two of them, differentiating and integrating termwise, and the interval bookkeeping each operation demands.
6.3 Taylor and Maclaurin SeriesTaylor polynomials built by matching derivatives at a point, Taylor's theorem with remainder, estimating that remainder, and the distinction between a Taylor series converging and converging to the function it came from.
6.4 Working with Taylor SeriesThe binomial series for any exponent, recognising the standard expansions in transformed form, evaluating integrals with no elementary antiderivative termwise, and solving differential equations by matching coefficients.
7.1 Parametric EquationsCurves described by a parameter rather than as graphs, sketching with orientation, eliminating the parameter and what that operation loses, and why a parametrisation is a choice rather than a property of the curve.
7.2 Calculus of Parametric CurvesThe slope as a ratio of derivatives against the parameter, second derivatives and concavity, arc length as the integral of a speed, area under a parametric curve, and where the formulas fail.
7.3 Polar CoordinatesLocating points by distance and direction, converting between coordinate systems and why one direction is harder, the non-uniqueness of polar names, and graphing the standard polar curves.
7.4 Area and Arc Length in Polar CoordinatesArea swept in circular sectors rather than rectangular strips, reading the limits from where the distance vanishes, area between two curves and the intersection problem, and arc length in polar form.
7.5 Conic SectionsThe three conics as slices of a cone and as focus-directrix loci, their standard rectangular equations, eccentricity as the single parameter distinguishing them, the unified polar equation, and identifying rotated conics from the discriminant.
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