Calculus II
Series, Taylor, vectors, multivariable functions and double integrals.
Textbook: Thomas' Calculus International Edition 12th Edition George Thomas, Maurice Weir, Joel Hass, Frank Giordano
- Week 1L'Hôpital's Rule, Improper Integrals
- L'Hôpital's rule and improper integralsYou learn to solve indeterminate limits with derivatives and to show whether integrals that run to infinity or have singularities converge. This is the foundation of the integral test for series.
- Week 2Sequences, Infinite Series
- Sequences and infinite seriesYou learn to find the limit of a sequence, to define a series through partial sums, and to compute the sum of telescoping series. The language of all convergence tests is set up here.
- Week 3Geometric Series, Divergence and Integral Tests
- Geometric series, the nth-term test and the integral testYou learn geometric series whose sum can be computed exactly, the nth-term test that shows divergence in one line, and the integral test that settles the fate of p-series.
- Week 4Ratio Test, Root Test, Comparison Test, Alternating Series Test
- Ratio, root, comparison and alternating series testsYou learn the ratio, root and comparison tests for positive-term series, the alternating series test for sign-changing series, and absolute/conditional convergence. At the end of the page there is a decision table showing which test to choose when.
- Week 5Power Series, Taylor Series
- Power series and Taylor seriesYou learn to find the radius and interval of convergence of a power series (endpoints included), to derive new series from the geometric series, and to write the Taylor/Maclaurin series of a function.
- Week 6Taylor Series, Vectors, Dot Product
- Applications of Taylor series: approximation, error, limits, sumsYou learn to approximate numerical values with a Taylor polynomial, to bound the error with Taylor's inequality and the alternating series estimate, to compute limits and integrals with series, and the binomial series.
- Vectors and the dot productYou learn to work with vectors in space, to find length and unit vectors, and to use the dot product for angles, orthogonality, projection and work. All the geometry of the multivariable part is written in this language.
- Week 7Cross Products, Planes and Surfaces
- Cross product, lines and planesYou learn to find a perpendicular vector, area and volume with the cross product; to write equations of lines and planes in space; and to solve intersection, angle and distance questions. An introduction to surfaces with cylinders is included.
- Week 8Planes and Surfaces, Level Curves, Functions in Several Variables
- Quadric surfaces, multivariable functions and level curvesYou learn to recognize the six basic quadric surfaces from their equations, to find the domain, range and level curves of a function of two variables, and to show that a two-variable limit does not exist with the two-path test.
- Week 9Partial Derivatives, Chain Rule
- Partial derivatives and the chain ruleYou learn to take partial derivatives and second-order derivatives of a multivariable function, to apply the chain rule with a tree diagram, and to find derivatives of implicit functions with the formula -F_x/F_y.
- Week 10Directional Derivatives, Gradient, Tangent Planes
- Gradient, directional derivative and tangent planeYou learn to compute the gradient vector, to find the rate of change in any direction with it (the directional derivative), to determine the direction and value of fastest increase, to write tangent planes and normal lines to surfaces, and to approximate with linearization and differentials.
- Week 11Maximum and Minimum Problems
- Local extrema and the second derivative testYou learn to find the critical points of a function of two variables, to classify them as local maximum, local minimum and saddle point with the second derivative test (the discriminant D), and to reduce application problems to a single variable.
- Week 12Maximum and Minimum Problems, Lagrange Multipliers
- Absolute extrema on a closed region and Lagrange multipliersYou learn to find the absolute largest and smallest values of a function on a closed and bounded region (interior critical points + boundary), and to solve extremum problems under a constraint with Lagrange multipliers.
- Week 13Double Integrals Over Rectangular and General Regions
- Double integrals: rectangular and general regionsYou learn to compute iterated integrals on a rectangular region, to write general regions as type I / type II and set up the limits, to reverse the order of integration, and to compute volume, area and average value with a double integral.
- Week 14Double Integrals in Polar Coordinates
- Double integrals in polar coordinatesYou learn to convert double integrals to polar coordinates (dA = r dr dθ) on regions bounded by disks, annuli, sectors and polar curves, to transform Cartesian integrals to polar, and to compute volume and area. The value of the Gaussian integral also comes out of this.