The student will:
- be proficient in techniques of solving integrals including improper integrals as well as be familiar with the Fundamental Theorem of Calculus.
- Have a thorough understanding of infinite series and tests for convergence, in particular power and Taylor series.
- Be able to parametrise space curves and find arc lengths and tangent lines.
- Be able to work effectively with functions of two variables including sketching, assessing for differentiability, determining directional derivatives and tangent planes, and finding extrema using methods including Lagrange multipliers.
- Sketch and describe regions in R2 and R3 in Cartesian, polar, cylindrical and spherical coordinates as applicable
- Compute double, surface and triple integrals, including in the applications of mass, area, volume and flux
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Sequences, series and summation: We shall define sequences and series and consider the convergence of infinite sequences and series. To do this we use a variety of different tests for convergence and also look at approximation errors. Series of particular interest are the p-series, geometric series, and Taylor and Maclaurin expansions. We shall consider summation techniques with particular interest in the relationship with Riemann sums.
Integration, single variable: We define integration in the context of areas under curves and use Riemann sums to determine definite integrals from first principles. We study the Fundamental Theorem of Calculus and use some integration techniques such as integration by substitution and integration by parts.
Integration, multivariable: Also starting from the concept of Riemann sums, we define and compute double and triple integrals in two- and three-dimensional space.
Three-dimensional geometry: Our primary interest here is in space curves and surfaces. We parametrise space curves, find arc lengths, and determine the intersection of surfaces as curves. We see the graphs of functions of two variables as surfaces and consider what it means for limits of such functions to exist and how to define continuity. In the context of surfaces we consider families of level curves, tangent planes, directional derivatives and optima. We close off this section with Taylor series in two variables and using Lagrange multipliers to determine optima.
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