Occurrence | Teaching cycle |
---|---|
A | Autumn Term 2019-20 to Summer Term 2019-20 |
The first years of all mathematics programmes are designed to give students a thorough grounding in a wide spectrum of mathematical ideas, techniques and tools in order to equip them for the later stages of their course. During first year, as well as consolidating, broadening and extending core material from pre-University study, we initiate a cultural transition to the rigorous development of mathematics which is characteristic at University. Students will develop both their knowledge of mathematics as a subject and their reasoning and communication skills, through lectures, tutorials, seminars, guided self-study, independent learning and project work. This development is addressed in all of our first year modules, although different modules have a different emphasis.
In addition to the above broad aims of the first year, this module focusses on ensuring that students have competence in a wide range of essential concepts, techniques and applications of differential and integral calculus, and differential equations.
Subject content
Differential calculus. Rules of differentiation (chain, product, quotient). Derivatives of standard functions (powers, polynomials, trigonometric). The exponential function: as solution of dy/dx=y with y(0)=1; as a power series, leading to e=1+1/2+1/3! + ··· and exp(x)=e^{x}; as the limit of Euler’s sequence. Logarithm as inverse. Derivatives of inverse functions via chain rule (and thus dy/dx=1/(dx/dy)). Local extrema, curve sketching.
Integral calculus. The definite integral. Anti-derivatives and the indefinite integral. Fundamental Theorem of Calculus. Rules and techniques for integration: partial fractions, by parts, by substitution. Improper integrals. Recursion formulae, the gamma function.
Hyperbolic functions. Conic sections as polynomial equations of degree 2 in two variables. Relationships between trigonometric and hyperbolic functions, connections with Algebra: the complex numbers, Euler’s formula.
Parametric curves. Vector-valued functions. Arc length, speed, velocity.
Ordinary differential equations. Classification, existence and uniqueness of solutions (Lipschitz-Picard condition), constants of integration. First-order: separable, linear (by integrating factor). Second-order: reduction of order; linear with constant coefficients, homogeneous and inhomogeneous. Coupled linear first-order equations.
Taylor series with integral form of remainder, statement of derivative form. Standard examples: trigonometric and hyperbolic functions, exponential and logarithmic series, binomial series and relation with binomial theorem. Estimation of remainder. Radius of convergence.
Fourier series. Trigonometric and complex exponential forms. Fourier coefficients, even and odd functions. Periodic extension, convergence and a statement of Dirichlet's theorem. Half-range series. Parseval's theorem. Examples, including evaluation of series, and the Fourier series representation of solutions to the heat equation
Functions of two variables. Surfaces as graphs, level curves. Partial derivatives: intuitive notion, statement of chain rule, examples. Directional derivatives derived from chain rule. Tangent plane as linear approximation to the surface at a point. Equality of mixed second partial derivatives. Proof of chain rule. The gradient vector: geometric interpretation, directional derivative, tangent planes. Vector fields; the potential function and its computation. Implicit differentiation: of functions of one variable and of scalar fields; tangent lines to level curves. Application of chain rule to coordinate transformations.
Double integrals. Surface area, volumes of revolution. Double integral as the volume under a surface. Evaluation over rectangular regions, as iterated integrals; changing order of integration. Integrals over more general regions and in polar coordinates; the Gaussian integral as example. Change of variables in double integrals, the Jacobian.
Academic and graduate skills
Academic skills: the application of rigorous mathematical techniques and ideas to the development of mathematics;the power of abstraction as a way of solving many similar problems at the same time;the development and consolidation of essential skills which a mathematician needs in their toolkit and needs to be able to use without pausing for thought.
Task | Length | % of module mark |
---|---|---|
University - closed examination Calculus - Spring |
1.5 hours | 0 |
University - closed examination Calculus - Summer |
3 hours | 100 |
5% of the final exam mark comes from coursework. Module is non-compensatable.
Competency exam is pass/fail and must be passed to pass the module, although multiple attempts will be allowed. Since a proper grasp of the material in this module is essential for success in any of our programmes involving it, the module is non-compensatable. 5% of the final exam mark comes from a coursework mark, calculated by taking the average over the best n-1 out of n assignments per term (with Spring-Summer counted as one term).
Task | Length | % of module mark |
---|---|---|
University - closed examination Calculus - Spring |
1.5 hours | 0 |
University - closed examination Calculus - Summer |
3 hours | 100 |
Current Department policy on feedback is available in the undergraduate student handbook. Coursework and examinations will be marked and returned in accordance with this policy.
The principal text for Calculus is:
Thomas' Calculus, is a long-standing text and pretty much any edition written since about 1970 will cover the material adequately,but do check the contents before you buy a second-hand copy. The library shelf S7 has many old examples which are still useful. There are many other Calculus texts which cover the same material and would be perfectly suitable. Here are three examples:
None of these texts cover Fourier series or partial differential equations, and most do not cover enough on ordinary differential equations for the whole module. For these topics, you will probably want to consult an additional textbook. Here are a few suggestions: