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# Calculus - MAT00001C

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• Department: Mathematics
• Module co-ordinator: Dr. Ian McIntosh
• Credit value: 30 credits
• Credit level: C
• Academic year of delivery: 2022-23
• See module specification for other years: 2021-22

## Module will run

Occurrence Teaching period
A Autumn Term 2022-23 to Summer Term 2022-23

## Module aims

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.

## Module learning outcomes

Subject content

• Functions and their properties. Domain and range, inverse functions. Limits and continuity (axiomatic as definitions). The derivative (from first principles). l'Hopital's rule.

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)=ex; 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 the evaluation of series.

• 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, chain rule. The gradient vector: geometric interpretation, directional derivative, tangent planes. Vector fields. Implicit differentiation: of functions of one variable and of scalar fields; tangent lines to level curves. Application of chain rule to coordinate transformations.

• Extrema. Local extrema. Taylor series in 2D up to quadratic terms, classification of critical points. Global extrema in closed regions; Lagrange multipliers.
• 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 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.

• Many of the techniques and ideas developed in this module are ones which graduates employed as mathematicians and in other numerate professions will use from day to day in their work. On top of this, students, through lectures, examples, classes, will develop their ability to assimilate, process and engage with new material quickly and efficiently.

## Assessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Calculus - Competency Exam
N/A 0
Closed/in-person Exam (Centrally scheduled)
Calculus - Summer
3 hours 100

### Special assessment rules

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.

### Reassessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Calculus - Competency Exam
N/A 0
Closed/in-person Exam (Centrally scheduled)
Calculus - Summer
3 hours 100

## Module feedback

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:

• Wier, Hass and Giordano, Thomas' Calculus, 11th, 12th or 13th Edition. Pearson/Addison-Wesley.

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: