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# Department of Mathematical Sciences

# MATH1061 Calculus I

Calculus is a fundamental part of mathematics and provides a foundation for all your future mathematical studies. This course will seek to consolidate and expand your knowledge of this topic and is designed to be completely accessible to the beginning calculus student. The three basic concepts of calculus will be covered, namely, limits, differentiation and integration. The emphasis of this module is on concrete methods for calculation, while the Analysis I module will revisit the above concepts and provide a deeper knowledge with a more formal approach.

First and second order ordinary differential equations are studied together with solution methods that are naturally associated with the techniques of integration.

Taylor and Fourier series are also covered, in preparation for their application in later modules.

Numerous exercises are provided to reinforce the material.

## Outline of course

Aim: To master a variety of methods for solving problems and acquire some skill in writing and explaining mathematical arguments.

### Term 1 (30 lectures)

• Elementary Functions of a Real Variable: Domain and range. Graphs of elementary functions. Even and odd functions. Exponential, trigonometric and hyperbolic functions. Algebraic combinations and composition. Injective, surjective and bijective functions. Theorem of inverse functions. Logarithm function as inverse of exponential function; inverse trigonometric functions.

• Limits and Continuity: Informal treatment of limits. Calculation of limits. Vertical and horizontal asymptotes. Continuity at a point and on intervals.

• Differentiation: Derivative as slope of tangent line. Differentiability and continuity. Product, quotient and chain rule. Implicit differentiation. Differential equations. Derivative as rate of change. Increasing and decreasing functions. Max-min problems.

• Integration: Antiderivatives. Fundamental theorem of calculus. Integration by parts and use of partial fractions to integrate rational functions. Integration of even/odd functions. Gaussian integration.

• Ordinary Differential Equations: First order: separable, exact, homogeneous, linear. Second order linear with constant coefficients, importance of boundary conditions, reduction to a set of rst order equations, treatment of homogeneous and inhomogeneous equations, particular integral and complementary function.

• Taylor's Theorem: Taylor polynomials. Statement of Taylor's theorem with Lagrange remainder. Calculation of limits using Taylor's series.

• Fourier Series: Convergence, periodic extension, sine and cosine series, half-range expansion. Parseval's theorem.

• Multiple Integration: iterated sums, double and triple integrals by repeated integration, volume enclosed by surface, Jacobians and change of variables.

### Term 2 (25 lectures)

• Functions of several variables: Plotting functions of two variables, sketches, contour maps. Continuity and differentiability. Chain rule. Cylindrical and Spherical Polar Coordinates. Taylor's Theorem for functions of more than one variable (statement only).
• Max/Min problems for functions of more than one variable: Stationary points, maxima, minima, saddle points via the Hessian matrix and its diagonalisation. Constrained variation and the use of Lagrange multipliers.
• Linear Differential Operators in one variable: 2nd order linear differential operators. Eigenvalue problems. Special polynomials as examples of eigenfunctions. Inner products on functions. Reality of eigenvalues and orthogonality of eigenfunctions for Hermitian operators. Fourier Series as an example.
• Linear PDEs and the Wave Equation: The wave equation in one dimension and its general solution. Principle of superposition for linear PDEs. Solution to the wave equation on a nite interval using the method of separation of variables. Generalisation to more than one spatial dimension.
• Fourier Transforms: Frequency analysis for non-periodic functions. Fourier transform as the limit of Fourier Series as the periodicity goes to in finity. Derivation of the inverse Fourier transform. Solving linear differential equations as an application.

### Prerequisites

For details of prerequisites, corequisites, excluded combinations, teaching methods, and assessment details, please see the Faculty Handbook.