This module is an introduction to multivariable calculus and differential equations, and focuses on aspects that facilitate pre-service teachers' understanding of the Functions and Calculus strand of the Senior Cycle curriculum, and consequently prepares them to teach it effectively.

Multivariable Calculus: review of vectors and introduction to vector functions; differentiation of vector functions and application of the derivative to determine the tangent to a curve and the arc length of a curve between two points; development of the notion of a derivative of a function of two or more variables and application of this knowledge to using partial derivatives of a function as a method to apply a linear approximation to a function (Taylor's expansion); interpretation of scalar and vector fields: grad, div and curl identities; determining the integral of a scalar or vector-valued function on curves. Differential Equations: order, degree, solution, boundary and initial conditions, graphs of solutions; examples from mechanics and population growth; First order ODEs: variable separable, homogeneous, linear and exact with applications; Second order differential equations: linear with constant coefficients with applications; Numerical solution of first order differential equations: The Euler method.

On successful completion of this module, students will be able to: • sketch contours and three-dimensional graphs and compute partial and directional derivatives. • approximate a function of two variables using Taylor's theorem. • evaluate double integrals. • recognise the type of a first order differential equation and apply the relevant method to find the general solution or the solution satisfying given initial conditions. • solve a first-order differential equation numerically. • solve a second-order linear differential equation with constant coefficients. • demonstrate and express an understanding of the concepts introduced in the module by providing examples in order to illustrate certain specified properties. • apply the content material to solve unfamiliar real world problems.

On successful completion of this module, students will be able to: • appreciate and value the importance of multivariable calculus and differential equations and their significance and relevance in solving problems in the real world. • experience mathematics as an enjoyable subject and through active participation in the lessons improve their overall attitude towards mathematics.

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The delivery of this mathematics module is in traditional lecture/tutorial format that integrates opportunities for active learning. Emphasis is placed on learning concepts and integrating content material within and across modules. Due attention is paid to applying mathematics learned to appropriate and relevant real world problems. Time is devoted in the module relating mathematics content to post-primary mathematics and translating topics into representations and approaches suitable for school teaching.