## ENGSCI 111

Mathematical Modelling 1

Semester 1, 2019

Staff

#### Contents

Calendar notes

Introduction to mathematical modelling. Differentiation and integration (polynomials, trigonometric, exponential, logarithmic, and rational functions). Integration by parts, substitution and partial fractions. Differential equations and their solutions (including Euler's method). Vector and matrix algebra, transformations, solving systems of linear equations. Modelling using probability.
Restriction: ENGSCI 211, 213, 311, 313, 314, MATHS 150, 153

#### Outcome mapping

##### Intended learning outcomes

Use Modelling with Uncertainty to solve real world problems. This includes the following subset of skills: • Apply Bayes Theorem to update the estimate of the probability for event A (given information about event B) • Identify whether two random variables are independent • Compute probabilities of events, making use of the definite integral of a probability density function and/or the cumulative distribution functions where relevant • Comprehend the parameters involved with the exponential, Uniform and Normal probability distributions • Compute the expected value (mean) of a variable and the expected value of a function of that variable by using definite integrals. • Sketch both frequency and probability trees for a scenario described by words and use those trees to determine conditional probability values, including the application of Bayes theorem via the tree. • Explain the features of a probability density function (probability distribution) including being able to graph them. • Explain the connection between probability density functions (probability distributions) and cumulative distribution functions, including being able to graph cdfs.

Calculate derivatives to solve problems involving rates of change and series. This includes the following subset of skills: • Apply implicit differentiation to determine rates of change for equations that are defined implicitly. • Derive difference approximations and apply them to tabulated data to compute numerical derivatives • Apply the chain rule to solve rate-of-change problems, including the use of algebraic manipulation and implicit differentiation when necessary. • Compute 2 and three term polynomial approximations of functions using Maclaurin, Taylor and binomial series. • Identify the range of validity for well known series.

Model real world problems with Ordinary Differential Equations and then solve them. This includes the following subset of skills: • Apply the prescribed boundary values of the dependent variable to find any unknown constants in the general solution to a boundary value problem ODE. • Identify whether an ODE is linear, homogeneous and determine the order of the ODE. • Formulate the Euler's method iterative algorithm for a given ODE and apply it to generate a numerical solution. • Solve a second order homogeneous ODE by looking for solutions of exponential form, and finding the necessary coefficients by substitution of this trial solution into the ODE. • Solve first-order ODE's by using direct integration, method of separation or integration factor, as appropriate. • Apply the prescribed values of the dependent variable at some specified time (typically time t=0) to find any unknown constants in the general solution to an initial value problem ODE. • Apply the integration factor method to find the general solution to applicable first-order ODEs. • Apply physical laws, such as Newton's 2nd Law, to describe the spatial or temporal evolution of a system using an Ordinary Differential Equation. • Apply the method of separation to find the general solution to applicable first-order, non-constant coefficient ODEs (i.e. separable ODEs). • Find the general solution to first-order ODE's and homogeneous second-order orders, using Method of Separation, Direct Integration, Integrating Factor and Method of Exponential Substitution, as appropriate. Then apply boundary\initial conditions, to solve the full boundary\initial-value problem. • Derive the appropriate second-order to describe physical systems (often using Newton's 2nd Law), and then solve using Method of Exponential Substitution.

Calculate definite and indefinite integrals, using the techniques of integrations by parts, substitution and partial fractions where needed. This involves the following subset of skills: • Compute definite integrals when given limits of integration. • Identify an appropriate method by which to determine the integral of a given function. • Compute integrals of products by using integration by parts • Compute integrals which cannot be found directly, by using integration by substitution • Compute integrals of any rational function by using partial fractions.

Use Vectors and Matrices to describe real world problems and then solve them. This includes the following subset of skills: • Sketch a set of points in a coordinate system, and compute their positions in an alternate coordinate system • Compute the cross product of two vectors, using two methods and explain the result in the context of geometry. • Compute the determinate of 2x2 and triangular matrices, explaining the geometric significance and have knowledge of the co-factor method. • Compute the dot product of two vectors, and using two methods and find the angle between two vectors. Use the dot product to compute the projection of one vector onto another. • Apply Gaussian elimination to solve systems of linear equations • Apply 2x2 geometric transformation matrices to a set of points. • Classify the type of solution a linear system of equations will have. • Apply LU factorisation to factorise a matrix and solve a system of linear equations using forward and backward substitution. • Formulate a Markov chain matrix for a given scenario, and solve for the steady state. • Compute the inverse of a 2x2 matrix and explain the connection between the inverse, the original matrix and the identity matrix. • Compute the product of two matrices (where possible), and determine when multiplication is not possible. • Compute the L2 norm (length) of a vector. • Formulate the equation of a plane from a point and a normal direction. • Identify a singular matrix and explain the significance what being singular implies in the context of matrix equations. • Classify the solutions, and compute the unique solution if one exists. • Formulate the state transition matrix for a given scenario that can be modelled by a Markov chain. • Compute the steady state of a Markov chain. • Identify triangular matrices and solve matrix equations that involve them by forward or backward substitution. • Draw vectors on a diagram. • Compute the projection of one vector onto another, and decompose a vector into parallel and perpendicular components. • Explain and describe vectors.

Formulate Mathematical Models via the use of diagrams, assumptions, data, physical laws, dimensional analysis and approximation. This includes the following subset of skills : • Use dimensional analysis to check a model for dimensional consistency • Formulate a mathematical model by using dimensional analysis • Use free body diagrams as an aid to formulating a mathematical model when applying Newton's 2nd law • Formulate simplified expressions which involve very small and/or very large values • Explain the modelling process and apply the process to create a model • Identify valid assumptions when modelling and explain the possible impact of those assumptions • Create simple differential equations to model physical situations • Formulate equations and then optimise to solve basic max/min modelling problems • Identify proportionality relationships from word questions and use them to create an appropriate model

#### Assessment

Coursework

No description given

Exam rules

Inclusive learning

Students are urged to discuss privately any impairment-related requirements face-to-face and/or in written form with the course convenor/lecturer and/or tutor.

Other assessment rules

No description given