We start the chapter with a mathematical model of how consumers might anticipate market trends and what effect this will have on the evolution of prices. This leads us to second-order differential equations. We then embark on describing how to solve linear constant-coefficient second-order differential equations. The general solution is the sum of the solution of a corresponding homogeneous equation and a particular solution. In an analogous way to the way in which second-order recurrence equations are solved, there is a general method for determining the solution of the homogeneous equation, involving the solution of a corresponding quadratic equation known as the auxiliary equation. We explain how to find particular solutions and how to use initial conditions. We also discuss the behaviour of the solutions obtained.

The derivative is introduced as an instantaneous rate of change and it is shown how this can be determined from first principles. Techniques (sum, product, quotient and composite function rules) are then explained and the connection with small changes is illustrated. Economic interpretations via marginals are given.

Elasticity of demand is introduced and it is shown how this characterises how revenue will change upon a price increase (the two distinct possibilities being represented by elastic and inelastic demand). Profit maximisation is considered in general, and it is shown that, when maximising profit, marginal revenue and marginal cost must be equal. Two very different cases are then studied: that in which the firm is a monopoly and that of perfect competition.

This chapter focuses on the learning and teaching of second language pronunciation. It starts by asking basic questions such as: Why is it so hard to sound like a “native speaker”? and Should second language learners even sound like native speakers? The chapter reviews the components of pronunciation including phoneme, segmentals, and suprasegmentals. It then explores the topics of comprehensibility, intelligibility, and accentedness, in order to consider which are most important for second language communication and learning. It argues that native-like pronunciation should not be the goal of pronunciation instruction; rather, the goal should be comprehensibility (i.e., how well a second language learner is understood by a listener). It reviews other issues related to pronunciation learning such as the distinction between receptive and productive knowledge, the role of identity in maintaining a first-language accent, and the role that biology plays in affecting second language pronunciation mastery. Finally, the chapter explores both explicit and implicit methods of pronunciation instruction.

This chapter is intended to review concepts that the reader has some familiarity with and introduce high level descriptions of linear marine systems analysis. An initial discussion on the similarity between mechanical vibration equations of motion and marine dynamical systems is made. Mechanical vibrations are defined as vibrations in the absence of fluids. Examples of static and dynamic coupling between the various modes of motion or degrees of freedom are presented. The differences between frequency domain and time domain representations are given by introducing the concept of response amplitude operators (RAO’s). Complex arithmetic and linear, second order differential equations are briefly reviewed. Two examples of mechanical vibrations that are relevant to marine dynamics are developed and solved. The first example has to do with base excitation, similar to what a high speed planing craft may experience in long waves. The second example addresses one method for vibration isolation/suppression, that may, or may not, be useful in shock/impact mitigation schemes.

This chapter addresses psychological individual differences that are upmost importance for second language teachers. It answers teachers’ everyday questions such as Why do some students never speak? and Why do some students give up so easily? The chapter begins by explaining some key information to understand learner psychology (e.g., trait-like vs. state-like) and argues that some psychological constructs are susceptible to instruction but some are not. The chapter then discusses multiple individual differences including L2 motivation, willingness to communicate, foreign language anxiety and enjoyment, metacognition, self-regulated learning, mindset, interaction mindset, and learner beliefs. Throughout the chapter, pedagogical recommendations for maximizing learner psychology for second language learning are shared. In addition to learner psychology, the chapter discusses teacher psychology (e.g., teacher cognition) and how it influences the success of second language teaching.

Optimisation of two-variable functions is motivated via the example of a firm producing two goods, where the concepts of complementary and substitute goods are discussed. The general idea of a critical point is discussed and it is explained that there can be different types of critical point: maxima, minima and saddle points. It is explained how the second partial derivatives may be used to determine what type of critical point one has. This is shown explicitly for the case in which the two-variable function is quadratic, and then stated in general. Examples involving profit-maximisation are given.

This chapter zeroes in on the similarities and differences between first and second language acquisition. First, the chapter breaks down the term “second language acquisition” by discussing each of those words. It revisits the components of language (grammar, vocabulary, pronunciation, and pragmatics) from second language acquisition perspectives. It then introduces different second language acquisition theories such as input processing theory, skill acquisition theory, usage-based theory, sociocultural theory, complex dynamic systems theory, translanguaging, and Monitor Theory. The applicability of those theories to classroom second language teaching is discussed.

The determinant of a 2 × 2 and a 3 × 3 matrix are defined explicitly, and a more general way of (defining and) calculating determinants of larger matrices is described, involving the use of row operations to transform a matrix to upper-triangular form. It is then explained that a non-zero determinant is equivalent to invertibility. Cramer's rule is presented and a general method (based on the co-factor matrix) is given for inverting 3 × 3 matrices, an alternative to the row operations procedure described in .

Continuing from the previous chapter, this chapter explores the powerful applications of diagonalisation. We demonstrate first how it can be used to determine the powers of a matrix, which can then be applied to solve a coupled system of recurrence equations. An alternative approach to solving such systems also uses diagonalisation, but uses it to effect a change of variable so that the corresponding system in the new variables is much simpler to solve (and can then be used to revert to the solution in the original variables). We show how an analogous approach can be used to solve coupled systems of differential equations. This closing chapter provides an interesting link between the calculus and linear algebra aspects of the course.

This chapter involves the input--output model. Here, there are several goods under production, and some of each is needed to meet the production of the others, and there is also an external demand for each good. The model involves a matrix known as the technology matrix and a related matrix know as the Leontief matrix. It is shown how to solve such problems and it is explained that, in general (under very reasonable conditions), there always will be a solution. It is also shown how to approximate the solution using powers of the technology matrix.

Previous chapters presented linear models for responses of marine systems in regular, harmonic waves and various probabilistic properties of random processes, e.g. ocean waves. This chapter combines the two topics - a system’s deterministic response in the frequency domain and the statistics of that system’s random response when excited by a random, irregular sea. Several models for ocean wave spectra are presented and input/output relations for linear systems subject to stochastic excitation developed. The ocean wave environment is described by a single-sided wave spectrum based on various empirical formulae: P-M spectrum (single parameter, wind speed or significant wave height for the North Atlantic); ISSC spectrum (two parameter, significant crossing period and wave height); JONSWAP spectrum (six parameter, fetch limited, typical of the North Sea); and the Ochi six parameter spectrum (combined wind and swell). Short crested seas are defined and their effects discussed. The output spectrum of a linear system subject to stochastic input is derived and its Gaussian PDF given. By invoking a narrow banded assumption, PDF’s of the output follow the Rayleigh most probable extremes.

This chapter discusses the necessary components of second language acquisition, that is, input, interaction, and output. While all language learning theories support the importance of input (written and spoken), they diverge in ways which input is connected to second language acquisition. The chapter then examines some of the second language acquisition theories, such as the interaction hypothesis, the noticing hypothesis, the cognitive-interactionist approach, the output hypothesis, and sociocultural theory, all of which explains how interaction leads to second language acquisition. The chapter moves onto specific pedagogical frameworks that support interaction, including communicative language teaching and task-based language teaching. Finally, the chapter delves into the role that output plays by discussing its functions for second language acquisition.