ECO 104 — INTRODUCTORY TO MATHEMATICS 2: Lecture Note

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LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Course code: ECO 104 Course title: Introductory Mathematics II Course credit load: 2 Units Series 1: Derivatives of Trigonometric Functions and Expansions Series 2: Sequences and Series in Economic Applications Series 3: Partial and Total Derivatives with Optimisation Series 4: Linear and Matrix Algebra in Economics Series 5: Mathematical Relationships and Graphical Analysis of Economic Concepts Suggested Outline Part 1: Derivatives of trigonometric functions • Definition of trigonometric functions in calculus • Rules for differentiation of sine, cosine, tangent, and reciprocal functions • Higher-order derivatives of trigonometric functions • Applications of trigonometric derivatives in economics (e.g., modelling cycles and periodic behaviour) Part 2: Expansions and approximations • Concept of expansions in mathematics • Binomial theorem and its applications • Taylor and Maclaurin series expansions • Use of expansions for approximating economic functions Part 3: Sequences and series in economic contexts • Definition of sequences and series • Arithmetic and geometric progressions • Convergence and divergence of series • Application of series in modelling economic growth and investment returns

LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Part 4: Integration of derivatives, expansions, and series in economic analysis • Linking derivatives and expansions to economic modelling • Practical examples of using series and expansions in forecasting • Case studies of mathematical tools applied to introductory economics Introduction Mathematics provides the language through which many economic ideas are expressed, and calculus in particular allows us to describe change and relationships with precision. Trigonometric functions, expansions, and series are not only central to mathematics but also have practical applications in economics, such as modelling cycles, approximating functions, and analysing growth patterns. By engaging with these concepts, you will strengthen your ability to translate economic statements into mathematical form and interpret mathematical results in economic terms. The aim of this Series is to equip you with the skills to differentiate trigonometric functions, apply expansions, and work with sequences and series, while also appreciating how these tools can be applied to economic analysis. We shall commence this Series by exploring the derivatives of trigonometric functions and their rules of differentiation. Next, we will move on to expansions, beginning with the binomial theorem and continuing with Taylor and Maclaurin series. Following that, we will examine sequences and series, focusing on arithmetic and geometric progressions as well as convergence and divergence. Finally, we will bring the Series to a close by linking these mathematical tools together, showing how derivatives, expansions, and series can be integrated into economic modelling and forecasting. Series Learning Outcomes Upon completing this Series, you should be able to: SLO 1.1 define trigonometric functions and demonstrate how to differentiate sine, cosine, tangent, and reciprocal functions, SLO 1.2 explain the rules for higher-order derivatives of trigonometric functions and apply them to simple problems, SLO 1.3 describe the concept of expansions and apply the binomial theorem to mathematical and economic expressions, SLO 1.4 demonstrate how to use Taylor and Maclaurin series expansions to approximate functions, SLO 1.5 define sequences and series and distinguish between arithmetic and geometric progressions, SLO 1.6 analyse the convergence and divergence of series with reference to economic contexts, and

LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS SLO 1.7 apply derivatives, expansions, and series together to represent and solve introductory economic statements and forecasting problems. 1.1 Derivatives of Trigonometric Functions 1.1.1 Definition of trigonometric functions Trigonometric functions are mathematical functions that relate the angles of a right-angled triangle to the ratios of its sides. The primary functions are sine (sin), cosine (cos), and tangent (tan). Their reciprocal functions are cosecant (csc), secant (sec), and cotangent (cot). These functions are periodic, meaning they repeat their values at regular intervals. This property makes them useful in economics for modelling cycles such as seasonal demand or fluctuations in investment. For example, the sine function oscillates between –1 and +1, representing a repeating pattern. In economics, this can be used to model demand that rises and falls over time. Fill in the blank: The reciprocal of sine is called __________. [] Diagram showing the unit circle with sine, cosine, and tangent labelled. Figure 1.1: Unit circle representation of trigonometric functions 1.1.2 Rules for differentiation of sine, cosine, tangent, and reciprocal functions Differentiation is the process of finding the rate of change of a function. For trigonometric functions, the derivatives follow predictable rules:

LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS • d/dx (sin x) = cos x • d/dx (cos x) = –sin x • d/dx (tan x) = sec²x • d/dx (cot x) = –csc²x • d/dx (sec x) = sec x tan x • d/dx (csc x) = –csc x cot x These rules are essential for solving problems involving periodic behaviour. For instance, if demand is modelled as a sine function, its rate of change can be found using the derivative. Fill in the blank: The derivative of tan x is __________. [Insert Table here] Table 1.1: Derivatives of basic trigonometric functions Function | Derivative sin x | cos x cos x | –sin x tan x | sec²x cot x | –csc²x sec x | sec x tan x csc x | –csc x cot x AI prompt suggestion: “Create a table listing trigonometric functions and their derivatives.” Search string: “derivatives of trigonometric functions table OER mathematics” 1.1.3 Higher-order derivatives of trigonometric functions A higher-order derivative is the derivative of a derivative. For trigonometric functions, these derivatives form cycles: • Differentiating sin x repeatedly produces: sin x → cos x → –sin x → –cos x → sin x. • Differentiating cos x repeatedly produces: cos x → –sin x → –cos x → sin x → cos x. This cyclical behaviour is particularly useful in modelling phenomena that repeat over time. For example, seasonal demand in agriculture or tourism can be represented using sine and cosine functions, and their derivatives show how quickly demand is rising or falling. Fill in the blank: Differentiating cos x four times returns the function back to __________.

LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS [] Diagram showing the cycle of derivatives for sin x and cos x. Figure 1.2: Cyclical nature of higher-order derivatives of sine and cosine 1.1.4 Applications of trigonometric derivatives in economics Trigonometric derivatives are not just abstract mathematics, they have practical applications in economics: • Cyclical demand: Sine and cosine functions can model demand that rises and falls periodically. • Seasonal production: Agricultural output often follows seasonal cycles, which can be represented mathematically. • Oscillating growth models: Investment returns or market cycles can be analysed using trigonometric derivatives. For instance, if consumer demand follows a sine curve, the derivative shows whether demand is increasing or decreasing at a given point in time. This helps economists and businesses anticipate changes and plan accordingly. Fill in the blank: Oscillating growth models can be analysed using __________ derivatives. [

LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Topic / Application Area Description of Use Derivative Use / Interpretation Modeling Seasonal Demand Many industries (tourism, agriculture, retail) face cyclical demand. If demand ($Q$) is modeled as $f(t) = \sin(t)$, the rate of change is seasonal. The derivative, $\frac{dQ}{dt} = \cos(t)$, provides the instantaneous rate of change of demand, crucial for forecasting inventory and staffing. Marginal Analysis of Periodic Revenue Companies with cyclic income (e.g., ski resorts, summer festivals) model their total revenue ($R$) using sine waves. Set Marginal Revenue ($MR = \frac{dR}{dt}$) to zero to find the exact timing of peaks and troughs in the seasonal cycle. Business Cycle Dynamics Macroeconomic models can use sine functions to represent the repeating phases of expansion and contraction (the Business Cycle). Higher-order derivatives are used to define the boundaries (turning points), stability conditions, and period (duration) of the cycle. Option Pricing & Cyclic Volatility In finance, certain models (less standard than Black-Scholes) may use trigonometric functions to model predictable, repeating patterns of volatility in asset prices (e.g., time-of-day or seasonal effects). Derivatives are used to calculate sensitivity (Greeks) related to these periodic components, which are essential for risk management. Trade and Commodity Price Cycles Commodity prices often follow seasonal or multi-year cycles. Models of these cycles assist with trade strategy. Derivatives are used to find the rate of acceleration or deceleration of price changes at various stages of the cycle. ] Box 1.1: Applications of trigonometric derivatives in economics Application | Description Cyclical demand | Sine and cosine functions model periodic demand Seasonal production | Trigonometric functions represent seasonal variations Oscillating growth | Derivatives help analyse fluctuations in growth models

LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Pilot questions 1.1 1. Define trigonometric functions and give two examples. 2. State the derivative of cos x. 3. What is the derivative of tan x? 4. Explain the cyclical nature

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