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Mathematics · 100 Level
This is the complete lecture note for MTH 102 Elementary Mathematics II, prepared for Mathematics students at 100 Level. You can read it online below, open it in the full viewer, or download the PDF for offline study. It pairs with the course's video lectures, so you can read and watch together.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Outline for Series 1 1.1 Functions of a Real Variable 1.1.1 Definition of a function of a real variable 1.1.2 Domain and range 1.1.3 Types of functions (algebraic, trigonometric, exponential, logarithmic) 1.2 Graphs of Functions 1.2.1 Plotting functions and interpreting graphs 1.2.2 Common graphs (linear, quadratic, trigonometric, exponential, logarithmic) 1.2.3 Transformations of graphs (translation, reflection, scaling) 1.3 Limits of Functions 1.3.1 Concept of a limit 1.3.2 Techniques for evaluating limits 1.3.3 One-sided limits and infinite limits 1.4 Continuity of Functions 1.4.1 Definition of continuity at a point 1.4.2 Types of discontinuities (removable, jump, infinite) 1.4.3 Relationship between limits and continuity Introduction Functions are the building blocks of calculus and analysis. They describe relationships between quantities and allow us to represent real-world phenomena mathematically. In this Series, we shall explore the concept of a function of a real variable, examine how graphs provide visual insights into functions, and study the fundamental ideas of limits and continuity. These concepts are essential because they form the foundation upon which differentiation and integration are built, and they will support your learning throughout the rest of the course. The aim of this Series is to equip you with the ability to define and describe functions of a real variable, interpret and sketch their graphs, evaluate limits, and explain the meaning of continuity. We shall commence this Series by introducing the definition of a function of a real variable, including its domain, range, and common types of functions. Next, we will move on to graphs of functions, where you will learn how to plot and interpret different types of graphs and transformations. Following that, we will study the concept of limits, focusing on techniques for evaluating them and understanding one-sided and infinite limits. Finally, we will conclude with continuity, examining its definition, types of discontinuities, and the relationship between limits and continuity.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Series Learning Outcomes Upon completing this Series, you should be able to: SLO 1.1 define a function of a real variable and identify its domain and range, SLO 1.2 describe different types of functions including algebraic, trigonometric, exponential, and logarithmic, SLO 1.3 demonstrate how to plot and interpret graphs of functions and apply transformations, SLO 1.4 evaluate limits of functions using appropriate techniques, including one-sided and infinite limits, and SLO 1.5 explain the concept of continuity, identify types of discontinuities, and analyse the relationship between limits and continuity. 1.1 Functions Of A Real Variable Functions are central to calculus because they describe how one quantity depends on another. In this Part, we will carefully define what a function of a real variable is, explore its domain and range, and examine common types of functions that you will encounter throughout the course. 1.1.1 Definition of a function of a real variable A function is a rule that assigns each element in a set called the domain to exactly one element in another set called the codomain. When the codomain is the set of real numbers, we call it a function of a real variable. For example, the function f(x)=x2takes any real number xand maps it to its square. Fill in the blank: A function of a real variable assigns each element in the domain to exactly __________ element in the codomain.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Mapping of a function of a real variable 1.1.2 Domain and range The domain of a function is the set of all input values for which the function is defined. The range is the set of all possible output values. For example, for f(x)=x, the domain is all real numbers x≥0, and the range is all real numbers y≥0. Fill in the blank: The set of all possible input values of a function is called the __________. Domain and range examples Content:
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS • f(x)=x2: Domain = all real numbers, Range = y≥0. • f(x)=x: Domain = x≥0, Range = y≥0. • f(x)=1x: Domain = all real numbers except 0, Range = all real numbers except 0.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS 1.1.3 Types of functions There are several important types of functions of a real variable: Algebraic functions: Built from polynomials, rational expressions, roots, etc. Example: f(x)=x3-2x+1. Trigonometric functions: Functions involving sine, cosine, tangent, etc. Example: f(x)=sin(x). Exponential functions: Functions where the variable appears in the exponent. Example: f(x)=2x. Logarithmic functions: Inverse of exponential functions. Example: f(x)=log(x). Fill in the blank: The inverse of an exponential function is called a __________ function. [Insert Table here] Caption: Table 1.1 Types of functions of a real variable ‘ General form Example Domain Range Algebraic Polynomial, rational x2+3x+2 All real numbers All real numbers Trigonometric sin,cos,tan sin(x) All real numbers Between -1 and 1 Exponential ax 2x All real numbers Positive real numbers
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Logarithmic log(x) log(x) Positive real numbers All real numbers Pilot Questions 1.1 A function of a real variable assigns each element in the domain to: A. More than one element in the codomain B. Exactly one element in the codomain C. No element in the codomain D. None of the above (Linked to SLO 1.1) The set of all possible input values of a function is called the: A. Range B. Domain C. Codomain D. Graph (Linked to SLO 1.1) The function f(x)=xhas domain: A. All real numbers B. x≥0C. x≤0D. None of the above (Linked to SLO 1.1) The inverse of an exponential function is called a: A. Polynomial function B. Rational function C. Logarithmic function D. None of the above (Linked to SLO 1.2) Which of the following is a trigonometric function? A. f(x)=x2+3xB. f(x)=sin(x)C. f(x)=2xD. f(x)=log(x)(Linked to SLO 1.2) 1.2 Graphs Of Functions Graphs provide a visual representation of functions, making it easier to understand their behaviour. By plotting points and observing patterns, you can see how a function changes as the input varies. In this Part, we will explore how to plot and interpret graphs, examine common types of graphs, and study transformations that alter their appearance. 1.2.1 Plotting functions and interpreting graphs To plot a function, you select values from the domain, calculate the corresponding outputs, and mark these points on a coordinate plane. Connecting the points reveals the overall shape of the function. For example, plotting f(x)=x2gives a parabola opening upwards. Fill in the blank: The graph of f(x)=x2is a __________ opening upwards.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Graph of the quadratic function f(x)=x2. 1.2.2 Common graphs Several functions have characteristic graphs: Linear functions: Straight lines, e.g., f(x)=2x+1. Quadratic functions: Parabolas, e.g., f(x)=x2-3x+2. Trigonometric functions: Periodic waves, e.g., f(x)=sin(x). Exponential functions: Rapid growth or decay, e.g., f(x)=2x. Logarithmic functions: Slow growth, e.g., f(x)=log(x). Fill in the blank: The graph of f(x)=sin(x)is a __________ wave. Common graphs of functions Function type Example Graph shape Key features
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