BUS-221 Quantitative Methods LECTURE 3

Learning Outcome Knowledge - Be familiar with basic mathematical techniques including: systems of linear equations Research - Retrieve and analyse information from directed sources for calculation and interpretation  Mentation - Analyse business case studies and make decisions based on quantitative data.

Topics Functions ▪ common functions: ln, exp, sin, cos, polynomial Linear equation systems

Equations, in Particular Linear Equations (1 of 12) An equation is a statement that two expressions are equal. The two expressions that make up an equation are called its sides. They are separated by the equality sign, =.

Equations, in Particular Linear Equations (2 of 12) Example 1 – Examples of Equations A variable is a symbol that can be replaced by any one of a set of different numbers. The most popular symbols are letters from the latter part of the alphabet, such as x, y, z, w and t.

Equations, in Particular Linear Equations (3 of 12) Equivalent Equations Two equations are said to be equivalent if they have exactly the same solutions. There are three operations that guarantee equivalence: Adding (subtracting) the same polynomial to (from) both sides of an equation. Multiplying (dividing) both sides of an equation by the same nonzero constant. Replacing either side of an equation by an equal expression.

Equations, in Particular Linear Equations (4 of 12) Operations That May Not Produce Equivalent Equations Multiplying both sides of an equation by an expression involving the variable. Dividing both sides of an equation by an expression involving the variable. Raising both sides of an equation to equal powers.

Equations, in Particular Linear Equations (5 of 12) A linear equation is also called a first-degree equation or an equation of degree one.

Equations, in Particular Linear Equations (6 of 12) Example 2 – Solving a Linear Equation

Equations, in Particular Linear Equations (7 of 12) Example 3 – Solving a Linear Equations

Equations, in Particular Linear Equations (8 of 12) Literal Equations Equations in which some of the constants are not specified, but are represented by letters such as a, b, c, or d, are called literal equations. The letters are called literal constants.

Equations, in Particular Linear Equations (9 of 12) Example 4 – Solving a Literal Equation

Equations, in Particular Linear Equations (10 of 12) Fractional Equations A fractional equation is an equation in which an unknown is in a denominator. Example 9 – Solving a Fractional Equation

Equations, in Particular Linear Equations (11 of 12) Example 5 – Literal Equation

Equations, in Particular Linear Equations (12 of 12) Radical Equations A radical equation is one in which an unknown occurs within a radical expression. Example 13 – Solving a Radical Equation

Lines (1 of 11) Slope of a Line

Lines (2 of 11) Example 1 – Price-Quantity Relationship

Lines (3 of 11) Example 1 – Continued

Lines (4 of 11) Equations of Lines

Lines (5 of 11) Example 2 – Determining a Line from Two Points

Lines (6 of 11) Example 5 – Find the Slope and y-Intercept of a Line

Lines (7 of 11) Example 3 – Converting Forms of Equations of Lines

Lines (8 of 11) Example 3 – Continued

Lines (9 of 11) Parallel and Perpendicular Lines Two lines are parallel if and only if they have the same slope or are both vertical. Moreover, any horizontal line and any vertical line are perpendicular to each other.

Lines (10 of 11) Example 4 – Parallel and Perpendicular Lines

Lines (11 of 11) Example 4 – Continued

Systems of Linear Equations (1 of 5) Two-Variable Systems There are three different linear systems: There are two methods to solve simultaneous equations: elimination by addition method elimination by substitution method

Systems of Linear Equations (2 of 5) Example 1 – Elimination-by-Addition Method Use elimination by addition to solve the system

Systems of Linear Equations (3 of 5) Example 2 – A Linear System with Infinitely Many Solutions

Systems of Linear Equations (4 of 5) Example 3 – Solving a Three-Variable Linear System

Systems of Linear Equations (5 of 5) Example 4 – Two-Parameter Family of Solutions

Systems (1 of 2) A system of equations with at least one nonlinear equation is called a nonlinear system. Example 1 – Solving a Nonlinear System

Nonlinear Systems (2 of 2) Example 1 – Solving a Nonlinear System

Exponential Functions (1 of 9) Rules for Exponents

Exponential Functions (2 of 9) Example 1 – Bacteria Growth

Exponential Functions (3 of 9) Example – Graphing Exponential Functions with 0 < b < 1 Graph the exponential function Solution:

Exponential Functions (4 of 9) Table 4.1 Properties of the Exponential Function 1. The domain of any exponential function is (−∞, ∞). The range of any exponential function is (0, ∞). 2. The graph of f(x) = bx has y-intercept (0,1). There is no x-intercept. 3. If b > 1, the graph rises from left to right. If 0 < b < 1, the graph falls from left to right. 4. If b > 1, the graph approaches the x-axis as x becomes more and more negative. If 0 < b < 1, the graph approaches the x-axis as x becomes more and more positive.

Exponential Functions (5 of 9) Example – Graph of a Function with a Constant Base Solution:

Exponential Functions (6 of 9) Exponential functions are involved in compound interest, whereby the interest earned by an invested amount of money (or principal) is reinvested so that it, too, earns interest.

Exponential Functions (7 of 9) Example – Population Growth The population of a town of 10,000 grows at the rate of 2% per year. Find the population three years from now.

Exponential Functions (8 of 9) Example – Population Growth

Exponential Functions (9 of 9) Example – Radioactive Decay

Logarithmic Functions (1 of 5) Each exponential function has an inverse. These functions, inverse to the exponential functions, are called logarithmic functions.

Logarithmic Functions (2 of 5) Example – Converting from Exponential to Logarithmic Form Exponential Form Logarithmic Form a. Since 52 = 25 it follows that log5 25 = 2 b. Since 34 = 81 log3 81 = 4 c. Since 100 = 1 log10 1 = 0

Logarithmic Functions (3 of 5) Example – Graph of a Logarithmic Function with b > 1

Logarithmic Functions (4 of 5) Example – Finding Logarithms

Logarithmic Functions (5 of 5) Example 7 – Finding Half-Life

Properties of Logarithms (1 of 8) The logarithm function has many important properties. For example:

Properties of Logarithms (2 of 8) Example 1 – Finding Logarithms by Using a Table Table 4.4 Common Logarithms x log x 2 0.3010 7 0.8451 3 0.4771 8 0.9031 4 0.6021 9 0.9542 5 0.6990 10 1.0000 6 0.7782 e 0.4343

Properties of Logarithms (3 of 8) Example 1 – Continued

Properties of Logarithms (4 of 8) Example – Writing Logarithms in Terms of Simpler Logarithms

Properties of Logarithms (5 of 8) Example – Continued

Properties of Logarithms (6 of 8) Example – Simplifying Logarithmic Expressions

Properties of Logarithms (7 of 8) Example – Continued

Properties of Logarithms (8 of 8) Example – Evaluating a Logarithm Base 5

Logarithmic and Exponential Equations (1 of 6) A logarithmic equation involves the logarithm of an expression containing an unknown. An exponential equation has the unknown appearing in an exponent.

Logarithmic and Exponential Equations (2 of 6) Example 1 – Oxygen Composition

Logarithmic and Exponential Equations (3 of 6) Example 1 – Continued

Logarithmic and Exponential Equations (4 of 6) Example – Using Logarithms to Solve an Exponential Equation

Logarithmic and Exponential Equations (5 of 6) Example – Predator-Prey Relation

Logarithmic and Exponential Equations (6 of 6) Example – Continued