for Business, Science, and Technology

It is written for the reader who needs a review of the very basics of arithmetic. The most notable changes are the addition of the new Chapters 6 and 7, end-of-chapter summaries and detailed solutions to all exercises.

Introduction

In this text, a left-aligned horizontal line indicates the beginning of a case, and a right-aligned horizontal line indicates the end of a case. Also, when one case immediately follows the previous case, the right-aligned line will be omitted.

Elementary Algebra Example 1.1

Algebraic Equations

Algebraic Equations Example 1.2

Elementary Algebra Example 1.4

In this case, this will be achieved if we subtract 5 from both sides of the given equation and after simplification we get the value of the unknown as We do this by multiplying both sides of the equation by. and after simplification we get

Laws of Exponents the given equation

Laws of Exponents

Elementary Algebra

Laws of Exponents

Excel has several financial functions that apply to annuities.* It is beyond the scope of this text to discuss them all. As a first step, we use the alternative designation of the square root; this is discussed in Appendix A, Page A−31.

Laws of Exponents Solution

Laws of Logarithms

Logarithms are very useful because laws (1.27) to (1.29) allow us to replace multiplication, division, and exponentiation with addition, subtraction, and multiplication, respectively. As we mentioned in Appendix A, the common (base 10) logarithm and the natural (base e) logarithm are most commonly used, where e is an irrational (infinite) number with the value ..

Laws of Logarithms Interest = earned interest

Elementary Algebra Then, in any cell we enter the formula

Common logarithms consist of an integer, called the characteristic, and an endless decimal, the mantissa*. Because mantissas appear as positive numbers in math tables, the negative sign is written above the attribute.

Quadratic Equations For negative numbers, the mantissa is the complement * of the mantissa given in math tables

Quadratic Equations

Quadratic Equations For negative numbers, the mantissa is the complement* of the mantissa given in the math tables.

Cubic and Higher Degree Equations

Cubic and Higher Degree Equations A cubic equation has the form

Measures of Central Tendency

The median of a sample is the value that separates the bottom half of the data from the top half. To find the mean, we sort the sample values in ascending (ascending) order.

An exponential moving average (EMA) takes a percentage of the most recent value and adds the exponential moving average of the previous value times 1 minus that percentage. Measures of Central TendencyFor the example* below, we'll use the Simulink Weighted Moving Average block.

Measures of Central Tendency For the example * below, we will use the Simulink Weighted Moving Average block

Elementary Algebra 1.8 Interpolation and Extrapolation

Let us consider the points shown in Figure 1.2 where, in general, the designation is used to indicate the intersection of lines parallel to and. Next, let's assume that a known value lies between and and we want to find the value that corresponds to the known value of .

Interpolation and Extrapolation shown in Figure 1.3

If we use linear interpolation to find the square of 5 using the data in the table above, we will find that. Infinite sequences and series and obviously this is a big mistake since the square of , no.

Figure 1.4. Graph for example 1.22 Solution:

Infinite Sequences and Series

Arithmetic Series

Geometric Series

Geometric Series (1.57)

Harmonic Series

Suppose we have had a list of temperature numbers for a certain day and time for years. Proportionsus assumes that the numbers are arbitrary, i.e. the temperature on a given day and time of one.

Proportions us assume the numbers are random, that is, the temperature on a particular day and time of one

Proportions

The golden ratio is that where the ratio of the shorter segment to the longer segment, i.e., is equal to the ratio of the longer segment to the entire segment, i.e. The golden rectangle is defined as a rectangle whose length and width are as shown in Figure 1.7.

Summary 1.14 Summary

Logarithms consist of a whole number called the characteristic and an infinite decimal called the mantissa. This means that the negative sign does not apply to the mantissa.

Summary

A geometric series (or geometric progression) is a sequence of numbers such that each number bears a constant ratio, called the common ratio, to the previous number. A sequence of numbers whose reciprocals form an arithmetic series is called a harmonic series (or harmonic progression).

Exercises 1.15 Exercises

The formula below computes the terms of an annuity with an initial payment and periodic pay- ments

Solve the following quadratic equations

Twenty houses were sold for the prices listed below where, for simplicity, the dollar ($) sign has been omitted

Solutions to End-of-Chapter Exercises 1.16 Solutions to End-of-Chapter Exercises

Elementary Algebra 1

Systems of Two Equations

Then we verify the answers with the worksheet below, where the values were entered into cells A1 to A20 in the order given and they were copied into B1 to B20 and were sorted. However, a solution can be found if we write two or more linear independent equations* with the same number of unknowns, and solve these simultaneously.

Intermediate Algebra

Systems of Two Equations

Intermediate Algebra 2.2 Systems of Three Equations

Let us write these as for brand A, for brand B and for brand C. Then the sales for each of the three-month periods can be represented by the following system of equations*. 2.14). In the next section we discuss matrix theory and methods for solving these types of systems of equations.

Matrices and Simultaneous Solution of Equations 2.3 Matrices and Simultaneous Solution of Equations

Two matrices and are said to be conformal for multiplication in the order only when the number of columns of matrix is equal to the number of rows of matrix. The product, which is not the same as the product, can be adapted to multiplication only if it is one and the matrix is a matrix.

Matrices and Simultaneous Solution of Equations

An identity matrix is a square matrix where all elements on the main diagonal are one and all other elements are zero. Let matrix be defined as the square matrix. then the determinant of , denoted as , is defined as. 2.17).

Matrices and Simultaneous Solution of Equations Example 2.6

If we remove its row and column elements, the determinant of the remaining square matrix is called the minor determinant and denoted as Calculate the determinant from the elements of the first row and their cofactors with respect to it.

Matrices and Simultaneous Solution of Equations Example 2.11

We note that the numerators (2.23) are determinants that are formed from Δ by substituting the known values , , and for the coefficients of the desired unknown.

Matrices and Simultaneous Solution of Equations Solution

Intermediate Algebra Example 2.15

If and are square matrices such that where is the identity matrix, is called the inverse of, denoted as, and is also called the inverse of, that is,. We can also use Microsoft Excel's MINVERSE (Matrix Inversion) and MMULT (Matrix Multiplication) functions to obtain the values of the three unknowns of Example 2.18.

Figure 2.3. Spreadsheet for Example 2.18

Intermediate Algebra 2.4 Summary

If the matrix is a square matrix and is a cofactor of , the adjoint of , denoted as , is defined as a square matrix below. The cofactors of the row (column) elements of , are the column (row) elements of .

Intermediate Algebra 2.5 Exercises

Perform the following computations, if possible. Verify your answers with Excel or MATLAB
Solve the following system of equations using Cramer’s rule. Verify your answers with Excel or MATLAB
Repeat Exercise 4 using the Gaussian elimination method
Solve the following system of equations using the inverse matrix method. Verify your answers with Excel or MATLAB

Exercises 8. Use Excel to find the unknowns for the system

Solutions to End − of − Chapter Exercises 1

Solutions to End−of−Chapter Exercises

Intermediate Algebra 4

Introduction

The science of geometry also includes analytic geometry, descriptive geometry, fractal geometry, non-Euclidean* geometry, and spaces of four or more dimensions. For relatively small distances, Euclidean geometry and non-Euclidean geometry are almost the same.

Fundamentals of Geometry

Plane Geometry Figures Solution

If is the hypotenuse of a right triangle, the area of this triangle is found from the relation. As we know, but we reject the negative value as it is unrealistic for this example.

Plane Geometry Figures A triangle that has two equal sides is called isosceles triangle. Figure 3.3 shows an isosceles triangle

Since the perpendicular line divides the isosceles triangle into two equal right triangles, it follows that.

Plane Geometry Figures For an equilateral triangle,

VIII Any triangle can be divided into four congruent triangles by connecting the midpoints of its three sides as shown in Figure 3.11. X Any isosceles triangle can be divided into two congruent triangles by one of its altitudes.

Figure 3.9. Two angles and a non−included side of one triangle are congruent to the two angles and the non−included side of the other triangle

Plane Geometry Figures

All congruent triangles are similar, but not all similar triangles are congruent
If the angles in two triangles are equal and the corresponding sides are the same size, the trian- gles are congruent
If the angles in two triangles are equal the triangles are similar
If the angles in two triangles are not equal the triangles are neither similar nor congruent

If the angles in two triangles are equal and the corresponding sides are the same size, the triangles are congruent. If the angles in two triangles are not equal, the triangles are neither similar nor congruent.

Plane Geometry Figures Property 5

Solid Geometry Figures

A solid with six planes, each a parallelogram, and each parallel to the opposite plane, is called a parallelepiped. The part of a cone between two parallel planes that intersects the cone, especially the part between the base and a plane parallel to the base, is called a frustum of a cone.

Solid Geometry Figures

Using Spreadsheets to Find Areas of Irregular Polygons The area of any polygon can be found from the relation

We arbitrarily choose the origin as our starting point and traverse the polygon counterclockwise as shown in Figure 3.35. The Excel spreadsheet in Figure 3.36 below calculates the area of any polygon with up to 10 vertices.

Using Spreadsheets to Find Areas of Irregular Polygons

Summary 3.5 Summary

A solid with six faces, each of which is a rectangle and each of which is parallel to the opposite face, is called a rectangular parallelepiped. A three-dimensional surface whose points are all equidistant from a fixed point is called a sphere.

Exercises 3.6 Exercises

Compute the area of the general triangle shown below in terms of the side and height

For an equilateral triangle below, prove that

Fundamentals of Geometry 5. Prove the Phythagorean theorem

Solutions to End−of−Chapter Exercises 3.7 Solutions to End − of − Chapter Exercises

Fundamentals of Geometry 3

Introduction

An angle is an angular unit of measurement with the vertex (the point at which the sides of an angle intersect) at the center of a circle and with sides that subtend (cut) part of the circumference. If the subtended arc is equal to a quarter of the total circumference, the angular unit is a right angle.

Fundamentals of Plane Trigonometry

Trigonometric Functions
Trigonometric Functions of an Acute Angle

Trigonometric Functions of an Any Angle

Trigonometric Functions of an Arbitrary Angle Now let's look at the rectangular Cartesian coordinate system of Figure 4.5, where the radius is located.

Figure 4.3. Quadrants in Cartesian Coordinates

Trigonometric Functions of an Any Angle Now, let us consider the rectangular Cartesian coordinate system of Figure 4.5 where the ray

Fundamental Relations and Identities

From these, we can derive the angles of other trigonometric functions using relations.

Fundamental Relations and Identities The formulas below give the values of special angles in both degrees and radians

Fundamental Relations and Identities (4.66)

Fundamental Relations and Identities

Fundamentals of Plane Trigonometry 4.6 Triangle Formulas

Inverse Trigonometric Functions

Area of Polygons in Terms of Trigonometric Functions

Area of polygons in terms of trigonometric functions Besides using Excel to find the values of trigonometric functions, one can also use MATLAB.

Area of Polygons in Terms of Trigonometric Functions Besides using Excel to find the values of trigonometric functions, one can also use MATLAB

Fundamentals of Plane Trigonometry 4.9 Summary

If the subtended arc is equal to the circumference, the unit of angle is a degree. The numerical values of the trigonometric functions can be found by using formulas called trigonometric identities.

Fundamentals of Plane Trigonometry 4.10 Exercises

Solutions to End−of−Chapter Exercises 4.11 Solutions to End − of − Chapter Exercises

Solutions to End−of−Chapter Exercises b

Introduction

Calculus is the branch of mathematics that deals with concepts such as the rate of change, the slope of a curve at a particular point, and the calculation of an area bounded by curves. The two branches into which elementary calculus is usually divided are differential calculus, based on the limits of ratios, and integral calculus, based on the limits of sums.

Differential Calculus

For example, the amount of postage required to send a package is related to its weight. Here the weight of the package is considered as the independent variable and the amount of postage is considered as the dependent variable.

Fundamentals of Calculus

The Derivative of a Function where the notation is used to indicate that the velocity is a function of time, denotes a

The Derivative of a Function

At this limit, the slope becomes tangent to the curve at the point. we denote this slope as. 5.8). We express the derivative of (5.8) as. 5.9) where is held fixed and varies and approaches zero but never becomes exactly zero; otherwise the right-hand side of (5.9) would be reduced to the indefinite* form.

The Derivative of a Function (5.12)

Henceforth, the terminology "derivative" will mean the first derivative, that is, if , is its derivative. For higher order derivatives we will use the terminology "second derivative", "third derivative" and so on.

The Derivative of a Function 2. If

This is the same answer as in Example 5.2, which was found by direct application of the definition of the derivative. In Column A of the spreadsheet shown in Figure 5.3, we enter the time in increments of 0.020 hours.

The Derivative of a Function

Thus (5.28) This result shows that after driving for , the distance has decreased from to miles. Maxima and Minima are derivatives that show the rate of increase in cost per unit increase in output.

Maxima and Minima is the derivative which indicates the rate of increase of cost per unit increase in produc-

Maxima and Minima

At these points the slope of the tangent to the curve is said to be positive. At these points the slope of the tangent to the curve is said to be negative.

Maxima and Minima Solution

Integral Calculus

Fundamentals of Calculus Note 5.2

Definite Integrals

The integral of the sum of two or more differentials is the sum of their integrals. If, that is, if is not equal to minus one, the integral of is obtained by adding one to the. exponent and division by the new exponent.

Definite Integrals

In (5.58), is the integral of , and the meaning of the notation is that we must first replace with the upper value , that is, we set to obtain , and from it we subtract the value , which is obtained by putting . The lower value in (5.58) is called the lower limit of integration, and the upper value is the upper limit of integration.

Figure 5.7. Plot for the function of Example 5.7.

So one has to leave their money in a bank or savings and loan for about 10 years for the original deposit to double if the interest rate. 5.62) is called a second-order differential equation. The resulting motion is called simple harmonic motion and is described by a second-order differential equation such as (5.62).

Summary 5.8 Summary

When evaluating indefinite integrals, we add the constant referred to as an arbitrary constant of integration. In definite integrals, the lower value is called the lower limit of integration, and the upper value is the upper limit of integration.

Exercises 5.9 Exercises

Find the derivatives of the following functions

Evaluate the following indefinite integrals

Evaluate the following definite integrals, that is, find the area under the curve for the lower and upper limits of integration

Solutions to End − of − Chapter Exercises 1

Solutions to End−of−Chapter Exercises Check with the MATLAB diff(f) function

Fundamentals of Calculus 3

Common Terms .1 Bond

Corporate Bond
Municipal Bond
Treasury Bond
Perpetuity
Perpetual Bond

In exchange for our money, the entity will issue us a certificate or bond, indicating the interest rate we must pay and when our borrowed funds will be returned (maturity date). Because it is tax-free, the interest rate is usually lower than on a taxable bond.

Mathematics of Finance and Economics

Convertible Bond
Treasury Note
Treasury Bill
Face Value
Par Value
Book Value
Coupon Bond
Zero Coupon Bond
Junk Bond
Bond Rating Systems
Promissory Note
Discount Rate
Prime Rate
Mortgage Loan
Predatory Lending Practices
Annuity
Ordinary Annuity
Sinking Fund
Interest

Simple Interest
Compound Interest

If no interest is mentioned, the maturity value of the note is its face value. However, if an interest payment is specified, the maturity value is the nominal value of the note plus accrued interest.

TABLE 6.1 Bond−Ratings Moody's Standard & Poor's Grade Risk

Mathematics of Finance and Economics Solution

What was the interest rate we paid, assuming interest was compounded annually. Therefore, the interest rate we paid is to the nearest tenth of one percent.

Mathematics of Finance and Economics For our example,

Effective Interest Rate

The effective interest rate is numerically equal to the interest earned on the principal during the year.

Mathematics of Finance and Economics 6.3 Sinking Funds

Annuities

If the interest rate is compounded annually, what is the principal in the fund at the end of the fourth year. Thus, the principal in the sinking fund at the end of the fourth year is

Mathematics of Finance and Economics 6.5 Amortization