Applications of Finite Mathematics

Applications of Finite Mathematics

Applications of Finite Mathematics

Gautami Devar

Applications of Finite Mathematics

Gautami Devar

ISBN - 9789361527173

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Preface

This is a book designed for senior secondary level students. It was a work of efforts to compile all the vast topics into a compressed book form. As an author we have focused on holding the standards of mathematics learning and gave our level best to mould the topics so that study of mathematics seems interesting and not a compulsion for the students and also managed to retain the essence of knowledge from a widely valued topic. Through this methodology, the students will acquire the necessary and will face interesting and approachable challenges towards the end of each of the learning’s. It is noticed that the different perspectives of people teaching varies on a very wide scale. Some need it to be of the most utility in the practical world and some need the knowledge to start from the basics, including every small or big aspect in the field of Mathematics. In this book, it is tried to satisfy the requirements of each perspective from the wide range of varieties.

Although, it was not easy to understand the topics required ­­­to study at a specific level and comprehend each of the topics at an operational level. But the prior goal as an author was to create a content that involves everyone of an age group doing mathematics. Also, another important step taken was to incorporate a language that is easily understood by all the people who may read the book.

The team that developed this text book consists of the experienced people from different background and with different perspective. This variation of perspectives helped in understanding the multiple aspects of different kinds of readers and their requirements and expectations with this textbook.

Also, another important thing that needs to be mentioned is the authors worked as a team and accepted varied ideas and advices from each other considering each one a knowledgeable person in different fields of interests. Doing this we also overcame the problem that arose by stretching ourselves to our maximum capacity and now we hope that we did justice to the challenges that persisted before us. The process of developing and reevaluating our content will remain continuous and will keep trying to make our book better with each upcoming editions. Suggestions and comments are welcomed.

Table of Contents

1. Functions and Relations 1

1.1 The Rectangular Coordinate System and Graphing Utilities 1

1.1.1 Plot points on a Rectangular Coordinate System 2

1.1.2 The Distance and Midpoint Formulas 3

1.1.3 Graph Equations by Plotting Points 8

1.1.4 Identify ­x and y-Intercepts 10

1.2 Circles 11

1.2.1 Write an Equation of a Circle in Standard Form 12

1.2.2 Write the General Form of an Equation of a Circle 14

1.3 Functions and Relations 15

1.3.1 Determine Whether a Relation is a Function 16

1.3.2 Apply Function Notation 18

1.3.3 Determine the x and y-intercepts of

a Function Defined by y = f(x) 19

1.3.4 Determine Domain and Range of a Function 19

1.4 Linear Equations in Two Variables and Linear Functions 21

1.4.1 Graph Linear Equations in Two Variables 21

1.4.2 Determine the Slope of a Line 23

1.4.3 Apply the Slope – Intercept Form of a Line 25

1.4.4 Compute Average Rate of Change 27

1.5 Applications of Linear Equations and Modeling 31

1.5.1 Apply the Point – Slope Formula 31

1.5.2 Determine the Slopes of Parallel and

Perpendicular Lines 33

1.5.3 Create a Linear Function in an Application 35

1.6 Transformations of Graphs 37

1.6.1 Recognize Basic Functions 37

1.6.2 Apply Vertical and Horizontal Translations 39

1.6.3 Apply Vertical and Horizontal Shrinking and Stretching 41

1.6.4 Apply Reflections across the x and y-Axes 45

1.6.6 Summarize Transformations of Graphs 47

1.7 Analyzing Graphs of Functions and Piecewise –

Defined Functions 50

1.7.1 Test for Symmetry 50

1.7.2 Identify Even and Odd Functions 53

1.7.3 Graph Piecewise – Defined Functions 55

1.7.4 Investigate Increasing, Decreasing, and

Constant Behavior of a Function 57

1.7.5 Determine Relative Minima and Maxima of

a Function 58

1.8 Algebra of Functions and Function Composition 61

1.8.1 Perform Operations on Functions 61

1.8.2 Evaluate a Difference Quotient 65

1.8.3 Compose and Decompose Functions 67

References 71

2. Linear Programming 72

Introduction 72

2.1 Linear Inequalities 73

2.2 Properties associated with Linear Inequalities 75

2.3 Graphing Linear Inequalities 77

2.4 Linear Programming Practical Problems 84

2.4.1 Furniture Manufacturing Problem 84

2.4.2 People’s Nutrition Problems 89

2.4.3 Packaging Problems 94

2.4.4 Investments and Funds Problems 98

2.4.4 Transportation and Shipping Problems 104

References 117

3. Vectors 118

3.1 Scalars and Vectors: Introduction 118

3.1.1 Magnitude of the Vector 120

3.1.2 Algebraic Properties 120

3.1.3 Classifications of the Vectors 127

3.2 Properties of Vector 128

3.2.1 Associative Property 128

3.2.2 Commutative Property 129

3.2.3 Plane of a Vector 131

3.2.4 Conditional Vectors 133

3.3 Coplanar Vectors 137

3.3.1 Laws in Vectors 139

3.3.2 Direction Cosines and their Ratios 140

3.3.3 Laws of Cosines and Sines 147

3.3.4 Projection of the Vector 147

3.4 Section Formula 153

3.5 Standard basis of R² and R³ 159

3.5.1 Linear Combination of the R² and R³ 160

3.5.2 Test for Dependency 162

3.5.3 Basis and its Properties 168

3.5.4 Span over the Basis 169

3.6 Vector Subspace 172

3.6.1 Operations under Vector Subspace 173

References 177

4. Linear and Non-linear Functions 178

4.1 Straight Lines 178

4.1.1 Introduction 178

4.1.2 Slope of Straight Line 178

4.1.3 Angle between two Straight Lines 179

4.1.4 Collinearity of three Points 180

4.1.5 Different forms of Line 181

4.1.6 Distance 191

4.1.7 Section Formula 191

4.1.7 Shifting of Axis 192

4.1.8 Rotation of Axis 193

4.2 Lines in Pair 194

4.2.1 Family of Lines 194

4.2.2 Homogeneous Equation of Second Degree 196

4.2.3 Angle Bisector 199

4.3 Non-linear Function 202

4.3.1 Introduction 202

4.3.2 Exponential Function 202

4.3.3 Logarithmic Functions 221

4.3.4 Conversion between Exponential &

Logarithmic Functions 234

References 240

5. Binomial Expansion, Sequence and Series 241

5.1 Binomial Expansion 241

5.1.1 Pascal’s Triangle 242

5.1.2 Factorial Notation 244

5.1.3 Binomial Theorem 246

5.2 Sequence 252

5.2.1 Finite and Infinite Sequences 252

5.2.2 Increasing and Decreasing Sequence 253

5.3 Series 256

5.3.1 Partial Sum 257

5.4 Arithmetic Sequence or Arithmetic Progression 260

5.4.1 General term (or nth term) of

an Arithmetic Sequence 262

5.4.2 Sum of an Arithmetic Sequence 263

5.4.3 Arithmetic Mean 267

5.4.4 Real life examples of Arithmetic Sequence 270

5.5 Geometric Sequence or Geometric Progression 272

5.5.1 General term (or nth term) of

a Geometric Sequence 275

5.5.2 Sum of a Geometric Sequence 276

5.5.3 Geometric Mean ( G. M. ) 281

5.5.4 Real life examples of Geometric Sequence 283

5.6 Relation between Arithmetic Mean and Geometric Mean 285

5.7 Convergence and Divergence of Series 287

5.7.1 Limit of a Sequence 287

5.7.2 Determining the Convergence and

Divergence of a Sequence 289

5.7.3 nth term test for the Divergence of

an Infinite Sequence 291

5.7.4 Test for the Convergence of Geometric Sequence 292

5.8 The p-Series and the Ratio Test for

the Convergence or Divergence of Series 296

5.8.1 The Harmonic Series 296

5.8.2 The p- Series 296

5.8.3 Test for the Convergence of a p- Series 297

5.8.4 Ratio Test for the Convergence and

Divergence of Series 298

5.8.5 Radius of Convergence 300

References 302

6. Permutations and Combinations 303

6.1 Permutations 304

6.1.1 Permutation of n Different Objects 305

6.1.2 Permutation of n Different Objects

when Repetition is Allowed 315

6.1.3 Permutation when the objects are not Distinct 316

6.2 Combinations 320

6.2.1 Combination Formulas 320

References 338

Index 339

Chapter

1.Functions and Relations

1.1 The Rectangular Coordinate System and Graphing Utilities

A century mathematician Rene Descartes found the technique of locating points in a plane with the help of a pair of coordinates. He intersected two perpendicular number lines and the point of their intersection became the origin. The lines make a rectangular coordinate system. Another name for it being the Cartesian coordinate system. The horizontal line is the x - axis and the line perpendicular to it is the y - axis. These two axes divide the plane into four quadrants.

1.1.1 Plot points on a Rectangular Coordinate System

Consider Figure 1, every point present in the plane can be distinctively recognized by using an ordered pair to specify its location with respect to the origin. In an ordered pair, the first co-ordinate represents the x - coordinate whereas the second co-ordinate represents the y- coordinate. The origin has the co-ordinates .

In Figure-2, 6 points have been plotted on the graph.

The point is positioned 4 units in the positive x direction (leftward) and 2 units in the negative y direction (downward).

1.1.2 The Distance and Midpoint Formulas

We know the distance between two points X and Y on a number line to be given by the formula or . However, to find the distance between two points on a co-ordinate plane is an entirely different matter.

Now, suppose that we have to find the distance between the points and . The required distance notified by d is shown in Figure-3. The dashed horizontal and vertical line segments form a right-angled triangle having hypotenuse equal to d.

Clearly,

From the Figure-3, the horizontal distance between the points is .

Similarly, the vertical distance between the points is .

Using the Pythagorean Theorem,

Now, d being the distance cannot have a negative value. So, the value of d is 5 units.

Thus, the distance between the points and is 5 units.

This process can be made generalized by labeling the points as and . Consult figure-4.

The horizontal base of the right angled triangle is or .

The vertical height of the right angled triangle is or .

On using the Pythagorean Theorem,

We can ignore the absolute value bars as for all real numbers a. Similarly, and .

1. Distance Formula

The distance between points and is given by

Example 1 How far apart are the points and from each other? Give the exact value and an approximation to decimal places.

Solution

Label the points and as and respectively.

Note: The choice of points for and will not affect the result obtained.

Apply the distance formula and simplify to obtain the desired answer,

The exact distance between the points is units. This is approximately equal to units.

Note The Pythagorean Theorem is useful in the determination of whether any triangle is right-angled or not. If the condition is satisfied, then the triangle having lengths a, b and hypotenuse c, is a right angled triangle.

Example 2 Assert if the set of points

form the vertices of a right angled triangle.

Solution

Apply the distance formula for finding the distance between each pair of points.

Similarly,

And,

Now, the longest line segment is MQ and should be the hypotenuse, c. Mark the shorter sides as a and b.

Verify the condition of Pythagorean Theorem that .

As the condition for the Pythagorean Theorem is not satisfied, so the points M, P and Q are not the vertices of a right-angled triangle.

2. Midpoint Formula

Now, we can also find the midpoint of a line segment between the two points and . The midpoint of a line segment is a point that is at equal distances from either ends of the line segments.

Consider Figure 5, the midpoint’s x - coordinate is the average of the x - coordinates of the end points and the midpoint’s y - coordinate is the average of the y - coordinates of the end points.

Therefore, a line segment having endpoints and has a midpoint

Example 3 Evaluate the midpoint of a line segment having and as its endpoints.

Solution

Label the points and as and respectively.

Apply the midpoint formula and simplify to obtain the solution.

Thus, the co-ordinates of the required midpoint of the line segment are .

1.1.3 Graph Equations by Plotting Points

A graph is often used for showing the relationship between two variables. Consider two variables, x and y, which are related in such a way that y is more than x. The equation that represents this relationship is . Solution of this equation in the variables x and y is called an ordered pair and when these ordered pairs are substituted into the equation, the condition of the equation is satisfied. For example, the ordered pairs of the equation are . All these ordered pairs satisfy the equation.

Similarly,

and

The solution set of the equation is the set of all the solutions to a particular equation. The graph of the equation is the graph of all solutions to a particular equation. Figure-6 represents the graph of .

This is called the point-plotting method of graphing the solution set of an equation. First values of x are selected and substituted in the equation for calculating the corresponding values of y. The points are then plotted on a graph to form an outline of the curve.

Example 4 Plot a graph by plotting the points of the equation .

Solution

Solve the equation for y in terms of x.

Randomly select the values of x and then calculate the corresponding values of y.

The graph is sketched by plotting the points one by one on the graph paper. The graphical representation of the equation has been shown in the figure beside.

1.1.4 Identify ­x and y-Intercepts

Graphs have two very important features and these are their x and y-intercepts. These are the points where the graph of a particular equation intersects the x and the y-axes.

The y-coordinate is always equal to zero on the x-axis and similarly the x-coordinate is always zero on the ­y-axis. Thus, an x-intercept is a point and a y-intercept is a point .

Example 5 For the provided equation , find the x-intercepts and the y-intercepts.

Solution

The equation is .

To find the x-intercepts, substitute in the equation and solve for x.