Basic Technical Mathematics with Calculus

Allyn J. Washington received his Masters Degree from Brown University, Providence, Rhode Island. Allyn taught mathematics at Trinity College, Hartford, Connecticut, after graduating Phi Beta Kappa from Trinity. He was Professor of Mathematics at Dutchess Community College, Poughkeepsie, New York, where he served as the Mathematics Department Head as well as Dean of the College. Allyn is a founding member of the New York State Mathematics Association of Two Year Colleges and also a founding member of the American Mathematics Association of Two Year Colleges. He was awarded an Honorary Doctorate Degree by State University of New York. He is presently listed in Who's Who in America. The Allyn J. Washington Center for Science and Art Building is named in his honor at Dutchess Community College, Poughkeepsie, New York. Allyn has authored several textbooks in Technical Mathematics for over 50 years, including this edition of Basic Technical Mathematics with Calculus.   Richard Evans received a Master’s Degree in mathematics from Binghamton University in Binghamton, New York. He is currently a professor of mathematics at Corning Community College, where he has taken an active role in the college’s governance system, including having chaired the mathematics department as well as the College Association. He is a long-standing member of the New York Mathematics Association of Two-Year Colleges and has attended and presented at many of these annual conferences. Rich has a strong background in applied mathematics, and has taken two sabbatical leaves involving real-world applications of math in business and industry. In his free time, Rich enjoys spending time with his family, playing guitar, and boating in the Finger Lakes.

Table of Contents

Basic Algebraic Operations

1.1 Numbers
1.2 Fundamental Operations of Algebra
1.3 Calculators and Approximate Numbers
1.4 Exponents and Unit Conversions
1.5 Scientific Notation
1.6 Roots and Radicals
1.7 Addition and Subtraction of Algebraic Expressions
1.8 Multiplication of Algebraic Expressions
1.9 Division of Algebraic Expressions
1.10 Solving Equations
1.11 Formulas and Literal Equations
1.12 Applied Word Problems


Geometry

2.1 Lines and Angles
2.2 Triangles
2.3 Quadrilaterals
2.4 Circles
2.5 Measurement of Irregular Areas
2.6 Solid Geometric Figures


Functions and Graphs

3.1 Introduction to Functions
3.2 More about Functions
3.3 Rectangular Coordinates
3.4 The Graph of a Function
3.5 Graphs on the Graphing Calculator
3.6 Graphs of Functions Defined by Tables of Data


The Trigonometric Functions

4.1 Angles
4.2 Defining the Trigonometric Functions
4.3 Values of the Trigonometric Functions
4.4 The Right Triangle
4.5 Applications of Right Triangles


Systems of Linear Equations Determinants

5.1 Linear Equations and Graphs of Linear Functions
5.2 Systems of Equations and Graphical Solutions
5.3 Solving Systems of Two Linear Equations in Two Unknowns Algebraically
5.4 Solving Systems of Two Linear Equations in Two Unknowns by Determinants
5.5 Solving Systems of Three Linear Equations in Three Unknowns Algebraically
5.6 Solving Systems of Three Linear Equations in Three Unknowns by Determinants


Factoring and Fractions

6.1 Factoring: Greatest Common Factor and Difference of Squares
6.2 Factoring Trinomials
6.3 The Sum and Difference of Cubes
6.4 Equivalent Fractions
6.5 Multiplication and Division of Fractions
6.6 Addition and Subtraction of Fractions
6.7 Equations Involving Fractions


Quadratic Equations

7.1 Quadratic Equations; Solution by Factoring
7.2 Completing the Square
7.3 The Quadratic Formula
7.4 The Graph of the Quadratic Function


Trigonometric Functions of Any Angle

8.1 Signs of the Trigonometric Functions
8.2 Trigonometric Functions of Any Angle
8.3 Radians
8.4 Applications of Radian Measure


Vectors and Oblique Triangles

9.1 Introduction to Vectors
9.2 Components of Vectors
9.3 Vector Addition by Components
9.4 Applications of Vectors
9.5 Oblique Triangles, the Law of Sines
9.6 The Law of Cosines


Graphs of the Trigonometric Functions

10.1 Graphs of y = a sin x and y = a cos x
10.2 Graphs of y = a sin bx and y = a cos bx
10.3 Graphs of y = a sin (bx + c) and y = a cos (bx + c)
10.4 Graphs of y = tan x, y = cot x, y = sec x, y = csc x
10.5 Applications of the Trigonometric Graphs
10.6 Composite Trigonometric Curves


Exponents and Radicals

11.1 Simplifying Expressions with Integer Exponents
11.2 Fractional Exponents
11.3 Simplest Radical Form
11.4 Addition and Subtraction of Radicals
11.5 Multiplication and Division of Radicals


Complex Numbers

12.1 Basic Definitions
12.2 Basic Operations with Complex Numbers
12.3 Graphical Representation of Complex Numbers
12.4 Polar Form of a Complex Number
12.5 Exponential Form of a Complex Number
12.6 Products, Quotients, Powers, and Roots of Complex Numbers
12.7 An Application to Alternating-current (ac) Circuits


Exponential and Logarithmic Functions

13.1 Exponential Functions
13.2 Logarithmic Functions
13.3 Properties of Logarithms
13.4 Logarithms to the Base
13.5 Natural Logarithms
13.6 Exponential and Logarithmic Equations
13.7 Graphs on Logarithmic and Semilogarithmic Paper


Additional Types of Equations and Systems of Equations

14.1 Graphical Solution of Systems of Equations
14.2 Algebraic Solution of Systems of Equations
14.3 Equations in Quadratic Form
14.4 Equations with Radicals


Equations of Higher Degree

15.1 The Remainder and Factor Theorems; Synthetic Division
15.2 The Roots of an Equation
15.3 Rational and Irrational Roots


Matrices; Systems of Linear Equations

16.1 Matrices: Definitions and Basic Operations
16.2 Multiplication of Matrices
16.3 Finding the Inverse of a Matrix
16.4 Matrices and Linear Equations
16.5 Gaussian Elimination
16.6 Higher-order Determinants


Inequalities

17.1 Properties of Inequalities
17.2 Solving Linear Inequalities
17.3 Solving Nonlinear Inequalities
17.4 Inequalities Involving Absolute Values
17.5 Graphical Solution of Inequalities with Two Variables
17.6 Linear Programming


Variation

18.1 Ratio and Proportion
18.2 Variation


Sequences and the Binomial Theorem

19.1 Arithmetic Sequences
19.2 Geometric Sequences
19.3 Infinite Geometric Series
19.4 The Binomial Theorem


Additional Topics in Trigonometry

20.1 Fundamental Trigonometric Identities
20.2 The Sum and Difference Formulas
20.3 Double-Angle Formulas
20.4 Half-Angle Formulas
20.5 Solving Trigonometric Equations
20.6 The Inverse Trigonometric Functions


Plane Analytic Geometry

21.1 Basic Definitions
21.2 The Straight Line
21.3 The Circle
21.4 The Parabola
21.5 The Ellipse
21.6 The Hyperbola
21.7 Translation of Axes
21.8 The Second-degree Equation
21.9 Rotation of Axes
21.10 Polar Coordinates
21.11 Curves in Polar Coordinates


Introduction to Statistics

22.1 Graphical Displays of Data
22.2 Measures of Central Tendency
22.3 Standard Deviation
22.4 Normal Distributions
22.5 Statistical Process Control
22.6 Linear Regression
22.7 Nonlinear Regression


The Derivative

23.1 Limits
23.2 The Slope of a Tangent to a Curve
23.3 The Derivative
23.4 The Derivative as an Instantaneous Rate of Change
23.5 Derivatives of Polynomials
23.6 Derivatives of Products and Quotients of Functions
23.7 The Derivative of a Power of a Function
23.8 Differentiation of Implicit Functions
23.9 Higher Derivatives


Applications of the Derivative

24.1 Tangents and Normals
24.2 Newton’s Method for Solving Equations
24.3 Curvilinear Motion
24.4 Related Rates
24.5 Using Derivatives in Curve Sketching
24.6 More on Curve Sketching
24.7 Applied Maximum and Minimum Problems
24.8 Differentials and Linear Approximations


Integration

25.1 Antiderivatives
25.2 The Indefinite Integral
25.3 The Area Under a Curve
25.4 The Definite Integral
25.5 Numerical Integration: The Trapezoidal Rule
25.6 Simpson's Rule


Applications of Integration

26.1 Applications of the Indefinite Integral
26.2 Areas by Integration
26.3 Volumes by Integration
26.4 Centroids
26.5 Moments of Inertia
26.6 Other Applications


Differentiation of Transcendental Functions

27.1 Derivatives of the Sine and Cosine Functions
27.2 Derivatives of the Other Trigonometric Functions
27.3 Derivatives of the Inverse Trigonometric Functions
27.4 Applications
27.5 Derivative of the Logarithmic Function
27.6 Derivative of the Exponential Function
27.7 L’Hospital’s Rule
27.8 Applications


Methods of Integration

28.1 The Power Rule for Integration
28.2 The Basic Logarithmic Form
28.3 The Exponential Form
28.4 Basic Trigonometric Forms
28.5 Other Trigonometric Forms
28.6 Inverse Trigonometric Forms
28.7 Integration by Parts
28.8 Integration by Trigonometric Substitution
28.9 Integration by Partial Fractions: Non-repeated Linear Factors
28.10 Integration by Partial Fractions: Other Cases
28.11 Integration by Use of Tables


Partial Derivatives and Double Integrals

29.1 Functions of Two Variables
29.2 Curves and Surfaces in Three Dimensions
29.3 Partial Derivatives
29.4 Double Integrals


Expansion of Functions in Series

30.1 Infinite Series
30.2 Maclaurin Series
30.3 Operations with Series
30.4 Computations by Use of Series Expansions
30.5 Taylor Series
30.6 Introduction to Fourier Series
30.7 More About Fourier Series


Differential Equations

31.1 Solutions of Differential Equations
31.2 Separation of Variables
31.3 Integrating Combinations
31.4 The Linear Differential Equation of the First Order
31.5 Numerical Solutions of First-order Equations
31.6 Elementary Applications
31.7 Higher-order Homogeneous Equations
31.8 Auxiliary Equation with Repeated or Complex Roots
31.9 Solutions of Nonhomogeneous Equations
31.10 Applications of Higher-order Equations
31.11 Laplace Transforms
31.12 Solving Differential Equations by Laplace Transforms



Appendix A Solving Word Problems Appendix B Units of Measurement Appendix C Newton’s Method Appendix D A Table of Integrals