Technical Mathematics with Calculus

Paul A. Calter is a Visiting Scholar at Dartmouth College and Professor Emeritus of Mathematics at Vermont Technical College. He is a book review editor of the Nexus Network Journal and has interests in both the fields of mathematics and art. He received his B.S. from Cooper Union and his M.S. from Columbia University, both in engineering, and his Masters of Fine Arts Degree from Norwich University. Calter has taught mathematics for over twenty-five years and is the author of ten mathematics textbooks and a mystery novel. He has been an active painter and sculptor since 1968, has had many solo shows and participated in dozens of group art shows, and has permanent outdoor sculptures at a number of locations. Calter developed a course called "Geometry in Art & Architecture," which he has taught at Dartmouth College and Vermont Technical College, and he has taught at Dartmouth College and Vermont Technical College, and he has given workshops and lectures on the subject. Calter's own art is concerned with astronomical and geometric themes; he searches for a link between the organic and geometric basis of beauty, what has been called the philosopher's stone of aesthetics.

1 Review of Numerical Computation. 1-1 The Real Numbers.

1-2 Addition and Subtraction.

1-3 Multiplication.

1-4 Division.

1-5 Powers and Roots.

1-6 Combined Operations.

1-7 Scientific Notation and Engineering Notation.

1-8 Units of Measurement.

1-9 Percentage.

Chapter 1 Review Problems.

2 Introduction to Algebra.

2-1 Algebraic Expressions.

2-2 Adding and Subtracting.

Polynomials 59 2-3 Laws of Exponents.

2-4 Product of Two Monomials.

2-5 Product of a Multinomial and a Monomial.

2-6 Product of Two Binomials.

2-7 Product of Two Multinomials.

2-8 Powers of Multinomials.

2-9 Removing Symbols of Grouping.

2-10 Quotient of Two Monomials.

2-11 Dividing a Polynomial by a Monomial.

2-12 Quotient of Two Polynomials.

Chapter 2 Review Problems.

3 Simple Equations and Word Problems.

3-1 Solving First-Degree Equations.

3-2 Solving Word Problems.

3-3 Uniform Motion.

3-4 Financial.

3-5 Mixtures.

3-6 Statics.

3-7 Work, Fluid Flow, and Energy Flow.

Chapter 3 Review Problems.

Functions.

4-1 Functions and Relations.

4-2 More on Functions.

Chapter 4 Review Problems.

5 Graphs.

5-1 Rectangular Coordinates.

5-2 Graphing an Equation.

5-3 Graphing a Function by Calculator.

5-4 The Straight Line.

5-5 Solving an Equation Graphically.

Chapter 5 Review Problems.

6 Geometry.

6-1 Straight Lines and Angles.

6-2 Triangles.

6-3 Quadrilaterals.

6-4 The Circle.

6-5 Volumes and Areas of Solids.

Chapter 6 Review Problems.

7 Right Triangles and Vectors.

7-1 The Trigonometric Functions.

7-2 Solution of Right Triangles.

7-3 Applications of the Right Triangle.

7-4 Introduction to Vectors.

7-5 Applications of Vectors.

Chapter 7 Review Problems.

8 Oblique Triangles and Vectors.

8-1 Trigonometric Functions of Any Angle.

8-2 Finding the Angle When the Trigonometric Function Is Kown.

8-3 Law of Sines.

8-4 Law of Cosines.

8-5 Applications.

8-6 Resultants of Nonperpendicular Vectors.

Chapter 8 Review Problems.

9 Systems of Linear Equations.

9-1 Systems of Two Linear Equations.

9-2 Other Systems of Equations.

9-3 Applications.

9-4 Systems of Three Linear Equations.

Chapter 9 Review Problems.

10 Matrices and Determinants.

10-1 Introduction to Matrices.

10-2 Solving Systems of Equations by the Unit Matrix Method.

10-3 Second-Order Determinants.

10-4 Higher-Order Determinants.

Chapter 10 Review Problems.

11 Factoring and Fractions.

11-1 Common Factors.

11-2 Difference of Two Squares.

11-3 Factoring Trinomials.

11-4 Other Factorable Expressions.

11-5 Simplification of Fractions.

11-6 Multiplication and Division of Fractions.

11-7 Addition and Subtraction of Fractions.

11-8 Complex Fractions.

11-9 Fractional Equations.

11-10 Literal Equations and Formulas.

Chapter 11 Review Problems.

12 Quadratic Equations.

12-1 Solving a Quadratic Equation by Calculator.

12-2 Solving a Quadratic by Formula.

12-3 Applications.

Chapter 12 Review Problems.

13 Exponents and Radicals.

13-1 Integral Exponents.

13-2 Simplification of Radicals.

13-3 Operations with Radicals.

13-4 Radical Equations.

Chapter 13 Review Problems.

14 Radian Measure, Arc Length, and Rotation.

14-1 Radian Measure.

14-2 Arc Length.

14-3 Uniform Circular Motion.

Chapter 14 Review Problems.

15 Trigonometric, Parametric, and Polar Graphs.

15-1 Graphing the Sine Wave by Calculator.

15-2 Manual Graphing of the Sine Wave.

15-3 The Sine Wave as a Function of Time.

15-4 Graphs of the Other Trigonometric Functions.

15-5 Graphing Parametric Equations.

15-6 Graphing in Polar Coordinates.

Chapter 15 Review Problems.

16 Trigonometric Identities and Equations.

16-1 Fundamental Identities.

16-2 Sum or Difference of Two Angles.

16-3 Functions of Double Angles and Half-Angles.

16-4 Evaluating Trigonometric Expressions.

16-5 Solving Trigonometric Equations.

Chapter 16 Review Problems.

17 Ratio, Proportion, and Variation.

17-1 Ratio and Proportion.

17-2 Similar Figures.

17-3 Direct Variation.

17-4 The Power Function.

17-5 Inverse Variation.

17-6 Functions of More Than One Variable.

Chapter 17 Review Problems.

18 Exponential and Logarithmic Functions.

18-1 The Exponential Function.

18-2 Logarithms.

18-3 Properties of Logarithms.

18-4 Exponential Equations.

18-5 Solving Logarithmic Equations.

Chapter 18 Review Problems.

19 Complex Numbers.

19-1 Complex Numbers in Rectangular Form.

19-2 Complex Numbers in Polar Form.

19-3 Complex Numbers on the Calculator.

19-4 Vector Operations Using Complex Numbers.

19-5 Alternating Current Applications.

Chapter 19 Review Problems.

20 Sequences, Series, and the Binomial Theorem.

20-1 Sequences and Series.

20-2 Arithmetic and Harmonic Progressions.

20-3 Geometric Progressions.

20-4 Infinite Geometric Progressions.

20-5 The Binomial Theorem.

Chapter 20 Review Problems.

21 Introduction to Statistics and Probability.

21-1 Definitions and Terminology.

21-2 Frequency Distributions.

21-3 Numerical Description of Data.

21-4 Introduction to Probability.

21-5 The Normal Curve.

21-6 Standard Errors.

21-7 Process Control.

21-8 Regression.

Chapter 21 Review Problems.

22 Analytic Geometry.

22-1 The Straight Line.

22-2 Equation of a Straight Line.

22-3 The Circle.

22-4 The Parabola.

22-5 The Ellipse.

22-6 The Hyperbola.

Chapter 22 Review Problems.

23 Derivatives of Algebraic Functions.

23-1 Limits.

23-2 Rate of Change and the Tangent.

23-3 The Derivative.

23-4 Rules for Derivatives.

23-5 Derivative of a Function Raised to a Power.

23-6 Derivatives of Products and Quotients.

23-7 Other Variables, Implicit Relations, and Differentials.

23-8 Higher-Order Derivatives.

Chapter 23 Review Problems.

24 Graphical Applications of the Derivative.

24-1 Equations of Tangents and Normals.

24-2 Maximum, Minimum, and Inflection Points.

24-3 Sketching, Verifying, and Interpreting Graphs.

Chapter 24 Review Problems.

25 More Applications of the Derivative.

25-1 Rate of Change.

25-2 Motion of a Point.

25-3 Related Rates.

25-4 Optimization.

Chapter 25 Review Problems.

26 Integration.

26-1 The Indefinite Integral.

26-2 Rules for Finding Integrals.

26-3 Simple Differential Equations.

26-4 The Definite Integral.

26-5 Approximate Area under a Curve.

26-6 Exact Area under a Curve.

Chapter 26 Review Problems.

27 Applications of the Integral.

27-1 Applications to Motion.

27-2 Applications to Eletric Circuits.

27-3 Finding Areas by Integration.

27-4 Volumes by Integration.

Chapter 27 Review Problems.

28 More Applications of the Integral.

28-1 Length of Arc.

28-2 Area of Surface of Revolution.

28-3 Centroids.

28-4 Fluid Pressure.

28-5 Work.

28-6 Moment of Inertia.

Chapter 28 Review Problems.

29 Trigonometric, Logarithmic, and Exponential Functions.

29-1 Derivatives of the Sine and Cosine Functions.

29-2 Derivatives of the Other Trigonometric Functions.

29-3 Derivatives of the Inverse Trigonometric Functions.

29-4 Derivatives of Logarithmic Functions.

29-5 Derivatives of the Exponential Functions.

29-6 Integrals of the Exponential and Logarithmic Functions.

29-7 Integrals of the Trigonometric Functions.

29-8 Average and Root Mean Square Values.

Chapter 29 Review Problems.

30 First-Order Differential Equations.

30-1 Definitions.

30-2 Solving a DE by Calculator, Graphically, and Numerically.

30-3 First-Order DE: Variables Separable.

30-4 Exact First-Order DE.

30-5 First-Order Homogeneous DE.

30-6 First-Order Linear DE.

30-7 Geometric Applications of First-Order DE.

30-8 Exponential Growth and Decay.

30-9 Series RL and RC Circuits.

Chapter 30 Review Problems.

31 Second-Order Differential Equations.

31-1 Second-Order DE.

31-2 Constant Coefficients and Right Side Zero.

31-3 Right Side Not Zero.

31-4 Mechanical Vibrations.

31-5 RLC Circuits.

Chapter 31 Review Problems.

Appendices.

A Summary of Facts and Formulas A-1.

B Conversion Factors A-27.

C Table of Integrals.

D Answers to Selected Problems.

Indexes.

Index to Applications.

General Index.