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Contents ix Real Vs. Imaginary Numbers 145 Complex Numbers 146 Complex Numbers Find New Roots 147 A Problem Using Simultaneous Equations 149 Questions and Problems 151 12 Mechanical Mathematics 155 What Is Force? 155 Units of Force 155 Acceleration, Speed, and Distance 157 Force and Work 159 Work and Energy 160 Energy and Power 160 Gravity as a Source of Energy 161 Weight as Force 162 Gravitational Measure of Work 163 Energy for Constant Acceleration 163 Potential and Kinetic Energy 164 Kinetic Energy and Speed 164 Acceleration at Constant Power 165 How a Spring Stores Energy 166 How a Spring Transfers Energy 167 Resonance Cycle 168 Travel and Speed in Resonant System 169 Questions and Problems 170 13 Ratio and Proportion 173 Fractions, Projections, Extremes, and Means 173 Proportions in Practice 175 Shape and Size 175 About Angles in Triangles 176 Use of Right Triangles 177 Angles Identified by Ratios 178 Theorem of Pythagoras 178 Names for Angle Ratios 179 Spotting the Triangle 180 Degree Measure of Angles 181 Three Applications of Trigonometry 182 Questions and Problems 184 14 Trigonometric and Geometric Calculations 187 Ratios for Sum Angles 187 Finding Tan (A + B) 188 Part 3 An Example: Ratios for 75º 189 Ratios for Angles Greater Than 90º 189 Ratios for Difference Angles 190 Ratios Through the Four Quadrants 191 Pythagoras in Trigonometry 192 Double Angles 193 Higher Multiple Angles 194 Properties of the Isosceles Triangle 195 Angles in a Circle 196 Angles in a Semicircle 198 Definitions 198 Questions and Problems 199 Analysis and Calculus 15 Systems of Counting 203 Mechanical Counters 203 Duodecimal Numbers 204 Conversion from Decimal to Duodecimal 204 Conversion from Duodecimal to Decimal 206 Octal Numbers 207 Hexadecimal numbers 207 Binary Numbers 207 Comparing the Values 208 Converting Decimal to Binary 210 Binary Multiplication 210 Binary Division 212 Indices 212 Negative and Fractional Exponents 215 Surds 216 Questions and Problems 216 16 Progressions, Permutations, and Combinations 219 Arithmetic Progression 219 Geometric Progression 220 Harmonic Progression 221 Arithmetic Series 222 Geometric Series 223 Convergence 224 Rate of Convergence 226 Permutations 227 Combinations 229
x Contents Powers of a Binomial 230 Alternative Notation for Multiplying Numbers 231 Questions and Problems 232 17 Introduction to Derivatives 235 Rates of Change 235 Infinitesimal Changes 235 Successive Differentiation 239 Differentiating a Polynomial 240 Successive Differentiation of Movement 242 Circular Measure of Angles 243 Derivatives of Sine and Cosine 244 Successive Derivatives of Sine 246 Questions and Problems 247 18 More About Differentiation 251 Frequency and Period of a Sine Wave 251 Sinusoidal Motion 252 Harmonic Motion 253 Linear and Nonlinear Relationships 255 Multiples and Powers 257 Functions 259 Derivative of Constant 260 Derivative of Sum of Two Functions 260 Derivative of Difference of Two Functions 260 Derivative of Function Multiplied by a Constant 261 Derivative of Product of Two Functions 261 Derivative of Product of Three Functions 261 Derivative of Quotient of Two Functions 262 Reciprocal Derivatives 262 Derivative of Function Raised to a Power 262 Chain Rule 262 Questions and Problems 263 19 Introduction to Integrals 265 Adding up the Pieces 265 The Antiderivative 265 Patterns in Calculations 266 The Constant of Integration 266 Definite Integrals 268 Finding Area by Integration 269 Area of a Circle 271 Curved Areas of Cylinders and Cones 272 Surface Area of a Sphere 272 Volumes of Wedges and Pyramids 273 Volumes of Cones and Spheres 275 Questions and Problems 276 20 Combining Calculus with Other Tools 279 Local Maximum and Minimum 279 Point of Inflection 281 Maximum and Minimum Slope 283 Maximum Area with Constant Perimeter 284 Box with Minimum Surface Area 285 Cylinder with Minimum Surface Area 287 Cone with Minimum Surface Area 288 Equations for Circles, Ellipses, and Parabolas 289 Directrix, Focus, and Eccentricity 291 The Ellipse and the Circle 292 Relationships Between Focus, Directrix, and Eccentricity 293 Focus Property of Parabola 294 Focus Property of Ellipse 295 Hyperbola: Eccentricity Greater Than 1 296 Geometry of Conic Sections 298 Questions and Problems 299 21 Alternative Coordinate Systems 301 Mathematician’s Polar Coordinates 301 Some Examples 303 Compressed Polar Coordinates 309 Mathematician’s Polar vs. Cartesian Coordinates 310 Navigator’s Polar Coordinates 311 Mathematician’s Vs. Navigator’s Polar Coordinates 313
Page 2 and 3: Mastering Technical Mathematics
Page 4 and 5: Mastering Technical Mathematics Thi
Page 6 and 7: To Tim, Samuel, and Tony
Page 8 and 9: For more information about this tit
Page 12 and 13: Contents xi Navigator’s Polar Vs.
Page 14 and 15: Contents xiii Law of Large Numbers
Page 16 and 17: Introduction This book is intended
Page 18 and 19: Copyright © 2008, 1999 by The McGr
Page 20 and 21: 1From Counting to Addition We’ve
Page 22 and 23: From Counting to Addition 5 Figure
Page 30 and 31: From Counting to Addition 13 WEIGHT
Page 34 and 35: 13. In the United States, common li
Page 36 and 37: 2Subtraction Just as addition is co
Page 38 and 39: Subtraction 21 the hundreds column,
Page 40 and 41: Subtraction 23 MAKING CHANGE This i
Page 42 and 43: Subtraction 25 Figure 2-6 Two metho
Page 44 and 45: 3Multiplication Suppose you go into
Page 46 and 47: Multiplication 29 In Fig. 3-2A, exa
Page 48 and 49: Multiplication 31 Figure 3-3 An exa
Page 50 and 51: Multiplication 33 Figure 3-6 It doe
Page 52 and 53: Multiplication 35 Figure 3-9 Subtra
Page 54 and 55: Multiplication 37 Figure 3-12 Multi
Page 56 and 57: Multiplication 39 3. Multiply the f
Page 58 and 59: 4Division Before the days of electr
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Division 43 Figure 4-2 A pictorial
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Division 45 MULTIPLICATION CHECKS D
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Division 47 Figure 4-6 At A, we can
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Division 49 In the example of Fig.
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Division 51 7/10 0.7 17/20 0.85 3
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Division 53 QUESTIONS AND PROBLEMS
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5Fractions Chapter 4 presented some
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Fractions 57 This line of reasoning
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Fractions 59 493/23 21.xxx . . . (
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Fractions 61 The numbers in the ori
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Fractions 63 5/12 25/60 3/10 18/6
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Fractions 65 “limiting values”
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Fractions 67 QUESTIONS AND PROBLEMS
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6Plane Polygons So far, this book h
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Plane Polygons 71 WHAT IS A SQUARE?
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Plane Polygons 73 inches is not the
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Plane Polygons 75 Figure 6-7 Two wa
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Plane Polygons 77 AREA OF AN OBTUSE
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Plane Polygons 79 Figure 6-11 Some
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Plane Polygons 81 QUESTIONS AND PRO
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7Time, Percentages, and Graphs You
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Figure 7-2 A racecar driver can cal
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When a percentage expresses a chang
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Time, Percentages, and Graphs 89 Fi
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Time, Percentages, and Graphs 91 Fi
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5-mile track. Each graph gives you
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Part 2 ALGEBRA, GEOMETRY, AND TRIGO
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8First Notions in Algebra As proble
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First Notions in Algebra 99 Figure
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First Notions in Algebra 103 (a b)
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To recap, here are the rules if you
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4. Suppose x 5 and y 7. Then what
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9Some “School” Algebra In this
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Some “School” Algebra 115 DIMEN
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Some “School” Algebra 117 4 u
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Some “School” Algebra 119 20x
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Some “School” Algebra 121 Figur
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Figure 9-9 Sometimes it’s easier
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9. Adding 1 to the numerator and de
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10Quadratic Equations Simultaneous
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Quadratic Equations 129 Figure 10-3
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What about the “negative solution
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11 Some Useful Shortcuts Have you n
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Some Useful Shortcuts 143 FINDING S
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used to derive square roots to seve
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Some Useful Shortcuts 147 Figure 11
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Some Useful Shortcuts 149 Figure 11
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Some Useful Shortcuts 151 which sim
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12. In each of the following pairs
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12 Mechanical Mathematics In this c
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Mechanical Mathematics 157 Figure 1
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is at (a times t) feet per second.
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If the mass is 1 kg, the force is 9
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13 Ratio and Proportion Thus far, t
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Ratio and Proportion 175 PROPORTION
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In any triangle, its three interior
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Ratio and Proportion 179 Figure 13-
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14 Trigonometric and Geometric Calc
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Trigonometric and Geometric Calcula
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PROPERTIES OF THE ISOSCELES TRIANGL
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Suppose you name the equal pairs of
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Copyright © 2008, 1999 by The McGr
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15 Systems of Counting The decimal
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Systems of Counting 205 Figure 15-2
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Systems of Counting 207 OCTAL NUMBE
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Systems of Counting 209 DECIMAL BIN
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are shortcut methods for performing
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1. Find the decimal equivalent of t
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16 Progressions, Permutations, and
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Progressions, Permutations, and Com
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17 Introduction to Derivatives The
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Figure 17-2 The derivative of the f
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is negative when x is negative, bec
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This rule holds true for all expres
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Introduction to Derivatives 243 CIR
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Introduction to Derivatives 245 Fig
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18 More about Differentiation The p
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More about Differentiation 253 Figu
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More about Differentiation 255 by t
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More about Differentiation 257 MULT
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In the powers table, 0 n is always
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DERIVATIVE OF FUNCTION MULTIPLIED B
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More about Differentiation 263 QUES
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19 Introduction to Integrals Now th
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Introduction to Integrals 267 Figur
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(it subtracts from itself), so you
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Introduction to Integrals 275 VOLUM
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1. Find the derivatives of the prod
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20Combining Calculus with Other Too
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plotting nine points, but to get a
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In a situation like this, you have
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Combining Calculus with Other Tools
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These principal axes can be regarde
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e reaches 1, the ellipse “breaks
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etween the plane and the double con
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21Alternative Coordinate Systems Th
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Alternative Coordinate Systems 303
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perpendicular to the other two. The
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22Complex Numbers Long ago, humanki
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Multiplying repeatedly by i works t
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Complex Numbers 327 Figure 22-4 The
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Complex Numbers 329 When you cube i
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Complex Numbers 331 Figure 22-9 The
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Complex Numbers 333 Figure 22-11 At
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Complex Numbers 335 QUESTIONS AND P
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Part 4 TOOLS OF APPLIED MATHEMATICS
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23Trigonometry in the Real World Tr
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The absolute accuracy (in fixed uni
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Trigonometry in the Real World 343
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objects, even when they are observe
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Now find the distance PV by applyin
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7. What is the range of the aircraf
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24Logarithms and Exponentials A log
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Logarithms and Exponentials 361 Fig
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Now try an example in which neither
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Let’s put a number into this form
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Logarithms and Exponentials 369 10
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Suppose x is some real number. The
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To demonstrate this, let b e, x 2
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“Inv” key followed by a “ln
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and so on. A decrease of 1 order of
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25Scientific Notation Tutorial In s
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Another alternative multiplication
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line that covers 3 orders of magnit
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Scientific Notation Tutorial 387 (3
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Scientific Notation Tutorial 389 (5
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equality symbols. It is universally
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Sometimes, the outcome of determini
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26Vectors You’ve already seen a f
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Vectors 397 MULTIPLICATION OF A VEC
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Vectors 399 The values of the angle
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Vectors 401 If a is multiplied by a
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Vectors 403 Figure 26-5 Direction a
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Vectors 405 This formula works in a
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Vectors 407 You are told that these
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Vectors 409 4. Consider the two vec
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27Logic and Truth Tables To gain a
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are true. If any one of the compone
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antecedent is (P & C). The “impli
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Logic and Truth Tables 417 X Y X &
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Logic and Truth Tables 419 (X 1 Y)
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Logic and Truth Tables 421 The same
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classes—a certain way that multip
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Table 27-9B Derivation of truth val
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Logic and Truth Tables 427 A X Y X
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Logic and Truth Tables 429 X Y Z X
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Logic and Truth Tables 431 X Y Z X
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28Beyond Three Dimensions As you ha
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Beyond Three Dimensions 435 Figure
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a little later! It “lives” for
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Beyond Three Dimensions 441 IMPOSSI
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Beyond Three Dimensions 443 Displac
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Beyond Three Dimensions 449 NO PARA
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such as the pyramid, cube, and sphe
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2. If there are no pairs of paralle
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29A Statistics Primer Statistics is
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A Statistics Primer 457 Figure 29-2
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A Statistics Primer 459 PARAMETER A
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A Statistics Primer 463 of dice, th
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To determine µ sc , check the tabl
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A Statistics Primer 471 making it m
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30Probability Basics Probability is
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Probability Basics 475 Event 1 Even
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Probability Basics 477 As with math
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Probability Basics 479 that both co
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Probability Basics 481 The notion o
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Probability Basics 483 To calculate
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Probability Basics 485 • 30 peopl
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Probability Basics 487 The resultin
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Probability Basics 489 Because the
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Probability Basics 491 Figure 30-11
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Probability Basics 493 views are re
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Final Exam Do not refer to the text
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Final Exam 497 (a) An even number t
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Final Exam 499 X Y Z ? F F F F F F
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Final Exam 501 (c) A harmonic progr
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Final Exam 503 (c) 1.8 10 6 km (d)
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Final Exam 505 47. Suppose an item
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Final Exam 507 Figure E-6 Illustrat
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Final Exam 509 (a) p q (b) p q 1
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Final Exam 511 (d) Roots of w (e) P
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Final Exam 513 82. Which of the fol
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Final Exam 515 (b) It decreases in
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Final Exam 517 95. In the situation
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Appendix 1 Solutions to End-of-Chap
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Solutions to End-of-Chapter Questio
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Figure A-30 Illustration for soluti
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Appendix 2 Answers to Final Exam Qu
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Appendix 3 Table of Derivatives Let
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Appendix 4 Table of Indefinite Inte
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Table of Indefinite Integrals 609 f
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Suggested Additional Reading Bluman
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Index absolute accuracy, 341 error,
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cosines, law of, 347-348 counter, m
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geometry elliptic, 449 Euclidean, 4
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exponential, 369-371 frequency, 255
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esonance, in system, 168-170, 255 r
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direction angles, 402-403 direction
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