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Solutions to End-of-Chapter Questions and Problems 577 Figure A-17 Illustration for solution to Prob. 3 in Chap. 21. Figure A-18 Illustration for solution to Prob. 4 in Chap. 21.
578 Mastering Technical Mathematics 5. If we know the rectangular coordinates (x 0 ,y 0 ), we can convert to mathematician’s polar coordinates ( 0 ,r 0 ) by using these formulas: 0 arctan (y 0 /x 0 ) if x 0 > 0 0 180 arctan (y 0 /x 0 ) if x 0 < 0 (for 0 in degrees) 0 arctan (y 0 /x 0 ) if x 0 < 0 (for 0 in radians) r 0 (x 02 y 02 ) 1/2 The solutions are found by substituting the relevant values x 0 and y 0 into these formulas, as follows. In these examples, it is easy to derive exact solutions, given that the input values are exact. (a) In this case, x 0 1 and y 0 1. We have 0 arctan (1/1) arctan 1 /4 and r 0 (1 2 1 2 ) 1/2 (1 1) 1/2 2 1/2 Therefore ( 0 ,r 0 ) (/4,2 1/2 ) (b) In this case, x 0 1 and y 0 1. We have 0 arctan [1/(1)] arctan (1) (/4) 3/4 and r 0 [(1) 2 1 2 ] 1/2 (1 1) 1/2 2 1/2 Therefore ( 0 ,r 0 ) (3/4,2 1/2 ) (c) In this case, x 0 1 and y 0 1. We have 0 arctan (1/1) arctan (1) /4 7/4
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Mastering Technical Mathematics
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Mastering Technical Mathematics Thi
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To Tim, Samuel, and Tony
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For more information about this tit
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Contents ix Real Vs. Imaginary Numb
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Contents xi Navigator’s Polar Vs.
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Contents xiii Law of Large Numbers
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Introduction This book is intended
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Copyright © 2008, 1999 by The McGr
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1From Counting to Addition We’ve
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From Counting to Addition 5 Figure
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From Counting to Addition 13 WEIGHT
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13. In the United States, common li
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2Subtraction Just as addition is co
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Subtraction 21 the hundreds column,
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Subtraction 23 MAKING CHANGE This i
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Subtraction 25 Figure 2-6 Two metho
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3Multiplication Suppose you go into
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Multiplication 29 In Fig. 3-2A, exa
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Multiplication 31 Figure 3-3 An exa
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Multiplication 33 Figure 3-6 It doe
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Multiplication 35 Figure 3-9 Subtra
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Multiplication 37 Figure 3-12 Multi
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Multiplication 39 3. Multiply the f
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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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Page 544 and 545: Solutions to End-of-Chapter Questio
Page 612 and 613: Figure A-30 Illustration for soluti
Page 620 and 621: Appendix 2 Answers to Final Exam Qu
Page 622 and 623: Appendix 3 Table of Derivatives Let
Page 624 and 625: Appendix 4 Table of Indefinite Inte
Page 626 and 627: Table of Indefinite Integrals 609 f
Page 632 and 633: Suggested Additional Reading Bluman
Page 634 and 635: Index absolute accuracy, 341 error,
Page 636 and 637: cosines, law of, 347-348 counter, m
Page 638 and 639: geometry elliptic, 449 Euclidean, 4
Page 640 and 641: exponential, 369-371 frequency, 255
Page 642 and 643: esonance, in system, 168-170, 255 r
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direction angles, 402-403 direction
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