Technical Mathematics with Calculus, Sixth Edition

Cover......Page 1
Title Page......Page 5
Copyright......Page 6
Contents......Page 13
OBJECTIVES......Page 19
Real and Imaginary Numbers......Page 20
Signed Numbers......Page 21
Symbols of Grouping......Page 22
Significant Digits and Accuracy......Page 23
Rounding......Page 24
Changing the Calculator Display......Page 27
Adding Signed Numbers......Page 28
Subtracting Negative Numbers by Calculator......Page 29
Adding and Subtracting Approximate Numbers......Page 30
Combining Exact and Approximate Numbers......Page 31
Loss of Significant Digits During Subtraction......Page 32
Multiplying Signed Numbers......Page 33
Multiplying Negative Numbers by Calculator......Page 34
Multiplication of Approximate Numbers......Page 35
Multiplying Exact and Approximate Numbers......Page 36
Dividing by Calculator......Page 37
Dividing Signed Numbers......Page 38
Zero......Page 39
Reciprocals by Calculator......Page 40
Powers......Page 41
Negative Base......Page 42
Principal Root......Page 43
Odd Roots of Negative Numbers by Calculator......Page 44
Parentheses......Page 47
Combined Operations with Approximate Numbers......Page 48
1-7 Scientific Notation and Engineering Notation......Page 50
Converting Numbers to Scientific Notation......Page 51
Converting Numbers to Engineering Notation......Page 52
Addition and Subtraction......Page 53
Division......Page 54
Scientific and Engineering Notation on the Calculator......Page 55
Conversion of Units......Page 59
Converting Areas and Volumes......Page 60
Using More Than One Conversion Factor......Page 61
Metric Prefixes......Page 62
Angle Conversions......Page 63
Substituting into Formulas......Page 65
Converting to Percent......Page 69
Finding the Amount When the Base and the Rate Are Known......Page 70
Finding the Base When a Percent of It Is Known......Page 71
Percent Change......Page 72
Percent Efficiency......Page 73
Percent Concentration......Page 74
CHAPTER 1 REVIEW PROBLEMS......Page 77
OBJECTIVES......Page 80
Constants and Variables......Page 81
Symbols of Grouping......Page 82
Monomials, Multinomials, and Polynomials......Page 83
Combining Like Terms......Page 85
Commutative Law of Addition......Page 86
Instructions Given Verbally......Page 87
Multiplying Powers......Page 90
Dividing Powers......Page 91
Product Raised to a Power......Page 92
Zero Exponent......Page 93
Negative Exponent......Page 94
Exponents on the Calculator......Page 95
Rules of Signs......Page 98
Multiplying Monomials......Page 99
2-5 Multiplying a Monomial and a Multinomial......Page 101
Multiplication by Calculator......Page 102
2-6 Multiplying a Binomial by a Binomial......Page 104
FOIL Rule......Page 105
2-7 Multiplying a Multinomial by a Multinomial......Page 106
2-8 Raising a Multinomial to a Power......Page 108
Rules of Signs......Page 110
Dividing a Monomial by a Monomial......Page 111
2-10 Dividing a Polynomial by a Monomial......Page 113
2-11 Dividing a Polynomial by a Polynomial......Page 116
CHAPTER 2 REVIEW PROBLEMS......Page 119
OBJECTIVES......Page 121
The Solution of an Equation......Page 122
Solving an Equation......Page 123
Simple Fractional Equations......Page 124
Equations with Approximate Numbers......Page 125
Strategy......Page 126
Solving an Equation Using a Calculator’s Equation Solver......Page 127
Implicit or Literal Equations......Page 128
Picture the Problem......Page 131
Estimate the Answer......Page 132
Don’t Give Up......Page 133
Number Puzzles......Page 134
Define Other Unknowns......Page 135
3-3 Uniform Motion Applications......Page 136
3-4 Money Problems......Page 139
Basic Relationships......Page 141
A Typical Mixture Problem......Page 142
Equations of Equilibrium......Page 145
Solving Rate Problems......Page 147
Work......Page 148
Fluid Flow and Energy Flow......Page 149
CHAPTER 3 REVIEW PROBLEMS......Page 151
OBJECTIVES......Page 154
Domain and Range......Page 155
Implicit and Explicit Forms......Page 156
Functional Notation......Page 157
Changing from Implicit to Explicit Form......Page 158
Functions by Calculator......Page 159
A Function as a Set of Point Pairs......Page 162
A Function as a Verbal Statement......Page 163
Substituting into a Function......Page 164
Composite Functions......Page 165
Inverse of a Function......Page 166
Finding Domain and Range......Page 168
CHAPTER 4 REVIEW PROBLEMS......Page 172
OBJECTIVES......Page 174
Graphing Point Pairs......Page 175
Applications: Graphing Empirical Data......Page 176
Graphing an Equation Given in Implicit Form......Page 179
Graphing a Relation......Page 180
Graphing an Equation Given in Implicit Form......Page 182
Exploring Your Graph: Special Operations on the Calculator......Page 183
Table of Point Pairs and Split Screen Display......Page 184
Slope......Page 185
Equation of a Straight Line......Page 186
Writing the Equation Given Two Points on the Line......Page 187
5-5 Solving an Equation Graphically......Page 190
CHAPTER 5 REVIEW PROBLEMS......Page 191
OBJECTIVES......Page 193
Angles......Page 194
Families of Lines and Transversals......Page 195
Triangles......Page 198
Area of a Triangle......Page 199
Congruent and Similar Triangles......Page 200
Right Triangles......Page 201
Special Right Triangles......Page 202
6-3 Quadrilaterals......Page 205
Sum of the Interior Angles......Page 206
Circumference and Pi (p)......Page 208
Area of a Circle......Page 209
Tangent, Secant, and Chord......Page 210
Semicircle......Page 211
Polyhedra......Page 214
Rectangular Parallelepiped and Cube......Page 215
The Pyramid......Page 216
The Cylinder......Page 219
The Cone......Page 220
The Sphere......Page 221
CHAPTER 6 REVIEW PROBLEMS......Page 223
OBJECTIVES......Page 225
Sine, Cosine, and Tangent......Page 226
Finding the Angle......Page 227
Solving a Right Triangle When One Side and One Angle Are Known......Page 230
Solving a Right Triangle When Two Sides Are Known......Page 232
7-3 Applications of the Right Triangle......Page 234
7-4 An Angle in Standard Position......Page 239
Representation of a Vector......Page 240
Components of a Vector......Page 241
Rectangular Components by Calculator......Page 242
Resultants by Calculator......Page 243
Force Vectors......Page 244
Impedance Vectors......Page 245
CHAPTER 7 REVIEW PROBLEMS......Page 247
OBJECTIVES......Page 249
8-1 Trigonometric Functions of Any Angle......Page 250
Trigonometric Functions of Any Angle by Calculator......Page 251
Reciprocal Relationships......Page 252
Cofunctions......Page 253
8-2 Finding the Angle When the Trigonometric Function Is Known......Page 254
Algebraic Signs of the Trigonometric Functions......Page 255
Finding the Angle by Calculator......Page 256
Inverse of the Cotangent, Secant, and Cosecant......Page 257
Derivation......Page 258
Solving a Triangle When Two Angles and One Side Are Known (AAS or ASA)......Page 259
Solving a Triangle When Two Sides and One Angle Are Given (SSA): The Ambiguous Case......Page 260
Derivation......Page 264
When to Use the Law of Sines or the Law of Cosines......Page 265
Using the Law of Cosines When Two Sides and the Included Angle Are Known......Page 266
Using the Law of Cosines When Three Sides Are Known......Page 267
8-5 Applications......Page 269
Finding x and y Components of a Vector......Page 273
Resultant of Two Vectors at Any Angle......Page 274
Resultants by Calculator......Page 275
CHAPTER 8 REVIEW PROBLEMS......Page 278
OBJECTIVES......Page 281
Solution to a System of Equations......Page 282
Approximate Graphical Solution to a System of Two Equations......Page 283
Solving a Pair of Linear Equations by the Addition-Subtraction Method......Page 284
Systems Having No Solution......Page 286
Solving Sets of Equations Symbolically by Calculator......Page 287
9-2 Applications......Page 288
Money Applications......Page 289
Applications Involving Mixtures......Page 290
Statics Applications......Page 291
Applications to Work, Fluid Flow, and Energy Flow......Page 292
Electrical Applications......Page 293
A System with Fractional Coefficients......Page 297
Fractional Equations with Unknowns in the Denominator......Page 298
Literal Equations......Page 299
Addition-Subtraction Method......Page 302
Substitution Method......Page 303
Fractional Equations......Page 304
Literal Equations......Page 305
CHAPTER 9 REVIEW PROBLEMS......Page 308
OBJECTIVES......Page 310
Dimensions......Page 311
The Unit Matrix and the Null Matrix......Page 312
Entering a Matrix into a Calculator......Page 313
10-2 Solving Systems of Equations by the Unit Matrix Method......Page 315
Value of a Determinant......Page 320
Determinants by Calculator......Page 321
Solving a System of Two Linear Equations by Determinants......Page 322
Evaluating a Higher-Order Determinant by Calculator......Page 326
Development by Minors......Page 327
Solving a System of Equations by Determinants......Page 329
Systems of More than Three Equations......Page 332
CHAPTER 10 REVIEW PROBLEMS......Page 334
OBJECTIVES......Page 337
Common Factors......Page 338
Checking......Page 339
Factoring by Calculator......Page 340
11-2 Difference of Two Squares......Page 341
Factoring the Difference of Two Squares......Page 342
Factoring Completely......Page 343
11-3 Factoring Trinomials......Page 344
Factoring by Trial and Error......Page 345
Using the Signs to Aid Factoring......Page 346
Grouping Method......Page 347
The Perfect Square Trinomial......Page 348
Sum or Difference of Two Cubes......Page 351
Division by Zero......Page 353
Mixed Form......Page 354
Simplifying a Fraction by Reducing to Lowest Terms......Page 355
Simplifying a Fraction by Manipulating Signs......Page 356
Multiplying Fractions......Page 358
Dividing Fractions......Page 359
Multiplying and Dividing Fractions by Calculator......Page 360
Least Common Denominator......Page 362
Combining Fractions with Different Denominators......Page 363
An Application......Page 365
11–8 Complex Fractions......Page 367
Solving a Fractional Equation by Calculator......Page 370
Equations with the Unknown in the Denominator......Page 371
11–10 Literal Equations and Formulas......Page 373
Solving Literal Equations and Formulas......Page 374
Literal Fractional Equations......Page 376
Formulas......Page 377
CHAPTER 11 REVIEW PROBLEMS......Page 378
OBJECTIVES......Page 381
Solving a Quadratic Graphically......Page 382
Solving a Quadratic Using a Calculator’s Equation Solver......Page 383
Implicit Functions......Page 384
An Application......Page 385
General Form of a Quadratic......Page 386
The Quadratic Formula......Page 387
Completing the Square......Page 388
Derivation of the Quadratic Formula......Page 389
12-3 Applications......Page 390
CHAPTER 12 REVIEW PROBLEMS......Page 395
OBJECTIVES......Page 397
Zero Exponents......Page 398
Products......Page 399
Quotients......Page 400
Relation between Fractional Exponents and Radicals......Page 403
Root of a Product......Page 404
Root of a Quotient......Page 405
Reducing the Radicand......Page 406
Removing Radicals from the Denominator......Page 407
Reducing the Index......Page 408
Adding and Subtracting Radicals......Page 410
Multiplying Radicals......Page 411
Dividing Radicals......Page 412
Solving a Radical Equation Using a Calculator’s Equation Solver......Page 416
Solving a Radical Equation Algebraically......Page 417
CHAPTER 13 REVIEW PROBLEMS......Page 421
OBJECTIVES......Page 423
Angle Conversion......Page 424
Angle Conversion by Calculator......Page 425
Trigonometric Functions of Angles in Radians......Page 426
The Inverse Trigonometric Functions......Page 427
Areas of Sectors and Segments......Page 428
14-2 Arc Length......Page 431
Angular Displacement......Page 434
Applications......Page 435
CHAPTER 14 REVIEW PROBLEMS......Page 438
OBJECTIVES......Page 440
Periodic Functions......Page 441
The Sine Wave: Period and Frequency......Page 442
Amplitude and Phase Shift Related......Page 443
Zeros and Instantaneous Value......Page 444
An Application......Page 445
Quick Manual Graph of the Sine Wave......Page 448
Writing the Equation When the Amplitude, Period, and Phase Shift Are Given......Page 450
Writing the Equation When the Curve Is Given......Page 451
Period......Page 453
Frequency......Page 454
Alternating Current......Page 456
Phase Shift......Page 457
The Cosine Wave......Page 459
Graphs of the Tangent, Cotangent, Secant, and Cosecant Functions......Page 460
The Inverse Trigonometric Functions......Page 461
Arc Cosine and Arc Tangent......Page 462
Graphing an Inverse Trigonometric Function by Calculator......Page 463
Graphing Parametric Equations by Calculator......Page 466
Graphing a Trigonometric Equation in Parametric Form......Page 467
Graphing an Equation in Polar Coordinates by Calculator......Page 470
Graphing an Equation in Polar Coordinates Manually......Page 471
Transforming Between Rectangular and Polar Coordinates by Calculator......Page 473
Transforming an Equation......Page 474
CHAPTER 15 REVIEW PROBLEMS......Page 477
OBJECTIVES......Page 479
Quotient Relations......Page 480
Pythagorean Relations......Page 481
Simplifying a Trigonometric Expression......Page 482
Simplifying a Trigonometric Expression by Calculator......Page 483
Proving a Trigonometric Identity......Page 484
16-2 Sum or Difference of Two Angles......Page 487
Difference of Two Angles......Page 488
Tangent of the Sum or Difference of Two Angles......Page 489
Sine of Twice an Angle......Page 492
Cosine of Twice an Angle......Page 493
Sine of Half an Angle......Page 494
Tangent of Half an Angle......Page 496
16-4 Evaluating a Trigonometric Expression......Page 499
An Application: Compound cuts:......Page 500
16-5 Solving a Trigonometric Equation......Page 502
Equations Containing a Single Trigonometric Function and a Single Angle......Page 503
Equations with One Angle But More Than One Function......Page 505
CHAPTER 16 REVIEW PROBLEMS......Page 507
OBJECTIVES......Page 509
Dimensionless Ratios......Page 510
Finding a Missing Term......Page 511
Mean Proportional......Page 512
Applications......Page 513
17-2 Similar Figures......Page 515
Areas of Similar Figures......Page 516
Volumes of Similar Solids......Page 517
17-3 Direct Variation......Page 519
Solving Variation Problems......Page 520
Applications......Page 521
Graph of the Power Function......Page 523
Solving Power Function Problems......Page 524
An Application......Page 525
17-5 Inverse Variation......Page 527
An Application......Page 529
Joint Variation......Page 531
Combined Variation......Page 532
An Application......Page 533
CHAPTER 17 REVIEW PROBLEMS......Page 536
OBJECTIVES......Page 539
Graph of the Exponential Function......Page 540
Compound Interest......Page 541
Exponential Growth......Page 542
Exponential Growth to an Upper Limit......Page 544
Time Constant......Page 545
Universal Growth and Decay Curves......Page 546
Definition of a Logarithm......Page 550
Converting between Logarithmic and Exponential Forms......Page 551
Common Logarithms......Page 552
Antilogarithms......Page 553
Natural Antilogarithms......Page 554
Products......Page 557
Quotients......Page 558
Powers......Page 559
Log of 1......Page 561
Base Raised to a Logarithm of the Same Base......Page 562
Change of Base......Page 563
Solution by Calculator......Page 565
Algebraic Solution......Page 566
Solving an Exponential Equation with Base e......Page 567
Half-Life......Page 568
Doubling Time......Page 569
Algebraic Solution......Page 572
An Application......Page 573
CHAPTER 18 REVIEW PROBLEMS......Page 578
OBJECTIVES......Page 580
Addition and Subtraction of Complex Numbers......Page 581
Multiplying by Imaginary Numbers......Page 582
Multiplying Complex Numbers......Page 583
Dividing Complex Numbers......Page 584
19-2 Complex Numbers in Polar Form......Page 586
Arithmetic Operations in Polar Form......Page 587
Quotients......Page 588
Powers......Page 589
Entering and Converting Complex Numbers on the Calculator......Page 590
Arithmetic Operations on Complex Numbers by Calculator......Page 591
Components of a Vector......Page 593
Resultants of Several Vectors......Page 594
Rotating Vectors in the Complex Plane......Page 596
Effective or Root Mean Square (rms) Values......Page 597
Alternating Current and Voltage in Complex Form......Page 598
Ohm’s Law for Alternating Current......Page 599
CHAPTER 19 REVIEW PROBLEMS......Page 602
OBJECTIVES......Page 604
General Term......Page 605
Generating a Sequence or Series Using the General Term......Page 606
Graphing a Sequence or Series Using a Recursion Formula......Page 607
Sequences and Series by Computer......Page 608
Convergence or Divergence of an Infinite Series......Page 609
General Term......Page 611
AP: Sum of n Terms......Page 612
Sums by Calculator......Page 613
Harmonic Progressions......Page 614
Harmonic Means......Page 615
General Term......Page 618
GP: Sum of n Terms......Page 619
Geometric Means......Page 620
Sum of an Infinite Geometric Progression......Page 622
Limit Notation......Page 623
A Formula for the Sum of an Infinite, Decreasing Geometric Progression......Page 624
Powers of a Binomial......Page 625
Factorial Notation......Page 626
Pascal’s Triangle......Page 627
General Term......Page 628
Fractional and Negative Exponents......Page 629
CHAPTER 20 REVIEW PROBLEMS......Page 632
OBJECTIVES......Page 635
Variables......Page 636
Ways to Display Data......Page 637
Displaying Statistical Data on the Calculator......Page 638
Graphical Representation of Data......Page 639
Frequency Distribution......Page 640
Frequency Polygon......Page 642
Stem and Leaf Plot......Page 643
Graphing Frequency Distributions on the Calculator......Page 644
Measures of Central Tendency: The Mean......Page 646
Weighted Mean......Page 647
Median......Page 648
Range......Page 649
Quartiles, Deciles, and Percentiles......Page 650
Variance......Page 651
Standard Deviation......Page 652
Numerical Description of Data on the Calculator......Page 654
21-4 Introduction to Probability......Page 656
Probability of a Single Event Occurring......Page 657
Probability of Two Events Both Occurring......Page 658
Probability of Two Mutually Exclusive Events Occurring......Page 659
Probability Distributions......Page 660
Probabilities as Areas on a Probability Distribution......Page 661
Binomial Experiments......Page 662
Binomial Distribution......Page 663
Binomial Distribution Obtained by Binomial Theorem......Page 664
Continuous Probability Distributions......Page 666
Normal Distribution......Page 667
Using the Normal Curve on the Calculator......Page 671
21-6 Standard Errors......Page 672
Frequency Distribution of a Statistic......Page 673
Standard Error of the Mean......Page 674
Predicting μ from a Single Sample......Page 675
Confidence Intervals......Page 676
Standard Error of a Proportion......Page 677
Control Charts......Page 679
The p Chart......Page 680
The – and Charts......Page 682
Curve Fitting......Page 687
Correlation Coefficient......Page 688
Method of Least Squares......Page 689
Linear Regression on the Calculator......Page 691
CHAPTER 21 REVIEW PROBLEMS......Page 692
OBJECTIVES......Page 697
Directed Distance......Page 698
Distance Formula......Page 699
Slopes of Horizontal and Vertical Lines......Page 700
Angle of Inclination......Page 701
Angle of Intersection Between Two Lines......Page 702
Applications......Page 704
Slope-Intercept Form......Page 705
Point-Slope Form......Page 706
Two-Point Form......Page 707
Equation of a Straight Line in General Form......Page 708
Definition of a Circle......Page 712
Standard Equation of a Circle: Center at Origin......Page 713
Standard Equation of a Circle: Center Not at the Origin......Page 714
Translation of Axes......Page 715
Changing from General to Standard Form......Page 716
An Application......Page 717
Standard Equation of a Parabola: Vertex at the Origin......Page 720
Standard Equation of a Parabola with Vertical Axis......Page 721
Standard Equation of a Parabola: Vertex Not at the Origin......Page 723
General Equation of a Parabola......Page 724
Completing the Square......Page 725
Definition of the Ellipse......Page 731
Distance to Focus......Page 732
Standard Equation of Ellipse: Center at Origin, Major Axis Horizontal......Page 733
Graphing an Ellipse with a Graphics Utility......Page 735
Standard Equation of Ellipse: Center at Origin, Major Axis Vertical......Page 736
Standard Equation of Ellipse: Center Not at Origin......Page 737
General Equation of an Ellipse......Page 738
Completing the Square......Page 739
Focal Width of an Ellipse......Page 740
Standard Equations of a Hyperbola with Center at Origin......Page 743
Asymptotes of a Hyperbola......Page 744
Manually Graphing a Hyperbola......Page 746
Standard Equation of Hyperbola: Center Not at Origin......Page 747
Hyperbola Whose Asymptotes Are the Coordinate Axes......Page 749
CHAPTER 22 REVIEW PROBLEMS......Page 751
OBJECTIVES......Page 755
Limit Notation......Page 756
Limits Found Numerically......Page 757
Limits by Computer Algebra System......Page 758
Limits Involving Zero or Infinity......Page 759
Limits of the Form 0/0......Page 761
When the Limit Is an Expression......Page 762
Tangent to a Curve......Page 764
Finding the Slope of the Tangent......Page 765
23-3 The Derivative......Page 768
Another Symbol for the Derivative......Page 769
Derivatives by the Delta Method......Page 770
More Symbols for the Derivative......Page 772
Approximate Derivatives by Calculator......Page 773
Graph of a Derivative......Page 774
Symbolic Differentiation by Calculator or Computer......Page 775
Derivative of a Constant Times a Power Function......Page 777
Power Function with Negative Exponent......Page 778
Derivative of a Sum......Page 779
Functions with Other Variables......Page 781
The Chain Rule......Page 784
The Power Rule......Page 785
Derivative of a Product......Page 788
Products with More Than Two Factors......Page 790
Derivative of a Quotient......Page 792
Derivatives with Respect to Other Variables......Page 795
Derivatives of Implicit Relations......Page 796
Differentials......Page 798
23-8 Higher-Order Derivatives......Page 801
Higher Derivatives by Calculator......Page 802
CHAPTER 23: REVIEW PROBLEMS......Page 803
OBJECTIVES......Page 805
24-1 Equations of Tangents and Normals......Page 806
Implicit Relations......Page 807
Angle of Intersection of Two Curves......Page 809
Increasing and Decreasing Functions......Page 810
Concavity......Page 811
Maximum and Minimum Points......Page 812
Testing for Maximum or Minimum......Page 814
Implicit Relations......Page 816
Inflection Points......Page 817
Summary......Page 818
24-3 Sketching, Verifying, and Interpreting Graphs......Page 820
Graphing Regions......Page 824
CHAPTER 24 REVIEW PROBLEMS......Page 825
OBJECTIVES......Page 827
Rate of Change with Respect to Time......Page 828
Current in a Capacitor......Page 829
Voltage Across an Inductor......Page 830
Beam Deflection......Page 831
Displacement and Velocity in Straight-Line Motion......Page 833
Acceleration in Straight-Line Motion......Page 834
Velocity in Curvilinear Motion......Page 835
Displacement Given by Parametric Equations......Page 836
Trajectories......Page 837
Rotation......Page 838
25–3 Related Rates......Page 841
Two Moving Objects......Page 843
25–4 Optimization......Page 847
CHAPTER 25 REVIEW PROBLEMS......Page 857
OBJECTIVES......Page 859
The Integral Sign......Page 860
Some Rules for Finding Integrals......Page 861
Checking an Integral by Differentiating......Page 865
Checking an Integral by Calculator......Page 866
Symbolic Integration by Calculator......Page 867
Integral of a Power Function......Page 869
Other Variables......Page 872
Miscellaneous Rules from a Table of Integrals......Page 874
Finding the Constant of Integration......Page 877
Solving a Simple Second-Order Differential Equation......Page 878
Another Application......Page 879
The Fundamental Theorem of Calculus......Page 881
Evaluating a Definite Integral......Page 882
Finding a Definite Integral by Calculator......Page 883
Estimating Areas......Page 885
A Numerical Technique: The Midpoint Method......Page 886
Exact Area by Integration......Page 889
Area Under a Curve by Calculator......Page 892
CHAPTER 26 REVIEW PROBLEMS......Page 893