Introduciton to calculus volume 1

Introduction to Calculus Volume I by J.H Heinbockel The regular solids or regular polyhedra are solid geometric figures with the same identical regular polygon on each face There are only five regular solids discovered by the ancient Greek mathematicians These five solids are the following the tetrahedron (4 faces) the cube or hexadron (6 faces) the octahedron (8 faces) the dodecahedron (12 faces) the icosahedron (20 faces) Each figure follows the Euler formula Number of faces + Number of vertices = Number of edges + F + V = E + Introduction to Calculus Volume I by J.H Heinbockel Emeritus Professor of Mathematics Old Dominion University c Copyright 2012 by John H Heinbockel All rights reserved Paper or electronic copies for noncommercial use may be made freely without explicit permission of the author All other rights are reserved This Introduction to Calculus is intended to be a free ebook where portions of the text can be printed out Commercial sale of this book or any part of it is strictly forbidden ii Preface This is the first volume of an introductory calculus presentation intended for future scientists and engineers Volume I contains five chapters emphasizing fundamental concepts from calculus and analytic geometry and the application of these concepts to selected areas of science and engineering Chapter one is a review of fundamental background material needed for the development of differential and integral calculus together with an introduction to limits Chapter two introduces the differential calculus and develops differentiation formulas and rules for finding the derivatives associated with a variety of basic functions Chapter three introduces the integral calculus and develops indefinite and definite integrals Rules for integration and the construction of integral tables are developed throughout the chapter Chapter four is an investigation of sequences and numerical sums and how these quantities are related to the functions, derivatives and integrals of the previous chapters Chapter five investigates many selected applications of the differential and integral calculus The selected applications come mainly from the areas of economics, physics, biology, chemistry and engineering The main purpose of these two volumes is to (i) Provide an introduction to calculus in its many forms (ii) Give some presentations to illustrate how powerful calculus is as a mathematical tool for solving a variety of scientific problems, (iii) Present numerous examples to show how calculus can be extended to other mathematical areas, (iv) Provide material detailed enough so that two volumes of basic material can be used as reference books, (v) Introduce concepts from a variety of application areas, such as biology, chemistry, economics, physics and engineering, to demonstrate applications of calculus (vi) Emphasize that definitions are extremely important in the study of any mathematical subject (vii) Introduce proofs of important results as an aid to the development of analytical and critical reasoning skills (viii) Introduce mathematical terminology and symbols which can be used to help model physical systems and (ix) Illustrate multiple approaches to various calculus subjects If the main thrust of an introductory calculus course is the application of calculus to solve problems, then a student must quickly get to a point where he or she understands enough fundamentals so that calculus can be used as a tool for solving the problems of interest If on the other hand a deeper understanding of calculus is required in order to develop the basics for more advanced mathematical iii efforts, then students need to be exposed to theorems and proofs If the calculus course leans toward more applications, rather than theory, then the proofs presented throughout the text can be skimmed over However, if the calculus course is for mathematics majors, then one would want to be sure to go into the proofs in greater detail, because these proofs are laying the groundwork and providing background material for the study of more advanced concepts If you are a beginner in calculus, then be sure that you have had the appropriate background material of algebra and trigonometry If you don’t understand something then don’t be afraid to ask your instructor a question Go to the library and check out some other calculus books to get a presentation of the subject from a different perspective The internet is a place where one can find numerous help aids for calculus Also on the internet one can find many illustrations of the applications of calculus These additional study aids will show you that there are multiple approaches to various calculus subjects and should help you with the development of your analytical and reasoning skills J.H Heinbockel September 2012 iv Introduction to Calculus Volume I Chapter Sets, Functions, Graphs and Limits Elementary Set Theory, Subsets, Set Operations, Coordinate Systems, Distance Between Two Points in the Plane, Graphs and Functions, Increasing and Decreasing Functions, Linear Dependence and Independence, Single-valued Functions, Parametric Representation of Curve, Equation of Circle, Types of Functions, The Exponential and Logarithmic Functions, The Trigonometric Functions, Graphs of Trigonometric Functions, The Hyperbolic Functions, Symmetry of Functions, Translation and Scaling of Axes, Inverse Functions, Equations of Lines, Perpendicular Lines, Limits, Infinitesimals, Limiting Value of a Function, Formal Definition of a Limit, Special Considerations, Properties of Limits, The Squeeze Theorem, Continuous Functions and Discontinuous Functions, Asymptotic Lines, Finding Asymptotic Lines, Conic Sections, Circle, Parabola, Ellipse, Hyperbola, Conic Sections in Polar Coordinates, Rotation of Axes, General Equation of the Second Degree, Computer Languages Chapter Differential Calculus 85 Slope of Tangent Line to Curve, The Derivative of y = f(x), Right and Left-hand Derivatives, Alternative Notations for the Derivative, Higher Derivatives, Rules and Properties, Differentiation of a Composite Function, Differentials, Differentiation of Implicit Functions, Importance of Tangent Line and Derivative Function f (x), Rolle’s Theorem, The Mean-Value Theorem, Cauchy’s Generalized Mean-Value Theorem, Derivative of the Logarithm Function, Derivative of the Exponential Function, Derivative and Continuity, Maxima and Minima, Concavity of Curve, Comments on Local Maxima and Minima, First Derivative Test, Second Derivative Test, Logarithmic Differentiation, Differentiation of Inverse Functions, Differentiation of Parametric Equations, Differentiation of the Trigonometric Functions, Simple Harmonic Motion, L´Hˆ opital’s Rule, Differentiation of Inverse Trigonometric Functions, Hyperbolic Functions and their Derivatives, Approximations, Hyperbolic Identities, Euler’s Formula, Derivatives of the Hyperbolic Functions, Inverse Hyperbolic Functions and their Derivatives, Relations between Inverse Hyperbolic Functions, Derivatives of the Inverse Hyperbolic Functions, Table of Derivatives, Table of Differentials, Partial Derivatives, Total Differential, Notation, Differential Operator, Maxima and Minima for Functions of Two Variables, Implicit Differentiation Chapter Integral Calculus 175 Summations, Special Sums, Integration, Properties of the Integral Operator, Notation, Integration of derivatives, Polynomials, General Considerations, Table of Integrals, Trigonometric Substitutions, Products of Sines and Cosines, Special Trigonometric Integrals, Method of Partial Fractions, Sums and Differences of Squares, Summary of Integrals, Reduction Formula, The Definite Integral, Fundamental theorem of integral calculus, Properties of the Definite Integral, Solids of Revolution, Slicing Method, Integration by Parts, Physical Interpretation, Improper Integrals, Integrals used to define Functions, Arc Length, Area Polar Coordinates, Arc Length in Polar Coordinates, Surface of Revolution, Mean Value Theorems for Integrals, Proof of Mean Value Theorems, Differentiation of Integrals, Double Integrals, Summations over nonrectangular regions, Polar Coordinates, Cylindrical Coordinates, Spherical Coordinates, Using Table of Integrals, The Bliss Theorem v Table of Contents Chapter Sequences, Summations and Products 271 Sequences, Limit of a Sequence, Convergence of a sequence, Divergence of a sequence, Relation between Sequences and Functions, Establish Bounds for Sequences, Additional Terminology Associated with Sequences, Stolz -Ces`aro Theorem, Examples of Sequences, Infinite Series, Sequence of Partial Sums, Convergence and Divergence of a Series, Comparison of Two Series, Test For Divergence, Cauchy Convergence, The Integral Test for Convergence, Alternating Series Test, Bracketing Terms of a Convergent Series, Comparison Tests, Ratio Comparison Test, Absolute Convergence, Slowly Converging or Slowly Diverging Series, Certain Limits, Power Series, Operations with Power Series, Maclaurin Series, Taylor and Maclaurin Series, Taylor Series for Functions of Two Variables, Alternative Derivation of the Taylor Series, Remainder Term for Taylor Series, Schlă omilch and Roche remainder term, Indeterminate forms à ∞, ∞ − ∞, 00 , ∞0 , 1∞ , Modification of a Series, Conditional Convergence, Algebraic Operations with Series, Bernoulli Numbers, Euler Numbers, Functions Defined by Series, Generating Functions, Functions Defined by Products, Continued Fractions, Terminology, Evaluation of Continued Fractions, Convergent Continued Fraction, Regular Continued Fractions, Euler’s Theorem for Continued Fractions, Gauss Representation for the Hypergeometric Function, Representation of Functions, Fourier Series, Properties of the Fourier trigonometric series, Fourier Series of Odd Functions, Fourier Series of Even Functions, Options, Chapter Applications of Calculus 363 Related Rates, Newton’s Laws, Newton’s Law of Gravitation, Work, Energy, First Moments and Center of Gravity, Centroid and Center of Mass, Centroid of an Area, Symmetry, Centroids of composite shapes, Centroid for Curve, Higher Order Moments, Moment of Inertia of an Area, Moment of Inertia of a Solid, Moment of Inertia of Composite Shapes, Pressure, Chemical Kinetics, Rates of Reactions, The Law of Mass Action, Differential Equations, Spring-mass System, Simple Harmonic Motion, Damping Forces, Mechanical Resonance, Particular Solution, Torsional Vibrations, The simple pendulum, Electrical Circuits, Thermodynamics, Radioactive Decay, Economics, Population Models, Approximations, Partial Differential Equations, Easy to Solve Partial Differential Equations Appendix A Units of Measurement Appendix B Background Material Appendix C Table of Integrals 452 454 466 Appendix D Solutions to Selected Problems Index 520 552 vi Chapter Sets, Functions, Graphs and Limits The study of different types of functions, limits associated with these functions and how these functions change, together with the ability to graphically illustrate basic concepts associated with these functions, is fundamental to the understanding of calculus These important issues are presented along with the development of some additional elementary concepts which will aid in our later studies of more advanced concepts In this chapter and throughout this text be aware that definitions and their consequences are the keys to success for the understanding of calculus and its many applications and extensions Note that appendix B contains a summary of fundamentals from algebra and trigonometry which is a prerequisite for the study of calculus This first chapter is a preliminary to calculus and begins by introducing the concepts of a function, graph of a function and limits associated with functions These concepts are introduced using some basic elements from the theory of sets Elementary Set Theory A set can be any collection of objects A set of objects can be represented using the notation S = {x | statement about x} and is read,“ S is the set of objects x which make the statement about x true ” Alternatively, a finite number of objects within S can be denoted by listing the objects and writing S = {S1 , S2 , , Sn } For example, the notation S = { x | x − > 0} can be used to denote the set of points x which are greater than and the notation T = {A, B, C, D, E} can be used to represent a set containing the first letters of the alphabet A set with no elements is denoted by the symbol ∅ and is known as the empty set The elements within a set are usually selected from some universal set U associated with the elements x belonging to the set When dealing with real numbers the universal set U is understood to be the set of all real numbers The universal set is usually defined beforehand or is implied within the context of how the set is being used For example, the universal set associated with the set T above could be the set of all symbols if that is appropriate and within the context of how the set T is being used The symbol ∈ is read “ belongs to ” or “ is a member of ” and the symbol ∈/ is read “ not in ” or “ is not a member of ” The statement x ∈ S is read “ x is a member of S ” or “ x belongs to S ” The statement y ∈ / S is read “ y does not belong to S ” or “y is not a member of S ” Let S denote a non-empty set containing real numbers x This set is said to be bounded above if one can find a number b such that for each x ∈ S , one finds x ≤ b The number b is called an upper bound of the set S In a similar fashion the set S containing real numbers x is said to be bounded below if one can find a number such that ≤ x for all x ∈ S The number is called a lower bound for the set S Note that any number greater than b is also an upper bound for S and any number less than can be considered a lower bound for S Let B and C denote the sets B = {x | x is an upper bound of S} and C = { x | x is a lower bound of S}, then the set B has a least upper bound ( u.b.) and the set C has a greatest lower bound (g .b.) A set which is bounded both above and below is called a bounded set Some examples of well known sets are the following The set of natural numbers N = {1, 2, 3, } The set of integers Z = { , −3, −2, −1, 0, 1, 2, 3, } The set of rational numbers Q = { p/q | p is an integer, q is an integer, q = 0} The set of prime numbers P = {2, 3, 5, 7, 11, } The set of complex numbers C = { x + i y | i2 = −1, x, y are real numbers} The set of real numbers R = {All decimal numbers} The set of 2-tuples R2 = { (x, y) | x, y are real numbers } The set of 3-tuples R3 = { (x, y, z) | x, y, z are real numbers } The set of n-tuples Rn = { (ξ1 , ξ2 , , ξn) | ξ1 , ξ2, , ξn are real numbers } where it is understood that i is an imaginary unit with the property i2 = −1 and decimal numbers represent all terminating and nonterminating decimals 544 Chapter πr , 5-1 V = 5-2 pv 1.4 = c, dp dt dv dt (a) (b) dV dr = 4πr , dt dt α = 4πr02 dp 1.4 dv v + p(1.4)v 0.4 =0 dt dt or dr dr α =⇒ = dt dt 4πr02 dp p dv = −1.4 dt v dt = −1.4 vp α = β v 1.4 p 5-3 +s s = , 10 5.5 5-4 if d dt = ft/s, (a) Let S1 = x − f > and S2 = y − f > 0, then 1 + = S1 + f S2 + f f ft/s 1 + = x−f +f y −f +f f or dy y2 = − r0 dt x Area of equilateral triangle with side x is Now substitute x = x0 and dx = r0 dt 5-5 √ A= x so that √ dA dx = x dt dt p = p0 e−α0 h [p0 ] = lbs/f t2 , dh = 10 f t/s dt (a) (b) ds 22 = dt 4.5 Simplify this expression and show f = S1 S2 (b) Differentiate the lens law and show 5-6 then show [α0 ] = 1/f t dp dh = p0 e−α0 h (−α0 ) dt dt Substitute for dh dt and find the pressure decreases with height dy y−y0 = α(x−x0 )2 has the derivative = 2α(x−x0 ) When x = ξ , the slope of the dx tangent is m = 2α(ξ −x0 ) and the equation of the tangent line is y −η = 2α(ξ −x0 )(x−ξ) dξ dθ dξ dθ 2α Here tan θ = m = 2α(ξ−x0 ) and if dξ = 1/2 cm/s, then sec2 θ = 2α or = dt 5-7 dt or dθ 2α = 2 dt + 4α (ξ − x0 ) 5-8 dT r0 = dt c0 5-9 dP = c0 r dt where c0 = V0 /T0 where c0 = P0 /T0 Solutions Chapter dt dt + tan θ dt 545 5-10 When h = 0, then h =200 − 16t2 f t dh v= = − 32 t f t/s dt dv d2 h a= = = − 32 f t/s2 dt dt √ √ t = 210/4 and v = −80 f t/s 5-11 pi 3 dh = 30/(π 420) dt π (a) V = πr3 − (2r−h)2 (r+h) = (3rh2 −h3 ) and (b) 5-12 (h − r)2 + R2 = r dV π dh dh = (3r 2h −3h2 ), dt dt dt r is a constant Differentiate this relation and show dR dh = − (h − r) r < h < 2r dt dt dR dh R =(r − h) 0 steadily increases, while v2 = dsdt2 > for t < 11/6, s2 increases and ds2 < for t > 11/6, then s2 decreases dt Solutions Chapter 551 ∂f ∂f 5-50 (a) = 2x − 2, and = when x = ∂x ∂x ∂f ∂f = 2y − and = when y = f (1, 2) = −25 is a minimum ∂y ∂y all (x, y) is a neighborhood of (1, 2) we have f (x, y) > f (1, 2) value, since for 5-51 −y0 and equation of line is x1 − x0 (−y0 ) y − y0 = (x − x0 ), when x = 0, y1 = y0 + (x1 − x0 ) m = slope = x0 y x1 −x0 Therefore = x21 + y12 = x21 + y0 + x0 y0 x1 − x0 , x0 , y0 f ixed 1/3 2/3 ∂ = 0, when x1 = x0 + x0 y0 This is the value of x1 which will produce the Show ∂x shortest line 5-52 x = 3, y = and x = 9, y = 12 Solutions Chapter 552 Index A abscissa absolute maximum 117 absolute value function 11 acceleration of gravity 369 addition 325 addition of series 326 adiabatic process 425 algebraic function 20 algebraic operations 325 alternating series test 293 amplitude 403 amplitude versus frequency 416 analysis of derivative 106 angle of intersection 104, 105 angle of intersection for lines 40 arc length 238 Archimedes 178 arctangent function 338 area between curves 220 area polar coordinates 240, 256 Area under a curve 215 arithmetic series 176 asymptotic lines 55, 68 axis of symmetry 59, 380 Cauchy product 326 Cauchy’s mean-value theorem 111 center of gravity 374 center of mass 374 centroid 375, 380 centroid of area 377 centroid of curve 384 centroid of composite shapes 383 chain rule differentiation 99 change of variables 220 characteristic equation 407 characteristic roots 407, 412 charge 419 Charles’s law 424 chemical kinetics 395 chemical reaction 395 circle 18, 59 circular functions 142 circular neighborhood 275 circumference of circle 239 closed interval comparison test 296, 298 complementary error function 237 complementary solution 414 composite function 98, 315 concavity 118 conditional convergence 325 conic sections 57 conic sections polar coordinates 70 conjugate hyperbola 69 conservation of energy 374 constant of integration 181 contained in continued fraction 332 continuity 116 continuous function 54, 88 convergence of a sequence 272 convergence of series 283 convergent continued fraction 336 coordinate systems cosine function 24 critical damping 414 current 419 curves 16 cycles per second 403 cycloid 134 cylindrical coordinates 256 B belongs to Bernoulli numbers 314, 328 Bessel functions 309 bimolecular reaction 397 binomial coefficients 356 binomial series 356 binomial theorem 92 Bonnet’s second mean value theorem 245 bounded increasing sequence 300 bounded sequence 324 bounded set bounds for sequence 275 Boyle’s law 365, 424 bracketing terms 295 C capacitance 419 cartesian coordinates Cauchy convergence 278, 287 Cauchy form for the remainder 313 Index 553 equations for line 36 equivalence error function 237, 310 escape velocity 369 estimation of error 291, 294, 298 Euler numbers 328 Euler-Mascheroni constant 310 Euler’s formula 147 Euler’s identity 409 evaluation of continued fraction 334 even and odd functions 358 even function of x 26 existence of the limit 278 exponential function 21, 113 exterior angle 40 extrema 119 extreme value 162 extremum 119 D d’Alembert ratio test 302 damped oscillations 413 damping force 405 de Moivre’s theorem 149, 168, 190 decreasing functions 12 definite integrals 213 derivative 87 derivative notation 90 derivative of a product 95 derivative of a quotient 97 derivative of the logarithm 111 derivative of triple product 96 derivatives of inverse hyperbolic functions 149 derivatives of trigonometric functions 131 determinants and parabola 62 difference between sets differential equations 399 differentials 101 differentiation of composite function 98 differentiation of implicit functions 102 differentiation of integrals 247 differentiation operators 90 differentiation rules 91 Dirac delta function 174 direction of integration 219 directrix 59 discontinuous function 54 disjoint sets distance between points distance from point to line 171 divergence of a sequence 273 divergence of series 283 domain of definition 33 double integrals 249 dummy summation index 282 dummy variable of integration 184 F finite oscillatory 277, 283 finite oscillatory sequence 277 finite sum 282 first derivative test 120 first law of thermodynamics 424 first mean value theorem for integrals 245 first moment 374 focal parameter 59 focus 59 Fourier cosine transform 235 Fourier exponential transform 235 Fourier series 339 Fourier sine transform 235 frequency of motion 403 full Fourier interval 345 function 271 function changes sign 108 functions 8, 20 functions defined by products 330 functions defined by series 330 functions of two variables 159 fundamental theorem of integral calculus 217 E elastic potential energy 403 electrical circuits 418 electromotive force 419 element of volume 226, 249 ellipse 63 empty set energy 372 epsilon-delta definition of limit 46 equality of sets equation of line 36 equation of state 425 G Gamma function 237 gas pressure 394 Gay-Lussac law 424 general equation of line 38 general equation of second degree 71 generalized mean value theorem for integrals 245 generalized mean-value theorem 111 Index 554 generalized second mean value theorem 245 generalized triangle inequality 300 geometric interpretations 273 geometric series 177, 287, 350, 357 graph compression 29 graph expansion 29 graph scaling 29 graphic compression 29 graphs graphs of trigonometric functions 24 intersection intersection of circles 105 intersection of two curves 105 interval neighborhood 275 interval notation interval of convergence 305 inverse functions 31, 128 inverse hyperbolic functions 153 inverse of differentiation 179 inverse operator 31 inverse trigonometric functions 34, 140 isothermal curves 425 iterative scheme 334 H half-life 427 harmonic series 283 harmonic series of order p 291 Heaviside 174 higher derivatives 90 higher order moments 385 higher partial derivatives 159 Hooke’s law 401 horizontal inflection point 118 horizontal line test 31 hyperbola 66 hyperbolic functions 25, 142, 149 hyperbolic identities 145 hypergeometric function 311 hypergeometric series 318 J jump discontinuity 43, 89, 107 K kinetic energy 372 Kirchoff’s laws 420 Kronecker delta 340 L L´Hˆ opital’s rule 138, 321 Lagrange form of the remainder 313 Laplace transform 235 latus rectum 59, 67 law of exponents 145 law of mass action 396 left-hand limits 40 left-handed derivative 89 Leibnitz 85 Leibnitz differentiation rule 168 Leibnitz formula 247 Leibnitz rule 248 length of curve 238 limit 46, 272 limit of a sequence 272 limit of function 42 limit point of sequence 277 limit theorem 50 limiting value 43 limits 40, 46, 304 linear dependence 13 linear homogeneous differential equation 410 linear independence 13 linear spring 403 lines 36 liquid pressure 393 local maximum 107, 117, 161 I implicit differentiation 106, 162 improper integrals 234 increasing functions 12 indefinite integral 180 indeterminate forms 43, 322 inductance 419 infinite oscillatory 283 infinite series 281 infinitesimals 41 inner product 339 integral notation 182 integral sign 181 integral test 288 integral used to define functions 236, 248 integration 179 integration by parts 209, 232 integration of derivatives 183 integration of polynomials 183 intercept form for line 38 intercepts 38 intermediate value property 54 intersecting lines 40, 104 Index 555 local minimum 107, 117, 161 logarithm base e 23 logarithmic differentiation 127 logarithmic function 21, 111 log-log paper 171 lower bound 2, 275 O odd function of x 26 one-to-one correspondence 271 one-to-one function 31, 33 open interval operator 90 operator box 90, 180 order of reaction 396 ordered pairs 33, 55 ordinate orientation of the surface 252 orthogonal intersection 40 orthogonal intersection 104 orthogonal lines 105 orthogonal sequence 340 orthonormal 340 oscillating sequence 277 overdamping 413 M Maclaurin Series 311, 315 mapping 271 maxima 107, 116, 161 mean value theorem for integrals 245, 289 mean value theorem 108, 245 mechanical resonance 412 method of undetermined coefficients 414 minima 107, 116, 161 mirror image 33 modification of series 324 moment of force 374 moment of inertia of solid 392 moment of inertial of area 390 moment of inertia of composite shapes 393 moments of inertia 385 momentum 367 monotone decreasing 107, 245, 277 monotone increasing 107, 245, 277 multiple-valued functions 14 multiplication 325 P parabola 60 parallel circuit 423 parallelepiped volume elements 252 parametric equations 129 parametric equation for line 38 parametric representation 17, 134 partial denominators 333 partial derivatives 158, 160 partial fractions 195, 284 partial numerators 333 partial sums 282 particular solution 414 period of oscillation 403 periodic motion 403 perpendicular distance 39 perpendicular lines 39 phase shift 403 piecewise continuous 345 piecewise continuous functions 11 piston 174 plane curves 14 plotting programs 14 point of inflection 118 point-slope formula 37, 88 polar coordinates polar coordinates 240, 255 polar form for line 39 polar graph 14 polynomial function 20, 95 positive monotonic 245 positive slope 37 N natural logarithm 23 natural logarithm function 236 necessary condition for convergence 286 negative slope 37 neighborhoods 275 Newton 85 Newton root finding 353 Newton’s law of gravitation 368 Newton’s laws 366 nonconvergence 277 nonrectangular regions 252 not in notation for limits 40, 42 notations for derivatives 90 nth term test 286 n-tuples null sequence 277 number pairs Index 556 potential energy 373 power rule 100 power rule for differentiation 99 power series 305 pressure 393 pressure-volume diagram 425 principal branches 35 product rule 95 products of sines and cosines 193 Proof of Mean Value Theorems 246 proper subsets properties of definite integrals 218 properties of integrals 181 properties of limits 50 p-series 291 Pythagorean identities 203 Pythagorean theorem Riemann sum 215 right circular cone 172 right-hand limits 40 right-handed derivative 89 Rolle’s theorem 107, 246 rotation of axes 30 rules for differentiation 91 S scale factors 29 scaling for integration 183 scaling of axes 28 Schlă omilch and Roche remainder term 320 secant line 85 second derivative test 120 second derivatives 90 second law of thermodynamics 424 second moments 385 sectionally continuous 43 semi-convergent series 325 semi-log paper 171 sequence of partial sums 282 sequence of real numbers 271 sequences and functions 274 series 281, 325 series circuit 421 set complement set operations set theory sets shearing modulus 418 shift of index 282 shifting of axes 28 shorthand representation 282 signed areas 220 simple harmonic motion 136, 403 simple pendulum 418 sine function 24 sine integral function 310 single-valued function 14, 31 slicing method 231 slope 37, 85 slope changes 120 slope condition for orthogonality 104 slope of line 36 slope-intercept form for line 38 slowly converging series 301 slowly diverging series 301 smooth curve 107 smooth function 116 Snell’s law 122 solids of revolution 225 Q quadrants quotient rule 97 R radioactive decay 426 radius of convergence 305 radius of Earth 369 range of function 33 rate of reaction 395 ratio comparison test 297 ratio test 302 rational function 20 ray from origin rectangular coordinates rectangular graph 14 reductio ad absurdum 276 reduction formula 211 reflection 29 refraction 123 regular continued fractions 336 related rates 363 relative maximum 107, 117, 161 relative minimum 107, 117, 161 remainder term 313, 319 Remainder Term for Taylor Series 319 representation of functions 337 resistance 419 resonance 412 resonance frequency 416 restoring force 403 reverse reaction rate 395 reversion of series 354 Index 557 special functions 309 special limit 304 special sums 176 special trigonometric integrals 194 spherical coordinates 257 spring-mass system 401 squeeze theorem 53, 275 Stolz -Ces´aro theorem 279 subscript notation 160 subsequence 277 subsets subtraction 325 subtraction of series 326 summation notation 175 summation of forces 401 sums and differences of squares 202 surface area 242 surface of revolution 242 symmetric functions 26 symmetry 26, 31, 380 total differential 159 transcendental function 21 transformation equations 28 translation of axes 28 transverse axis 67 triangular numbers 284 trigonometric functions 24, 129 trigonometric substitutions 189 truncation of series 286 two point equations of line 36 two-point formula 37 U union units of measurement 367 universal set upper bound 2, 275 using table of integrals 258 V Venn diagram vertex 59 voltage drop 419 volume of sphere 172 volume under a surface 253 T table of centroids 382 table of derivatives 156 table of differentials 157 table of integrals 186, 208 table of moments of inertia 390 tangent function 24 tangent line 86 Taylor series 311, 315, 318 Taylor series two variables 315 telescoping series 284, 351 terminology for sequences 277 thermodynamics 424 torque 374 torsional vibrations 417 total derivative 160 W weight function 340 weight of an object 369 work 371 work done 403 Z zero slope 37 zeroth law of thermodynamics 424 Index Public Domain Images Courtesy of Wikimedia.Commons ... -1. 8 -1. 6 -1. 4 -1. 2 -1. 0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1. 0 1. 2 1. 4 1. 6 1. 8 2.0 2.00 1. 44 0.96 0.56 0.24 0.00 -0 .16 -0.24 -0.24 -0 .16 0.00 0.24 0.56 0.96 1. 44 2.00 2.64 3.36 4 .16 5.04 6.00... -2.00 -1. 75 -1. 50 -1. 25 -1. 00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1. 00 1. 25 1. 50 1. 75 2.00 4.0000 3.0625 2.2500 1. 5625 1. 0000 0.5625 0.2500 0.0625 0.0000 0.0625 0.2500 0.5625 1. 0000 1. 5625... or (1. 11) (x − h)2 + (y − k)2 = ρ Figure 1- 14 Circle centered at (h, k) Equations of the form x2 + y + αx + βy + γ = 0, α, β, γ constants (1. 12) can be converted to the form of equation (1. 11)