Index

0

not an infinitesimal, 82 Ω, 85 α, . = , 108 107 |α|, 119 ω level, 83 (GSL), 184 3 R’s principle, 107

A (capital alpha), 110 absolute value corner, 194 derivative, 193 algebraic function, 158 approximation cos, 231 definition of, 94 fractions, 98, 99 inside continuous function, 170 products, 99 sin, 230 square root, 99 approximation principle, 97 violation of, 97, 100, 102, 116, 177 arbitrary infinitesimal, 108 positive, 119 arctic circle, 236 argument trig function, 64 aspect ratio, 36, 221 asymptote vertical, 122 average rate of change formula, 182 average velocity, 178 example, 179 formula, 179

bisection method, 165

cancellation, 50 cancellation error, 51 Cauchy, Augustin Louis, 146 closed under multiplication, 87 combinations of continuous functions, 166 combinations of functions, 38 complex number, 80 composition of continuous functions, 169 of functions, 39 conjugate, 100 connect-the-dot game, 143, 155, 194 constant multiple rule derivative, 204 continuity, 143 combinations, 166 compositions, 169 cosine, 168 definition, 145, 156 elementary functions, 158 evaluating limits, 160 linear functions, 155 quadratic functions, 157 rational functions, 157 sine, 168 trigonometric functions, 169 verifying at x = b, 145 continuous at x = b, 145 corner not differentiable, 194 on graph, 194 cos continuity, 168 of infinitesimal, 168 cosine approximation formula, 231 counterexample, 45 cube root estimation, 224 cubic polynomial, 23 cusp, 195

Dawson, C. Bryan, 94, 114 definition

continuous at x = b, 145 differentiable at x = k, 189 dependent variable, 12 derivative, 189 absolute value, 193 as a function, 190 constant, 202 constant multiple rule, 204 cos x, 232 definition, 189 difference rule, 206 dot notation, 197 first, 207 general power rule, 212 graph, 218 interpretations, 190

Leibniz notation, 197, 207, 222 line, 203 notation, 196 polynomial, 206 power rule, 204, 211, 225 prime notation, 196 product rule, 208, 225 quotient rule, 210 rate of change, 190 second, 207 sin x, 232 single-term denominator, 212 slope of tangent line, 190 sum rule, 205 third, 207 trig, 233 velocity, 190 difference rule derivative, 206 differentiable definition, 189 implies continuous, 193 polynomials, 207 rational functions, 211 differential rules, 225 differentials, 225

digital library of functions, 25 direct substitution property, 160 direction of momentum, 77, 173 discontinuity, 144 identify algebraically, 148 identify graphically, 146 infinite, 144 jump, 144 oscillatory, 150 removable, 145 discriminant, 57 distributive property, 71 division by zero almost, 110 not always infinite, 111 one-sided limits, 123 two-sided limit, 122 DNE, 119 domain, 11 natural, 11 double-prime, 207

EAR procedure, 108 in limit, 114 eighteenth century, 225 elementary function continuous, 158 equation of line point-slope, 78, 174 equation of motion, 179 equinox, 236 equivalence relation, 94 error in approximation, 91 percent relative, 92 relative, 92 estimating a limit graphically, 129 estimating a limit numerically, 135 estimation using linearization, 224 evaluating limits using continuity, 160 evaluating using continuity fails, 161 evaluation symbol, 197

factor, 50 first derivative, 207 formula average rate of change, 182 average velocity, 179 cosine approximation, 231 derivative, 189

instantaneous rate of change, 182 instantaneous velocity, 181 local linearity, 193 quadratic, 57 relative error, 92 sine approximation, 230 slope of tangent line, 175 squaring a binomial, 46 trig derivatives, 233 from GSL, 184 function, 11 algebraic, 158 algebraic representation, 12 argument, 64 as a machine, 11, 39, 105, 109 combinations, 38 composition, 39 difference, 38 digital library, 25 domain, 11 elementary, 158 graphical representation, 13 hyperreal input, 106, 109 infinite input, 111 linear, 22, 155 numerical representation, 12 piecewise-defined, 18 polynomial, 22 power, 24 product, 38 quadratic, 157 quotient, 38 range, 11 reciprocal, 25 root, 24 signum, 29 sum, 38 symbolic representation, 12 verbal representation, 12 vertical line test, 14 function argument error, 60

general power rule derivative, 212 Granville, Smith, and Longley, 184 graphical estimate limit, 129

harmonic motion, 235 headlight beam, 77, 173 higher level, 87

definition of, 87 highway kilometer markers, 178 mile markers, 178 hole in graph, 115 none, 118 horizontal reflection, 34 horizontal shift, 33 horizontal stretching, 36 hours of daylight, 236 hyperreal, 79 hyperreal number, 81, 87

imaginary number, 80 increment, 221 independent variable, 12 indeterminate form, 148 infinite negative, 107, 110 positive, 107, 110 infinite discontinuity, 144 infinite number, 85 infinitesimal, 79, 81, 82 arbitrary, 108 definition of, 82 inside-outside principle, 60 instantaneous rate of change formula, 182 instantaneous velocity, 180 formula, 181 same formula as slope, 180 integer, 79 Intermediate Value Theorem, 161 irrational number, 80

jump, 119 jump discontinuity, 144

kilometer zero, 178

Lagrange, Joseph Louis, 196 Leibniz, Gottfried Wilhelm, 197 level, 83 ω, 83 diagram, 86 higher, 87 lower, 83, 87 same, 95 lim, 114 limit, 113 definition of, 114, 120

direct substitution property, 160

DNE, 119 evaluate using continuity, 160 from left, 119 from right, 119 graphical approach, 129 graphical interpretation, 114 hole in graph, 115 infinite, 121, 122 interval of values, 151 jump in graph, 119 no formula, 139 numerical estimate, 135 one-sided, 119 value at b irrelevant, 129 vertical asymptote, 122 limit example absolute value, 120 compound fraction, 117 infinite, 121 one-sided, 120 piecewise-defined, 127 polynomial, 118 rational function, 114 square root, 116 squeeze theorem, 139 trig, 116, 230 using continuity, 160 using continuity fails, 161 linear function, 22 continuous, 155 linearization, 223 local linearity, 221 formula, 193 infinitesimal scale, 222 real scale, 221 local linearity formula, 193 local linearity theorem, 197 Ło´s, Jerzy, 89 lower level, 83, 87 definition of, 87

mathematical modeling, 14 mental estimation, 224 merry-go-round, 77, 173 motion equation of, 179 harmonic, 235

National Institute of Standards and

Technology, 25 natural domain, 11 negative number, 79 nineteenth century, 225 NIST Digital Library of Mathematical

Functions, 25 nondifferentiability recognizing graphically, 195 nonstandard analysts, 89 northern hemisphere, 236 number complex, 80 hyperreal, 79, 81, 87 imaginary, 80 infinite, 85 infinitesimal, 79, 81 integer, 79 irrational, 80 negative, 79 rational, 79 real, 80 whole, 79 number line, 82 numerical analysis, 136 numerical estimate bisection method, 165 limit, 135 limitations of, 137 solution to equation, 164

one-sided limit, 119 definition of, 120 oscillatory discontinuity, 150

parabola, 23 parentheses errors, 67 particle moving on number line, 179 party trick, 224 period, 37 piecewise-defined function, 18 plot points, 143 point of tangency, 78, 174, 176 point-slope equation of line, 78, 174 polynomial, 22 coefficient, 22 cubic, 23 degree, 22 parabola, 23 smooth, 207 polynomial equation factoring, 59 polynomial function

continuity, 167 position function, 179 power function, 24 reciprocal function, 25 root function, 24 power rule derivative, 204, 211 differential version, 225 general, 212 negative integer exponents, 211 positive integer exponents, 204 precalculus, 79 product rule, 208 differential version, 225

quadratic equation factoring, 55 quadratic formula, 57 solving, 57 taking square roots, 58 quadratic formula, 57 quadratic function continuity, 157 quadratic in form, 245 quotient rule, 210

range, 11 rapid response, 89 rate of change average, 182 derivative, 190 instantaneous, 181 price, 181 rational function continuity, 157, 167 smooth, 211 rational number, 79 rational roots theorem, 59 real estate, 60 real number, 80 reciprocal of infinite number, 85 of infinitesimal, 85 reciprocal function, 25 recognizing nondifferentiability, 195 reflection horizontal, 34 vertical, 34 removable discontinuity, 145 render real result, 107 rendering, 107

Riggs, Troy, 107 Robinson, Abraham, 89 root of equation, 163 root function, 24 root-finding, 163

same level, 95 sandwich theorem, 138 secant line, 78, 174 second derivative, 207 setminus notation, 18 sgn(x), 29 shift horizontal, 33 vertical, 31 signum function, 29 Silly Putty, 35 sin continuity, 168 of infinitesimal, 168 sine approximation formula, 230 sketching functions limits, 132 sketching graph hole in graph, 115 jump in graph, 121 vertical asymptote, 122, 123 slope derivative, 190 negative, 219 positive, 218 tangent line, 175 zero, 216, 218 slope formula, 78, 174 slope of tangent line example parabola, 175 rational function, 176 square root, 177 smooth, 194 polynomials, 207 rational functions, 211 socks and shoes, 40 solstice, 236 southern hemisphere, 236 speedometer, 178, 180 spring-mass system, 235 square root estimation, 224 square root of a product, 49 square root of a sum error, 48 squaring a binomial error, 45 squaring a binomial formula, 46 squeeze theorem, 138 example, 139 stretching horizontal, 36 vertical, 36 stuck, 110, 127 sum rule derivative, 205 synthetic division, 59

Tahlequah, OK, 236 tangent line equation, 176, 215 equation (vertical), 195 horizontal, 216 linearization, 223 slope, 175 to circle, 77, 173

to curve, 77, 173 vertical, 194 term, 51, 93 third derivative, 207 throw away terms, 95 transfer principle, 88 trigonometric functions argument, 64 derivatives, 233 tropic of cancer, 236

U. S. national debt, 96

velocity, 179 average, 178 derivative, 190 instantaneous, 180 negative, 179 positive, 179 verifying continuity at x = b, 145 versine, 238 vertical asymptote, 122 vertical line test, 14 vertical reflection, 34 vertical shift, 31 vertical stretching, 36 vertical tangent, 192, 194 violated the approximation principle, 97, 100, 102, 116, 177

whole number, 79

zahlen, 79 zero product property, 55