Calculus Know-It-ALL This page intentionally left blank Calculus Know-It-ALL Beginner to Advanced, and Everything in Between Stan Gibilisco New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto Copyright © 2009 by The McGraw-Hill Companies, Inc All rights reserved Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher ISBN: 978-0-07-154932-5 MHID: 0-07-154932-3 The material in this eBook also appears in the print version of this title: ISBN: 978-0-07-154931-8, MHID: 0-07-154931-5 All trademarks are trademarks of their respective owners Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of 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THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE McGraw-Hill and its licensors not warrant or guarantee that the functions contained in the work will meet your requirements or that its operation will be uninterrupted or error free Neither McGraw-Hill nor its licensors shall be liable to you or anyone else for any inaccuracy, error or omission, regardless of cause, in the work or for any damages resulting therefrom McGraw-Hill has no responsibility for the content of any information accessed through the work Under no circumstances shall McGraw-Hill and/or its licensors be liable for any indirect, incidental, special, punitive, consequential or similar damages that result from the use of or inability to use the work, even if any of them has been advised of the possibility of such damages This limitation of liability shall apply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or otherwise To Tim, Tony, Samuel, and Bill This page intentionally left blank About the Author Stan Gibilisco is an electronics engineer, researcher, and mathematician He is the author of Algebra Know-It-ALL, a number of titles for McGraw-Hill’s Demystified series, more than 30 other technical books and dozens of magazine articles His work has been published in several languages vii This page intentionally left blank Contents Preface xv Acknowledgment xvii Part Differentiation in One Variable Single-Variable Functions Mappings Linear Functions Nonlinear Functions 12 “Broken” Functions 14 Practice Exercises 17 Limits and Continuity 20 Concept of the Limit 20 Continuity at a Point 24 Continuity of a Function 29 Practice Exercises 33 What’s a Derivative? 35 Vanishing Increments 35 Basic Linear Functions 40 Basic Quadratic Functions 44 Basic Cubic Functions 48 Practice Exercises 52 ix APPENDIX F Table of Derivatives The letter a denotes a general constant, f and g denote functions, x denotes a variable, and e represents the exponential constant (approximately 2.71828) Function f (x ) a ax ax n ln |x | ln |g (x )| x −a ex ax a g (x ) e ax e g (x ) xe x sin x cos x tan x csc x sec x cot x Arcsin x Arccos x Arctan x 778 Derivative f ′ (x ) a nax n −1 x −1 g −1(x ) g ′ (x ) −ax (−a −1) ex a x ln |a | [a g (x )] [ g ′(x )] ln |a | ae ax [e g (x )] [ g ′ (x )] e x + xe x cos x −sin x sec x − csc x cot x sec x tan x − csc x (1 − x )−1/2 −(1 − x )−1/2 (1 + x 2)−1 APPENDIX G Table of Integrals Letters a and b denote general constants, c denotes a constant of integration, f denotes a function, x denotes a variable, and e represents the exponential constant (approximately 2.71828) Function Indefinite integral f (x ) a x ax ax ax ax ax −1 ax −2 ax −3 ax −4 ax n for n ≠ −1 (ax + b )−1 (ax + b )n for n ≠ −1 x (ax + b )1/2 x (ax + b )−1/2 x (ax + b )−2 (x + a 2)1/2 (x − a 2)1/2 (a − x 2)1/2 (x + a 2)−1/2 (x − a 2)−1/2 (a − x 2)−1/2 ∫ f (x ) dx c x+c ax + c (1/2) x + c (1/2) ax + c (1/3) ax + c (1/4) ax + c (1/5) ax5 + c a ln |x | + c −ax −1 + c (−1/2) ax −2 + c (−1/3) ax −3 + c (n + 1)−1 ax n +1 + c a −1 ln |ax + b | + c (an + a)−1 (ax + b )n +1 + c (1/15) a −2 (6ax − 4b )(ax + b )3/2 + c (1/3) a −2 (2ax − 4b )(ax + b )1/2 + c b (a 3x + a 2b )−1 + a −2 ln |ax + b | + c (x /2) (x + a 2)1/2 + (1/2) a ln |x + (x + a 2)1/2| + c (x /2) (x − a 2)1/2 − (1/2) a ln |x + (x − a 2)1/2| + c (x /2) (a − x 2)1/2 + (1/2) a Arcsin (a −1x ) + c ln |x + (x + a 2)1/2| + c ln |x + (x − a 2)1/2| + c Arcsin (a −1x ) + c 779 780 Table of Integrals (x + a 2)−1 for a > (x − a 2)−1 (a − x 2)−1 for |a | > |x | (x + a 2)−2 (x − a 2)−2 (a − x 2)−2 for |a | > |x | x (x + a 2)1/2 x (x − a 2)1/2 x (a − x 2)1/2 x (x + a 2)−1/2 x (x − a 2)−1/2 x (a − x 2)−1/2 x (x + a 2)−1 x (x − a 2)−1 x (a − x 2)−1 for |a | > |x | x (x + a 2)−2 x (x − a 2)−2 x (a − x 2)−2 for |a | > |x | ex e ax a e bx x e ax x e bx ln |x | x n ln |x | for n ≠ −1 x −1 ln x sin x cos x tan x csc x sec x cot x sin ax cos ax tan ax csc ax sec ax cot ax sin2 x cos2 x tan2 x csc x sec x cot x sin2 ax cos2 ax tan2 ax a −1 Arctan (a −1x ) + c (1/2) a −1 ln |(x + a)−1 (x − a)| + c (1/2) a −1 ln |(a − x )−1 (a + x )| + c (2a 2x + 2a )−1 x + (1/2) a −3 Arctan (a −1x ) + c (−x ) (2a 2x − 2a 4)−1 − (1/4) a −3 ln |(x + a)−1 (x − a)| + c −x (2a − 2a 2x 2)−1 + (1/4) a −3 ln |(a − x )−1 (a + x )| + c (1/3) (x + a 2)3/2 + c (1/3) (x − a 2)3/2 + c (−1/3) (a − x 2)3/2 + c (x + a 2)1/2 + c (x − a 2)1/2 + c −(a − x 2)1/2 + c (1/2) ln |x + a 2| + c (1/2) ln |x − a 2| + c −(1/2) ln |a − x 2| + c (−2x − 2a 2)−1 + c −(1/2) (x − a 2)−1 + c (1/2) (a − x 2)−1 + c ex + c a −1 e ax + c b −1a e bx + c a −1x e ax − a −2 e ax + c b −1x e bx − 2b−2x e bx + 2b−3 e b x + c x ln |x | − x + c (n + 1)−1 x (n +1) ln |x | − (n + 1)−2 x (n +1) + c (1/2) ln2 |x | + c − cos x + c sin x + c ln |sec x | + c ln |tan (x /2)| + c ln |sec x + tan x | + c ln |sin x | + c −a −1 cos ax + c a −1 sin ax + c a −1 ln |sec ax | + c a −1 ln |tan (ax /2)| + c a −1 ln |tan (p /4 + ax /2)| + c a −1 ln |sin ax | + c (1/2) {x − [(1/2) sin (2x )]} + c (1/2) {x + [(1/2) sin (2x )]} + c tan x − x + c − cot x + c tan x + c − cot x − x + c (1/2) x − (1/4) a −1 (sin 2ax ) + c (1/2) x + (1/4) a −1 (sin 2ax ) + c a −1 tan ax − x + c Appendix G 781 csc ax sec ax cot ax x sin ax x cos ax x sin ax x cos ax (sin x cos x )−2 (sin x cos x )−1 sin x cos x sin2 x cos2 x (sin ax cos ax )−2 (sin ax cos ax )−1 sin ax cos ax sin2 ax cos ax sec x tan x Arcsin x Arccos x Arctan x −a −1 cot ax + c a −1 tan ax + c −a −1 cot ax − x + c a −2 sin ax − a −1x cos ax + c a −2 cos ax + a −1x sin ax + c 2a −2x sin ax + (2a −3 − a −1x 2) cos ax + c 2a −2x cos ax + (a −1x − 2a −3) sin ax + c −2 cot 2x + c ln |tan x | + c (1/2) sin2 x + c (1/8) x − (1/32) sin 4x + c −2a −1 cot 2ax + c a −1 ln |tan ax | + c (1/2) a−1 sin2 ax + c (1/8) x − (1/32) (a −1) sin 4ax + c sec x + c x Arcsin x + (1 − x 2)1/2 + c x Arccos x − (1 − x 2)1/2 + c x Arctan x − (1/2) ln |1 + x 2| + c This page intentionally left blank Suggested Additional Reading • • • • • • • • • Bachman, D Advanced Calculus Demystified New York: McGraw-Hill, 2007 Gibilisco, S Mastering Technical Mathematics, 3rd ed New York: McGraw-Hill, 2008 Gibilisco, S Technical Math Demystified New York: McGraw-Hill, 2006 Huettenmueller, R Algebra Demystified New York: McGraw-Hill, 2003 Huettenmueller, R College Algebra Demystified New York: McGraw-Hill, 2004 Huettenmueller, R Pre-Calculus Demystified New York: McGraw-Hill, 2005 Krantz, S Calculus Demystified New York: McGraw-Hill, 2003 Krantz, S Differential Equations Demystified New York: McGraw-Hill, 2005 Olive, J Maths: A Student’s Survival Guide, 2d ed Cambridge, England: Cambridge University Press, 2003 783 This page intentionally left blank Index A absolute maximum, 30, 138–139, 182 absolute minimum, 30, 138–139, 182 acceleration, 128–130, 178–179 antiderivative basic vs general, 222 concept of, 205–207 of basic linear function, 208, 330 of basic nth-degree function, 209–210, 330–331 of basic quadratic function, 208–209 of constant function, 207–208, 329–331 of zero function, 205–206, 329 antidifferentiation See antiderivative apex of paraboloid, 417 approximation error, 194 Arccosine function, derivative of, 384–387, 504 Arcsine function, derivative of, 381–384, 504 arc breaking into chords, 297–300 length, 297–310, 363–369 arc-in-a-box method, 303, 364–365 Arctangent function, 157–158 area approximating, 192–194 between line and curve, 262–267, 350–353 between two curves, 267–274, 353–355 defined by curve, 191–198, 328–329 negative, 191–192, 326–327 positive, 191–192, 326–327 argument of function, asymptotes of hyperbola, 393 average value of function, 200–201, 329 axis of paraboloid, 418 B basic vs general antiderivative, 222 bell-shaped curve, 201 bijection, definition of, 13 “broken” function, 12–14 C Cartesian space, 417–419, 471–472, 533 chain rule, 99–103, 173, 378, 395–396, 398 child function, 116 chord breaking arc into, 297–300 definition of, 297–298 circle equation of, 391, 505–506 graph of, 391 implicit differentiation of equation for, 394–396 unit, definition of, 19 coefficient leading, 32, 80, 143, 170 in polynomial, 32, 72, 80, 142 785 786 Index composite function, 99–100, 229 concavity of graph, 138–139, 182 cone in Cartesian xyz–space, 444–445, 523–524 right circular, 444–448, 523–523 slant height, 445, 523 slant-surface area integral, 446–448, 523 volume integral, 446–448, 524 constant function, 9, 40–41, 155, 207–208, 329–331 constant of integration, definition of, 211 constant, stand-alone, 32 constant term, 79 continuity of a function, 29–33 left-hand, 25–27, 159–161 at a point, 24–27 right-hand, 24–25, 159–161 “total,” at a point, 27 continuous function, 27–32 convergent integral, 237 coordinates rectangular, definition of, cosecant function,124 cosine function derivative of, 111–112 integral of, 277–282, 355–356 inverse of, 384–387 principal branch of, 385, 503–504 unit circle model, 109–110 cotangent function, 125 counterexample, in proof, 102 crest, in wave function, 109–110 cubic function continuity of, 31–32 derivative of basic, 48–52 general form of, 31 graph of, 144–146 curve, singular, 270–274 curves, area between, 267–274 cylinder in Cartesian xyz-space, 442–443, 522–523 lateral-surface area integral, 442–443, 522 right circular, 442–445, 522–523 volume integral, 444–445, 522–523 D definite integral, 195, 215–218 dependent variable, 3, 154–155 derivative of Arccosine function, 384–387, 504 of Arcsine function, 381–384, 504 basic concepts of, 35–54 of basic cubic function, 48–52 of basic linear function, 40–44 of basic quadratic function, 44–48 chain rule for finding, 99–103, 173, 378 of cosine function, 109–112 definition of, 52, 163 determining existence of, 55–69 fifth, 133 fourth, 133–135 higher, 126–137 implicit, 390–403, 506–511 of inverse function, 377–389 as limit, 38, 52 multiplication-by-constant rule, 84–87, 171 of natural exponential function, 114–118 of natural logarithm function, 118–124, 181 nth, 133, 179 partial, 416–427, 515–518 of polynomial function, 71–83 power rule, 71–75, 168–169 product rule, 87–90, 171–172 quotient rule, 95–99, 173 as rate of change, 37 real-power rule, 106–108, 173–174 reciprocal rule, 90–94, 172–173 second, 126–130, 178 of sine function, 109–111, 179–181 sum rule, 75–79, 169–170 third, 130–133 as tool for graph analysis, 138–153 two-way, 394–402 difference function, 266, 268–269, 272–273 rule for partial derivatives, 422, 516–517 rule for second partials, 430, 519 differentiable function, definition of, 59, 166 Index 787 differential definition of, 38, 163 in integral, 195, 212 differential equation elementary first-order ordinary, 490–493, 535–537 elementary second-order ordinary, 493–499, 537–540 differentiation See derivative direction numbers, 419 discontinuity, 28, 55–57, 156 discontinuous function, 32–33 displacement vs speed, 198–200 distance formula, 305–306 divergent integral, 237 domain of mapping, definition of, 3–4, 154 restricted, 33 double integral in two variables, 458–462, 528–535 E e, definition of, 114–115 ellipse equation of, 392–393, 506, 508 graph of, 392 implicit differentiation of equation for, 396–399 semi-axes of, 392 Euler’s constant, 114–115, 177 exponent negative, 107–108 real-number, 124, 210 exponential constant, 114–115, 177 exponential function, derivative of, 114–118 extrema of graph, 138–139 F family of functions, 490, 536–538 fifth derivative, 133 five-space, Cartesian, 419 fluxion, 37 four-space, Cartesian, 419 fourth derivative, 133–135 fractions, partial, in integral, 318–323 function(s) Arccosine, 384–387, 504 Arcsine, 381–384, 504 Arctangent, 157–158 average value of, 200–201, 329 “broken,” 14–17 child, 116 composite, 99–100, 229 constant, 9, 40–41, 155, 207–208, 329–331 continuity of, 29–33 continuous, 29–32 cosine, 109–112, 277–282 cosecant, 124 cotangent, 125 cubic, 31–32, 48–52 definition of, 5, 154 difference, 266, 268–269, 272–273 differentiable, definition of, 59, 166 discontinuous, 32–33 family of, 490, 536–538 inverse of, 91, 155, 377–378, 502–505 limit of, 21–22 linear, basic, 9–11, 29–30, 40–44, 208, 330 multiplied by constant, 84–85 natural exponential, 114–118, 282–288 nondifferentiable, 63–69, 164–165 nonlinear, 12–14 nth-degree polynomial, 79–82, 209–210 of another function, 99–100, 122 parent, 116 partial, 420 quadratic, basic, 30–31, 44–48, 208–209 quotient, 95 polynomial, 71–83 reciprocal, 15–16, 90–91, 289–295 secant, 125 sine, 108–111, 277–282 singular, 119 step, 17–18 tangent, 16–17, 123 zero, 41, 205–206, 329 zeros of, 142 Fundamental Theorem of Calculus, 215, 332 788 Index G general vs basic antiderivative, 222 geodesic, 450 geometric volume vs mathematical volume, 458, 528, 535 gradient definition of, of line, graph absolute maximum of, 138–139 absolute minimum of, 138–139 analysis with derivatives, 138–153 concavity of, 138–139, 182 of cubic function, 144–146 extrema of, 138–139 inflection point of, 139–141 local maximum of, 138–139 local minimum of, 138–139 of quadratic function, 141–144 of sine function, 147–151 graphs, area between, 262–276, 350–355 great circle, 450 H higher derivatives, 126–137 horizontal-line test, 8, 155–156 hyperbola asymptotes of, 393 equation of, 392–393, 506, 509–510 graph of, 393 implicit differentiation of equation for, 399–402 semi-axes of, 393 unit, 393 hypotenuse, 299 I implicit differentiation concept and process, 390–403, 506–511 of equation for circle, 394–396 of equation for ellipse, 396–399 of equation for hyperbola, 399–402 improper integral, 234–249, 337–342 indefinite integral, 211–215 independent variable, 3, 154–155 indeterminate expression, 404 induction, mathematical, 77–78, 170 infinite sequence, limit of, 20–21 infinitesimal calculus, 46 inflection point, 139–141, 182–183 integral convergent, 237 of cosine function, 277–282, 355–356 definite, 195, 215–218 of difference function, 268–269, 272–273 differential in, 195, 212 divergent, 237 double, in two variables, 458–462, 528–525 improper, 234–249, 337–342 indefinite, 211–215 with infinite interval, 244–248 iterated, in two variables, 462–466 multiple definite, 455–456 multiplication–by–constant rule, 213, 250–251, 331–332, 342–343 of natural exponential function, 282–288 negation rule, 213, 250–251 by partial fractions, 318–323 by parts, 313–317, 370–372 principle of linearity for, 311–313, 369–370 of polynomial function, 250–261 of reciprocal function, 289–295 repeated, in one variable, 455–458, 526–528 reversal rule, 221–224, 332–333 Riemann, 194 of sine function, 277–282, 355–356 with singularity in interval, 238–244 split–interval rule, 224–229, 334 substitution rule, 229–232, 334–337 sum rule, 213, 250–251, 331–332, 342–343 surface–area, 442–454 volume, 442–454, 458–466, 468–489 integration See integral inverse of function, 91, 155, 377–378, 502–505 function, derivative of, 377–389 multiplicative, 91 of relation, 6–8 iterated integral in two variables, 462–466 intercept definition of, 10 variable bounds in, 234–238 Index 789 JK jerk, 132 L law of the mean, 300–301, 363–364 leading coefficient, 32, 80, 143, 170 legitimate function, Leibniz, Gottfried Wilhelm, 20 l’Hôpital principle for expressions that tend toward 0/0, 404–408, 511–513 for expressions that tend toward ±∞ / ±∞, 408–411, 512–515 for expressions that tend toward · (+∞), 411–412, 513 for expressions that tend toward · (−∞), 411–412, 514 for expressions that tend toward +∞ − (+∞), 412–413, 513 for expressions that tend toward +∞ · 0, 413, 513 for expressions that tend toward −∞ · 0, 413, 513 for expressions that tend toward −∞ + (+∞), 413, 513–514 for expressions that tend toward +∞ + (−∞), 413, 513–514 for expressions that tend toward (−∞) − (−∞), 413, 513–514 limit definition of, 20 of function, 21–22 of indeterminate expression, 404–415 of infinite sequence, 20–21 left-hand, 25–26 l’Hôpital principles for finding, 404–415 multiplication-by-constant rule for, 23 right-hand, 24 sum rule for, 22–23 linear equations, simultaneous, 322 linear function antiderivative of basic, 208, 330 continuity of, 29–30 definition of, derivative of basic, 40–44 graph of, 10–11 standard form for, 10 linearity, principle for integral, 311–313, 369–370 local maximum, 31, 109–110, 138–139, 182 minimum, 31, 109–110, 138–139, 182 logarithm base in, 118 function, derivative of, 118–124 function, higher derivatives of, 136 natural, 118–124 M many-to-one, mapping, 3–8 mathematical induction, 77–78, 170 mathematical volume vs “real-world” volume, 458, 528, 535 maximum absolute, 30, 138–139, 182 local, 31, 109–110, 138–139, 182 mean in normal distribution, 201–202 law of, 300–301, 363–364 minimum absolute, 30, 138–139, 182 local, 31, 109–110, 138–139, 182 mixed partial derivative for three-variable function, 438–440, 520–521 for two-variable function, 431–434, 519–521 monomial power function, 80 multiple definite integrals, 455–456 multiplication-by-constant rule for derivatives, 84–87, 171 for integrals, 213, 250–251, 331–332, 342–343 for limits, 24 for partial derivatives, 422, 516 for second partials, 430, 518 multiplicative inverse, 91 N natural exponential, derivative of, 114–118 natural exponential, integral of, 282–288 natural logarithm, derivative of, 118–124, 181 natural logarithm, higher derivatives of, 136 negation rule for integrals, 213, 250–251 negative area, 191–192, 326–327 790 Index negative-integer power, 107–108 nested functions, 122 Newton, Isaac, 20 nondifferentiable function characteristics of, 164–165 examples of, 63–69 nonlinear function definition of, 12 graph of, 12–14 normal distribution, 201–202 nth-degree basic function, function, 209–210, 330–331 polynomial function, 79–82, 209–210 nth derivative, 133, 179 O one-to-many, ordered pair, 3–4 ordered quadruple, 419 ordered triple, 416 ordinary differential equation elementary first-order, 490–493, 535–537 elementary second-order, 493–499, 537–540 P parabola, 30–31, 141–144 paraboloid, 417–418 parent function, 116 partial derivative, 416–441, 515–518 partial fractions, in integral, 318–323 partial functions, 420 partial sum, in series, 33 parts, integration by, 313–317, 370–372 phase, in wave, 112 polynomial function derivative of, 71–83 nth-degree, 79–82 positive area, 191–192, 326–327 power function, monomial, 80 negative-integer, 107–108 rational-number, 107–108, 174 real-number, 124, 174, 210 reciprocal, 107–108 rule for derivatives, 71–75, 168–169 principal branch of cosine function, 385, 503–504 of sine function, 382, 503 principle of linearity, 311–313 product of functions, 87–88 of function and constant, 84–85 rule for derivatives, 87–90, 171–172 Pythagorean formula, 299–300 Pythagorean theorem, 450 Q quadratic function antiderivative of basic, 208–209 continuity of, 30–31 derivative of basic, 44–48 general form of, 30 graph of, 141–144 quotient of functions, 95 rule for derivatives, 95–99, 173 R radian, 108 range of mapping, definition of, 3–4, 154 rational-number power, 107–108, 174 real-number power, 124, 174, 210 real-power rule for derivatives, 106–108, 173–174 “real-world” volume vs mathematical volume, 458, 528, 535 reciprocal function, 15–16, 289–295 power, 107–108 rule for derivatives, 90–94, 172–173 rectangular coordinates, definition of, relation definition of, 4, 154 inverse of, 6–8 two-way, 390–394 repeated integral in one variable, 455–458, 526–528 restricted domain, 33 reversal rule for integrals, 221–224, 332–333 Riemann, Bernhard, 194, 326 Index 791 right circular cone in Cartesian xyz-space, 444–445, 523–524 slant height, 445, 523 slant-surface area integral, 446–448, 523 volume integral, 446–448, 524 right circular cylinder in Cartesian xyz-space, 442–443, 522–523 lateral-surface area integral, 442–443, 522 volume integral, 444–445, 522–523 rise over run, 10 S second derivative, 126–130, 178 second partial derivative for three-variable function, 434–438 for two-variable function, 428–431, 518–521 semi-axes of ellipse, 392 of hyperbola, 393 series, 189–191 simultaneous linear equations, 322 sine function derivative of, 109–111, 179–181 graph of, 147–151 higher derivatives of, 135, 179–181 integral of, 277–282, 355–356 inverse of, 113, 381–384 principal branch of, 382, 503 unit circle model, 108–110 singular curve, 270–274 singular function, 119, 270–274 singularity definition of, 116 in function, 116, 119 in interval for integral, 238–244 slope definition of, of line, between two points, 35–38 speed, 128–130, 178–179, 198–200 sphere in Cartesian xyz-space, 448–449, 525–526 geodesic on, 450 great circle on, 450 surface-area integral, 449–450, 452–453, 525–526 volume integral, 451–453, 525–526 split-interval rule for integrals, 224–229, 334 stand-alone constant, 32, 79 standard deviation, 201–202 step function, 17–18 substitution rule for integrals, 229–232, 334–337 sum of functions, 75–76 partial, in series, 33 sum rule for derivatives, 75–79, 169–170 for integrals, 213, 250–251, 331–332, 342–343 for limits, 22–23 for partial derivatives, 422, 516 for second partials, 430, 518–519 summation notation, 189–191, 325–326 superscript −1, meanings of, 378, 502 surface-area integral, 442–454 T tangent function, 16–17, 123 line, definition of, 35–36, 162 plane, 419 third derivative, 130–133 three-space, Cartesian, 416, 471–472, 533 “times sign,” alternatives, 112–113 topographical map, 417 topography, 416–417 “total” continuity at a point, 27 trough, in wave function, 109–110 true function, two-way derivative, 394–402 two-way relation, 390–394 U unit circle definition of, 19 equation of, 391 implicit differentiation of equation for, 394–396 model for sine and cosine, 108–110 unit hyperbola, 393 V variable, dependent, 3, 154–155 variable, independent, 3, 154–155 792 Index vertex of paraboloid, 417 vertical-line test, 8, 155, 157, 418–419, 515–516 volume integral, 442–454, 458–466, 468–489 mathematical vs “real–world,” 458, 484, 528 wave (Cont.) phase, 112 sine, 109–111 XYZ W wave cosine, 111–112 function, crest in, 109–110 function, trough in, 109–110 xyz-space, structure of 471–472, 533 y-intercept, definition of, 10 zero function antiderivative of, 205–206, 329 definition of, 41 zeros of function, 142 .. .Calculus Know-It-ALL This page intentionally left blank Calculus Know-It-ALL Beginner to Advanced, and Everything in Between Stan... intentionally left blank Calculus Know-It-ALL This page intentionally left blank PART Differentiation in One Variable This page intentionally left blank CHAPTER Single-Variable Functions Calculus is the... school year But don’t hurry When you’ve finished this book, I recommend Calculus Demystified by Steven G Krantz and Advanced Calculus Demystified by David Bachman for further study If Chap 29 of