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Calculus This is the free digital calculus text by David R Guichard and others It was submitted to the Free Digital Textbook Initiative in California and will remain unchanged for at least two years The book is in use at Whitman College and is occasionally updated to correct errors and add new material The latest versions may be found by going to http://www.whitman.edu/mathematics/california_calculus/ This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike License To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/ or send a letter to Creative Commons, 543 Howard Street, 5th Floor, San Francisco, California, 94105, USA If you distribute this work or a derivative, include the history of the document This text was initially written by David Guichard The single variable material (not including infinite series) was originally a modification and expansion of notes written by Neal Koblitz at the University of Washington, who generously gave permission to use, modify, and distribute his work New material has been added, and old material has been modified, so some portions now bear little resemblance to the original The book includes some exercises from Elementary Calculus: An Approach Using Infinitesimals, by H Jerome Keisler, available at http://www.math.wisc.edu/~keisler/calc.html under a Creative Commons license Albert Schueller, Barry Balof, and Mike Wills have contributed additional material This copy of the text was produced at 16:02 on 5/31/2009 I will be glad to receive corrections and suggestions for improvement at guichard@whitman.edu Contents Analytic Geometry 1.1 1.2 1.3 1.4 Lines Distance Between Two Points; Circles Functions Shifts and Dilations 14 Instantaneous Rate Of Change: The Derivative 2.1 2.2 2.3 2.4 2.5 The slope of a function An example Limits The Derivative Function Adjectives For Functions 19 19 24 26 35 40 v vi Contents Rules For Finding Derivatives 3.1 3.2 3.3 3.4 3.5 The Power Rule Linearity of the Derivative The Product Rule The Quotient Rule The Chain Rule 45 45 48 50 53 56 Transcendental Functions 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 63 Trigonometric Functions The Derivative of sin x A hard limit The Derivative of sin x, continued Derivatives of the Trigonometric Functions Exponential and Logarithmic functions Derivatives of the exponential and logarithmic functions Limits revisited Implicit Differentiation Inverse Trigonometric Functions 63 66 67 70 71 72 75 80 84 89 Curve Sketching 5.1 5.2 5.3 5.4 5.5 Maxima and Minima The first derivative test The second derivative test Concavity and inflection points Asymptotes and Other Things to Look For 93 93 97 99 100 102 Contents vii Applications of the Derivative 6.1 6.2 6.3 6.4 6.5 Optimization Related Rates Newton’s Method Linear Approximations The Mean Value Theorem 105 105 118 127 131 133 Integration 7.1 7.2 7.3 139 Two examples The Fundamental Theorem of Calculus Some Properties of Integrals 139 143 150 Techniques of Integration 8.1 8.2 8.3 8.4 8.5 8.6 Substitution Powers of sine and cosine Trigonometric Substitutions Integration by Parts Rational Functions Additional exercises 155 156 160 162 166 170 176 viii Contents Applications of Integration 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11 177 Area between curves Distance, Velocity, Acceleration Volume Average value of a function Work Center of Mass Kinetic energy; improper integrals Probability Arc Length Surface Area Differential equations 177 182 185 192 195 200 205 210 220 222 227 10 Sequences and Series 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11 10.12 Sequences Series The Integral Test Alternating Series Comparison Tests Absolute Convergence The Ratio and Root Tests Power Series Calculus with Power Series Taylor Series Taylor’s Theorem Additional exercises 233 234 240 244 249 251 254 256 259 261 263 267 271 Contents ix A Introduction to Maple A.1 A.2 Getting Started Algebra A.2.1 Numbers A.2.2 Variables and Expressions A.2.3 Evaluation and Substitution A.2.4 Solving Equations A.3 Plotting A.4 Calculus A.4.1 Limits A.4.2 Differentiation A.4.3 Implicit Differentiation A.4.4 Integration A.5 Adding text to a Maple session A.6 Printing A.7 Saving your work A.8 Getting Help 275 276 277 279 280 284 284 285 285 275 276 282 284 286 286 286 287 B Selected Answers 289 Index 303 292 Appendix B Selected Answers 3.5.25 3(8x − 2)/(4x2 − 2x + 1)2 sin x √ √ + x cos x x cos x 4.4.3 − sin x (2x + 1) sin x − (x2 + x) cos x 4.4.4 sin2 x − sin x cos x 4.4.5 − sin2 x 4.4.2 3.5.26 −3x2 + 5x − 3.5.27 120x4 − 576x3 + 888x2 − 480x + 96 3.5.28 −2/(x − 1)2 3.5.29 4x/(x2 + 1)2 3.5.30 (x2 − 6x + 7)/(x − 3)2 3.5.31 −5/(3x − 4)2 4.5.1 cos2 x − sin2 x 3.5.32 60x4 + 72x3 + 18x2 + 18x − 4.5.2 − sin x cos(cos x) 3.5.33 (5 − 4x)/((2x + 1) (x − 3) ) 4.5.3 tan x + x sec2 x √ x tan x 3.5.35 56x6 + 72x5 + 110x4 + 100x3 + 60x2 + 28x + 4.5.4 sec2 x(1 + sin x) − tan x cos x (1 + sin x)2 3.5.36 y = 23x/96 − 29/96 4.5.5 − csc2 x 3.5.37 y = − 2x/3 4.5.6 − csc x cot x 3.5.38 y = 13x/2 − 23/2 4.5.7 3x2 sin(23x2 ) + 46x4 cos(23x2 ) 3.5.39 y = 2x − 11 √ √ 20 + 5 3.5.40 y = √ x+ √ 4+ 5 4+ 4.5.8 3.5.34 1/(2(2 + 3x)2 ) 4.5.9 −6 cos(cos(6x)) sin(6x) 4.5.10 sin θ/(cos θ + 1)2 4.1.1 2nπ − π/2, any integer n 4.5.11 5t4 cos(6t) − 6t5 sin(6t) 4.1.2 nπ ± π/6, any integer n √ √ 4.1.3 ( + 6)/2 √ √ 4.1.4 −(1 + 3)/(1 − 3) 4.5.12 3t2 (sin(3t) + t cos(3t))/ cos(2t) + 2t3 sin(3t) sin(2t)/ cos2 (2t) 4.1.11 t = π/2 4.3.1 4.3.2 7/2 4.3.3 3/4 4.3.4 √ 4.3.5 − 2/2 4.3.6 4.3.7 √ √ √ 4.4.1 sin( x) cos( x)/ x 4.5.13 nπ/2, any integer n 4.5.14 π/2 + nπ, any integer n √ √ 4.5.15 3x/2 + 3/4 − 3π/6 √ √ 4.5.16 3x + − 3π/3 √ √ 4.5.17 3x/2 − 3π/4 4.5.18 π/6 + 2nπ, 5π/6 + 2nπ, any integer n 4.7.1 ln(3)x3x cos x − sin x 4.7.2 ex 2x 4.7.3 2e 293 Appendix B Selected Answers 4.7.4 ex cos(ex ) 4.7.5 cos(x)esin x 4.7.6 xsin x cos x ln x + sin x x 4.7.7 3x2 ex + x3 ex √ √ √ √ 4.9.13 ( 3, 3), (− 3, −2 3), √ √ √ √ (2 3, 3), (−2 3, − 3) √ √ 4.9.14 y = 7x/ − 8/ 1/3 1/3 1/3 1/3 4.9.15 y = (−y1 x+y1 x1 +x1 y1 )/x1 4.9.16 (y − y1 )/(x − x1 ) = (2x31 + 2x1 y12 − x1 )/(2y13 + 2y1 x21 + y1 ) x 4.7.8 + ln(2) 4.7.9 −2x ln(3)(1/3)x 4.7.10 e4x (4x − 1)/x2 4.7.11 (3x2 + 3)/(x3 + 3x) 4.7.12 − tan(x) 4.7.13 (1 − ln(x2 ))/(x2 ln(x2 )) 4.10.3 −1/(1 + x2 ) 2x 4.10.4 √ − x4 ex 4.10.5 + e2x 5.1.1 at x = 1/2 4.7.14 sec(x) 5.1.2 at x = −1, max at x = 4.7.15 xsin(x) (cos(x) ln(x) + sin(x)/x) 5.1.3 max at x = 2, at x = 4.7.20 e 5.1.4 at x = ±1, max at x = 4.8.1 5.1.5 at x = 4.8.2 ∞ 5.1.6 none 4.8.3 5.1.7 none 4.8.4 4.8.5 5.1.8 at x = 7π/12 + kπ, max at x = −π/12 + kπ, for integer k 4.8.7 y = and y = −1 5.1.9 none 4.9.1 x/y 5.1.10 local max at x = 4.9.2 −(2x + y)/(x + 2y) 5.1.11 local at x = 49 4.9.3 −(3x2 + y − 2xy)/(2xy − 3y − x2 ) 5.1.12 local at x = 4.9.4 sin(x) sin(y)/(cos(x) cos(y)) √ √ 4.9.5 − y/ x 5.1.15 one 5.2.1 at x = 1/2 4.9.6 (y sec2 (x/y)−y )/(x sec2 (x/y)+y ) 5.2.2 at x = −1, max at x = 4.9.7 (y − cos(x + y))/(cos(x + y) − x) 5.2.3 max at x = 2, at x = 4.9.8 −y /x2 5.2.4 at x = ±1, max at x = 4.9.9 5.2.5 at x = 4.9.11 y = 2x ± 5.2.6 none 4.9.12 y = x/2 ± 5.2.7 none 294 Appendix B Selected Answers 5.2.8 at x = 7π/12 + kπ, max at x = −π/12 + kπ, for integer k 5.2.9 none 5.2.10 max at x = 0, at x = ±11 5.2.11 at x = −3/2, neither at x = 5.2.13 at nπ, max at π/2 + nπ 5.2.14 at 2nπ, max at (2n + 1)π 5.2.15 at π/2+2nπ, max at 3π/2+2nπ 5.3.1 at x = 1/2 5.3.2 at x = −1, max at x = √ 5.4.4 concave up when x < −1/ or √ x > 1/ 3, concave down when √ √ −1/ < x < 1/ 5.4.5 concave up when x < or x > 2/3, concave down when < x < 2/3 5.4.6 concave up when x < 0, concave down when x > 5.4.7 concave up when x < −1 or x > 1, concave down when −1 < x < or 0 6.1.4 w = l = · 52/3 , h = 52/3 , h/w = 1/2 6.1.3 P/4 × P/4 Appendix B Selected Answers 6.1.5 √ 100 × √ √ 100 × 100, h/s = 6.1.6 w = l = 21/3 V 1/3 , h = V 1/3 /22/3 , h/w = 1/2 6.1.7 1250 square feet 6.1.8 l2 /8 square feet 6.1.9 $5000 6.1.10 100 6.1.11 r2 √ 6.1.30 The ratio of the volume of the sphere to the volume of the cone is √ 1033/4096 + 33/4096 17 ≈ 0.2854, so the cone occupies approximately 28.54% of the sphere √ √ 6.1.31 P should be at distance c a/( a + √ b) from charge A 6.1.32 1/2 6.1.12 h/r = 6.1.33 $7000 6.1.13 h/r = 6.1.14 r = 5, h = 40/π, h/r = 8/π 6.1.15 8/π 6.1.16 4/27 6.1.17 Go direct from A to D 6.1.18 (a) 2, (b) 7/2 √ √ √ 3 1 6.1.19 × + × − 6 12 √ 6.1.20 (a) a/6, (b) (a+b− a − ab + b2 )/6 6.1.21 1.5 meters wide by 1.25 meters tall 6.1.22 If k ≤ 2/π the ratio is (2 − kπ)/4; if k ≥ 2/π, the ratio is zero: the window should be semicircular with no rectangular part 6.1.23 a/b √ √ √ 6.1.24 w = 2r/ 3, h = 2r/ √ 6.1.25 1/ ≈ 58% 6.1.26 18 × 18 × 36 6.1.27 r = 5/(2π)1/3 ≈ 2.7 cm, h = · 25/3 /π 1/3 = 4r ≈ 10.8 cm 750 6.1.28 h = π 7502 2π 6.1.29 h/r = 295 1/6 2π 7502 6.1.34 There is a critical point when sin θ1 /v1 = sin θ2 /v2 , and the second derivative is positive, so there is a minimum at the critical point 6.2.1 1/(16π) cm/s 6.2.2 3/(1000π) meters/second 6.2.3 1/4 m/s 6.2.4 −6/25 m/s 6.2.5 80π mi/min √ 6.2.6 ft/s 6.2.7 20/(3π) cm/s 6.2.8 13/20 ft/s √ 6.2.9 10/2 m/s 6.2.10 75/64 m/min 6.2.11 145π/72 m/s 6.2.12 25π/144 m/min √ 6.2.13 π 2/36 ft3 /s 6.2.14 tip: ft/s, length: 5/2 ft/s 1/3 , r = 6.2.15 tip: 20/11 m/s, length: 9/11 m/s √ 6.2.16 380/ − 150 ≈ 69.4 mph 6.2.17 81 km/hr 296 Appendix B Selected Answers 6.2.18 18 m/s √ 6.2.19 136 475/19 ≈ 156 km/hr 6.2.20 −50 m/s 7.2.1 (16/3)x3/2 + C 7.2.2 t3 + t + C √ 7.2.3 x + C 7.2.4 −2/z + C 6.2.21 68 m/s √ 7.2.5 ln s + C 6.2.22 3800/ 329 ≈ 210 km/hr √ √ √ 6.2.23 820/ 329 + 150 57/ 47 ≈ 210 km/hr 7.2.6 (5x + 1) /15 + C 7.2.7 (x − 6)3 /3 + C 6.2.24 4000/49 m/s 7.2.8 2x5/2 /5 + C 6.2.25 (a) x = a cos θ − a sin θ cot(θ + β) = √ a sin β/ sin(θ + β), (c) x˙ ≈ 3.79 cm/s 7.2.9 −4/ x + C 7.2.10 4t − t2 + C, t < 2; t2 − 4t + + C, 6.3.1 x3 = 1.475773162 t≥2 6.3.2 2.15 7.2.11 87/2 6.3.3 3.36 7.2.12 6.3.4 2.19 or 1.26 7.2.13 ln(10) 6.4.1 ∆y = 65/16, dy = 7.2.14 e5 − 6.4.2 ∆y = 11/10 − 1, dy = 0.05 7.2.15 34 /4 6.4.4 dV = 32π/25 7.2.16 26 /6 − 1/6 6.5.1 c = 1/2 7.2.17 x2 − 3x √ 6.5.2 c = 18 − 7.2.18 2x(x4 − 3x2 ) 6.5.6 x3 /3 + 47x2 /2 − 5x + k 7.2.19 ex 6.5.7 arctan x + k 6.5.8 x4 /4 − ln x + k 6.5.9 − cos(2x)/2 + k 7.1.1 10 7.1.2 35/3 7.1.3 x2 7.1.4 2x2 7.1.5 2x2 − 7.1.6 2b2 − 2a2 7.1.7 rectangles: 41/4 = 10.25, rectangles: 183/16 = 11.4375 7.1.8 23/4 7.2.20 2xex 7.3.1 It rises until t = 100/49, then falls The position of the object at time t is s(t) = −4.9t2 + 20t + k The net distance traveled is −45/2, that is, it ends up 45/2 meters below where it started The total distance traveled is 6205/98 meters 2π 7.3.2 sin t dt = 0 √ 7.3.3 net: 2π, total: 2π/3 + 7.3.4 7.3.5 17/3 Appendix B Selected Answers 297 7.3.6 A = 18, B = 44/3, C = 10/3 8.3.1 − ln | csc x + cot x| + C 8.1.1 −(1 − t)10 /10 + C 8.1.5 (sin4 x)/4 + C 8.3.2 − csc x cot x/2 − (1/2) ln | csc x + cot x| + C √ √ 8.3.3 x x2 − 1/2 − ln |x + x2 − 1|/2 + C √ 8.3.4 x + 4x2 /2 + √ (9/4) ln |2x + + 4x2 | + C 8.1.6 −(100 − x2 )3/2 /3 + C 8.3.5 −(1 − x2 )3/2 /3 + C 8.1.2 x5 /5 + 2x3 /3 + x + C 8.1.3 (x2 + 1)101 /202 + C 8.1.4 −3(1 − 5t)2/3 /10 + C − x3 /3 + C 8.1.7 −2 8.1.8 sin(sin πt)/π + C 8.1.9 1/(2 cos2 x) = (1/2) sec2 x + C 8.1.10 − ln | cos x| + C 8.3.9 − arctan x − 1/x + C √ 8.3.10 arcsin(x/2) − x − x2 /2 + C √ √ √ 8.3.11 arcsin( x) − x − x + C √ 8.3.12 (2x2 + 1) 4x2 − 1/24 + C 8.1.11 8.1.12 tan2 (x)/2 + C 8.1.13 1/4 8.1.14 − cos(tan x) + C 8.4.1 cos x + x sin x + C 8.1.15 1/10 √ 8.1.16 3/4 8.4.2 x2 sin x − sin x + 2x cos x + C 8.1.17 (27/8)(x2 − 7)8/9 8.1.18 −(3 + 1)/14 8.4.3 (x − 1)ex + C 8.4.4 (1/2)ex + C 8.4.5 (x/2) − sin(2x)/4 + C = (x/2) − (sin x cos x)/2 + C 8.1.19 8.1.20 f (x)2 /2 8.2.1 x/2 − sin(2x)/4 + C 8.2.2 − cos x + (cos x)/3 + C 8.2.3 3x/8 − (sin 2x)/4 + (sin 4x)/32 + C 8.3.6 arcsin(x)/8 − sin(4 arcsin x)/32 + C √ 8.3.7 ln |x + + x2 | + C √ 8.3.8 (x + 1) x2 + 2x/2 − √ ln |x + + x2 + 2x|/2 + C 8.2.4 (cos x)/5 − (cos x)/3 + C 8.2.5 sin x − (sin3 x)/3 + C 8.4.6 x ln x − x + C 8.4.7 (x2 arctan x + arctan x − x)/2 + C 8.4.8 −x2 cos x + 2x sin x + cos x + C 8.4.9 x2 /4−(cos2 x)/4−(x sin x cos x)/2+ C 8.2.8 −2(cos x)5/2 /5 + C 8.4.10 x/4 − (x cos2 x)/2 + (cos x sin x)/4 + C √ √ √ 8.4.11 x arctan( x)+arctan( x)− x+C √ √ √ 8.4.12 sin( x) − x cos( x) + C 8.2.9 tan x − cot x + C 8.4.13 sec x csc x − cot x + C 8.2.6 x/8 − (sin 4x)/32 + C 8.2.7 (sin3 x)/3 − (sin5 x)/5 + C 8.2.10 (sec x)/3 − sec x + C 8.5.1 − ln |x − 2|/4 + ln |x + 2|/4 + C 298 Appendix B Selected Answers 8.5.2 −x3 /3 − 4x − ln |x − 2|+ ln |x + 2| + C 8.5.3 −1/(x + 5) + C 8.5.4 −x − ln |x − 2| + ln |x + 2| + C 8.5.5 −4x + x3 /3 + arctan(x/2) + C 8.5.6 (1/2) arctan(x/2 + 5/2) + C 8.5.7 x2 /2 − ln(4 + x2 ) + C 8.5.8 (1/4) ln |x + 3| − (1/4) ln |x + 7| + C 8.5.9 (1/5) ln |2x − 3| − (1/5) ln |1 + x| + C 8.5.10 (1/3) ln |x| − (1/3) ln |x + 3| + C (t + 4)4 +C 8.6.1 (t2 − 9)5/2 8.6.2 +C 8.6.3 8.6.4 8.6.5 8.6.6 8.6.7 8.6.8 8.6.9 8.6.10 8.6.11 8.6.12 8.6.13 (et + 16)2 +C cos t − cos3 t + C tan t +C ln |t2 + t + 3| + C ln |1 − 4/t2 | + C tan(arcsin(t/5)) + C = 25 t √ +C 25 25 − t2 2√ sin 3t + C t tan t + ln | cos t| + C √ et + + C 3t sin 2t sin 4t + + +C 32 ln |t| ln |t + 3| − +C 3 −1 +C =− sin arctan t −1 8.6.15 +C 2(1 + tan t)2 8.6.14 8.6.16 8.6.17 8.6.18 8.6.19 8.6.20 8.6.21 8.6.22 8.6.23 8.6.24 8.6.25 8.6.26 8.6.27 8.6.28 9.1.1 + t2 /t + C (t2 + 1)5/2 (t2 + 1)3/2 − +C et sin t − et cos t +C (t3/2 + 47)4 +C − +C 3/2 3(2 − t ) (2 − t2 )1/2 ln | sin(arctan(2t/3))| +C = (ln(4t2 ) − ln(9 + 4t2 ))/18 + C (arctan(2t))2 +C ln |t + 3| ln |t − 1| + +C 4 cos7 t cos5 t − +C −1 +C t−3 −1 +C ln t t2 (ln t)2 t2 ln t t2 − + +C 2 t (t − 3t + 6t − 6)e + C √ √ 5+ ln(2t + − 5) + 10√ √ 5− ln(2t + + 5) + C 10 √ 2/15 9.1.2 1/12 9.1.3 9/2 9.1.4 4/3 9.1.5 2/3 − 2/π √ 9.1.6 3/π − 3/(2π) − 1/8 Appendix B Selected Answers 9.1.7 1/3 √ 9.1.8 10 5/3 − 9.4.2 4/3 9.1.9 500/3 9.4.4 π/4 9.4.3 1/A 9.1.11 1/5 9.4.5 −1/3, √ √ 9.4.6 −4 1224 ft/s; −8 1224 ft/s 9.1.12 1/6 9.5.1 ≈ 5, 305, 028, 517 N-m 9.1.10 9.2.1 1/π, 5/π 9.5.2 ≈ 4, 457, 854, 041 N-m 9.2.2 0, 245 9.5.3 367, 500π N-m 9.2.3 20, 28 9.5.4 49000π + 196000/3 √ 9.2.4 (3 − π)/(2π), (18 − 12 + π)/(4π) 9.2.5 10/49 meters, 20/49 seconds 9.2.6 45/98 meters, 30/49 seconds 9.2.7 25000/49 meters, 1000/49 seconds 9.2.8 s(t) = cos t, v(t) = − sin t, maximum distance is 1, maximum speed is 9.2.9 s(t) = − sin(πt)/π + t/π, v(t) = − cos(πt)/π + 1/π, maximum speed is 2/π 9.2.10 s(t) = t2 /2 − sin(πt)/π + t/π, v(t) = t − cos(πt)/π + 1/π 2 9.2.11 s(t) = t /2 + sin(πt)/π − t/π, v(t) = t + cos(πt)/π − 1/π 9.5.5 2450π N-m 9.5.6 0.05 N-m 9.5.7 6/5 N-m 9.5.8 3920 N-m 9.5.9 23520 N-m 9.6.1 15/2 9.6.2 9.6.3 16/5 9.6.5 x ¯ = 45/28, y¯ = 93/70 9.6.6 x ¯ = 0, y¯ = 4/(3π) 9.6.7 x ¯ = 1/2, y¯ = 2/5 9.6.8 x ¯ = 0, y¯ = 8/5 9.6.9 x ¯ = 4/7, y¯ = 2/5 9.6.10 x ¯ = y¯ = 1/5 9.3.5 8π/3 9.6.11 x ¯ = 0, y¯ = 28/(9π) 9.3.6 π/30 9.6.12 x ¯ = y¯ = 28/(9π) 9.3.7 π(π/2 − 1) 9.7.1 ∞ 9.3.8 (a) 114π/5 (b) 74π/5 (c) 20π (d) 4π 9.7.2 1/2 9.3.9 16π, 24π 9.7.4 diverges 9.7.3 diverges 9.3.11 πh (3r − h)/3 9.7.5 9.3.13 2π 9.7.6 diverges 9.4.1 2/π; 2/π; 9.7.7 299 300 Appendix B Selected Answers 9.7.8 diverges 10.3.4 converges 9.7.9 π/6 10.3.5 converges 9.7.10 diverges, 10.3.6 converges 9.7.11 diverges, 10.3.7 diverges 9.7.12 diverges, no CPV 10.3.8 converges 9.7.13 π 10.3.9 N = 9.7.14 80 mph: 90.8 to 95.3 N 90 mph: 114.9 to 120.6 N 100.9 mph: 144.5 to 151.6 N 9.8.13 through 16 √ 9.9.1 (22 22 − 8)/27 9.9.2 ln(2) + 3/8 9.9.3 a + a3 /3 √ √ 9.9.4 ln(( + 1)/ 3) 10.1.1 10.3.10 N = 10 10.3.11 N = 1687 10.3.12 any integer greater than e200 10.4.1 converges 10.4.2 converges 10.4.3 diverges 10.4.4 converges 10.4.5 0.90 10.4.6 0.95 10.1.3 10.5.1 converges 10.1.4 10.5.2 converges 10.1.5 10.5.3 converges 10.1.6 10.2.1 lim n2 /(2n2 + 1) = 1/2 n→∞ 10.2.2 lim 5/(21/n + 14) = 1/3 n→∞ ∞ 10.2.3 If converges so does n n=1 ∞ = n=1 n ∞ , but the latter n n=1 in fact diverges 10.5.4 diverges 10.5.5 diverges 10.5.6 diverges 10.5.7 converges 10.5.8 diverges 10.5.9 converges 10.5.10 diverges 10.6.1 converges absolutely 10.2.4 −3/2 10.6.2 diverges 10.2.5 11 10.6.3 converges conditionally 10.3.1 diverges 10.6.4 converges absolutely 10.3.2 diverges 10.6.5 converges conditionally 10.3.3 converges 10.6.6 converges absolutely Appendix B Selected Answers 301 ∞ 10.6.7 diverges (−1)n (n + 1)(x − 1)n , R = 10.10.6 n=0 10.6.8 converges conditionally 10.7.5 converges ∞ 10.10.7 + 10.7.6 converges ∞ 10.7.7 converges 1+ 10.7.8 diverges n=1 10.8.2 R = ∞, I = (−∞, ∞) 10.8.3 R = e, I = (−e, e) ∞ n=0 ∞ 10.8.5 R = 0, converges only when x = 10.10.10 10.8.6 R = 1, I = (−6, −4) 10.9.1 the alternating harmonic series ∞ (n + 1)xn n=0 ∞ (n + 1)(n + 2)xn (n + 1)(n + 2) n 10.9.4 x ,R=1 n=0 ∞ −1 10.9.5 C + xn+2 (n + 1)(n + 2) n=0 n 2n (−1) x /(2n)!, R = ∞ n=0 ∞ (−1)n n=0 ∞ 10.10.4 x2 x4 x6 x12 + − + ··· + 24 720 12! 10.11.2 1000; 10.11.1 − x3 2x5 + , error ±4.26 15 10.12.1 diverges n=1 (x − 1)n ,R=1 n ∞ 10.10.5 ln(2) + (−1)n−1 n=1 10.12.3 converges 10.12.4 diverges 10.12.5 diverges 10.12.7 converges 10.12.9 converges (x − 5)n ,R=5 5n+1 (−1)n−1 10.12.2 converges 10.12.8 converges xn /n!, R = ∞ n=0 ∞ 10.10.3 n=0 10.12.6 diverges ∞ 10.10.2 (−1)n xn+1 /n! 10.11.3 x + n=0 ∞ 10.10.1 (−1)n x4n+1 /(2n)! 10.10.9 10.8.4 R = e, I = (2 − e, + e) 10.9.3 (2n − 1)! xn , R = − 1)! n! 22n−1 (n 10.10.8 x + x3 /3 10.8.1 R = 1, I = (−1, 1) 10.9.2 · · · · · (2n − 1) n x = n n!2 n=1 10.12.10 converges 10.12.11 converges 10.12.12 converges 10.12.13 converges 10.12.14 converges (x − 2)n ,R=2 n n2 10.12.15 converges 302 Appendix B Selected Answers 10.12.16 converges 10.12.17 diverges 10.12.18 (−∞, ∞) 10.12.19 (−3, 3) 10.12.20 (−3, 3) 10.12.21 (−1, 1) 10.12.22 radius is 0—it converges only when x=0 √ √ 10.12.23 (− 3, 3) 10.12.24 (−∞, ∞) ∞ 10.12.25 (ln(2))n n x n! n=0 ∞ 10.12.26 (−1)n n+1 x n + n=0 ∞ 10.12.27 x2n+1 2n + n=0 10.12.28 + x/2 + ∞ (−1)n+1 n=2 · · · · · (2n − 3) n x 2n n! ∞ (−1)n x2n 10.12.29 n=0 ∞ 10.12.30 (−1)n 2n+1 x 2n + n=0 ∞ 10.12.31 π = (−1)n n=0 2n + Index A absolute extremum, 105 algebraic precedence, xii alternating harmonic series, 249 antiderivative, 145 arccosine, 90 arcsine, 89 asymptote, 10, 102 B bell curve, 214 bounded function, 40 C Cauchy Principal Value, 207 center of mass, 202 centroid, 203 chain rule, 56 chord, 20 circle area, xiii circumference, xiii equation of, xiii, 7, unit, completing the square, xii composition of functions, 13 concave down, 101 concave up, 100 cone lateral area, xiii surface area, xiii volume, xiii continuous, 42, 51 convergent sequence, 235 convergent series, 241 cosines law of, xiii critical value, 94 cumulative distribution function, 213 cylinder lateral area, xiii surface area, xiii volume, xiii D definite integral, 149 dependent variable, 10 derivative, 22 dot notation, 119 Leibniz notation, 36 difference quotient, 20 differentiable, 42 differential, 132 differential equation, 227 first order, 228 separable, 228 303 304 Index discrete probability, 210 divergence test, 243 divergent sequence, 235 divergent series, 241 domain, dot notation, 119 E ellipse equation of, xiii ellipsoid, 227 escape velocity, 208 even function, 215 expected value, 210 exponential distribution, 214 exponential function, 72 Extreme Value Theorem, 107 F Fermat’s Theorem, 94 floor, 51 frustum, 223 function, bounded, 40 differentiable, 42 even, 215 implicit, 84 linear, rational, 170 unbounded, 41 function composition, 13 Fundamental Theorem of Algebra, 97 Fundamental Theorem of Calculus, 143 G Gabriel’s horn, 209, 227 geometric series, 241 global extremum, 105 greatest integer, 51, 97 H harmonic series, 244 alternating, 249 Hooke’s Law, 198 I implicit differentiation, 84 implicit function, 84 improper integral, 206 convergent, 206 diverges, 206 indefinite integral, 149 independent variable, 10 inflection point, 101 initial value problem first order, 228 integral improper, 206 of sec x, 164 of sec3 x, 165 properties of, 153 integral sign, 145 integral test, 247 integration by parts, 167 Intermediate Value Theorem, 42 interval of convergence, 260 inverse function, 72 inverse sine, 89 K kinetic energy, 208 L L’Hˆ opital’s Rule, 81 lateral area, 117 law of cosines, xiii law of sines, xiii Leibniz notation, 36 limit, 29 limit at infinity, 81 limit of a sequence, 235 linear approximation, 131 linearity of the derivative, 48 local extremum, 93 local maximum, 93 local minimum, 93 logarithm, 72 logarithmic function, 72 logistic equation, 231 M Maclaurin series, 264 mean, 215 Mean Value Theorem, 134 Index median, 214 mode, 215 moment, 201 quadratic formula, xii quotient rule, 53 convergent, 235 decreasing, 238 divergent, 235 increasing, 238 monotonic, 238 non-decreasing, 238 non-increasing, 238 of partial sums, 241 series, 234 p-series, 247 absolute convergence, 255 alternating harmonic, 249 conditional convergence, 255 convergent, 241 divergent, 241 geometric, 241 harmonic, 244 integral test, 247 interval of convergence, 260 Maclaurin, 264 radius of convergence, 260 Taylor, 266 sines law of, xiii sphere surface area, xiii volume, xiii squeeze theorem, 67 standard deviation, 216 standard normal distribution, 214 standard normal probability density function, 214 subtend, 64 sum rule, 49 R T radian measure, 63 radius of convergence, 260 random variable, 211 rational function, 56, 170 related rates, 118 Rolle’s Theorem, 134 tangent line, 20 Taylor series, 266 Toricelli’s trumpet, 209, 227 torque, 200 torus, 227 transcendental function, 63 triangle inequality, 31 trigonometric identities, xiii N Newton, 197 Newton’s law of cooling, 227 normal distribution, 219 O one sided limit, 34 optimization, 105 P p-series, 247 physicists, 119 point-slope formula, xiii power function, 45 power rule, 45, 79, 87 precedence of algebraic operations, xii probability density function, 211 probability distribution, 213 product rule, 50, 51 generalized, 53 properties of integrals, 153 Q S separation of variables, 229 sequence, 234 bounded, 239 bounded above, 239 bounded below, 239 305 U unbounded function, 41 uniform distribution, 213 uniform probability density function, 213 unit circle, 306 Index W witch of Agnesi, 56 work, 195