Calculus- Early Transcendentals, 2021a

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598 Selected Exercise Answers 16.5.1 both are −45π/4 16.5.2 a 2 bc + ab 2 c + abc 2 16.5.3 e 2 − 2e + 7/2 16.5.4 3 16.5.5 384π/5 16.5.6 π/3 16.5.7 10π 16.5.8 π/2 16.6.2 25 √ 21/4 16.6.3 π √ 21 16.6.4 π(5 √ 5 − 1)/6 16.6.5 4π √ 2 16.6.6 πa 2 /2 16.6.7 2πa(a − √ a 2 − b 2 ) 16.6.8 π((1 + 4a 2 ) 3/2 − 1)/6 16.6.9 πa 2 − 2a 2 16.6.10 πa 2√ 1 + k 2 /4 16.6.11 A √ 1 + a 2 + b 2 16.6.12 A √ k 2 + 1 16.6.13 8a 2 16.7.1 (0,0,3/8) 16.7.2 (11/20,11/20,3/10) 16.7.3 (0,0,1364/425) 16.7.4 on center axis, h/3abovethebase 16.7.5 16 16.7.6 7 16.7.7 −π 16.7.8 −2/e 16.7.9 πb 2 (−4b 4 − 3b 2 + 6a 2 b 2 + 6a 2 )/6 16.7.10 9280 kg/s 16.7.11 24ε 0 16.8.1 −3π 16.8.2 0 16.8.3 −4π 16.8.4 3π 16.8.5 A(p(c − b)+q(a − c)+a − b)
Index absolute extrema, 172, 206 absolute maximum, 172 absolute minimum, 172 multivariate, 477 absolute value, 14 properties, 14 acceleration vector, 533 normal component, 533 tangential component, 533 antiderivative, 233 arc length parameterization, 528 asymptote horizontal, 202 vertical, 202 Cartesian coordinates, 401, 444 centerofmass,499 central angle, 33 centroid, 501 chain rule, 137 multivariate, 465 characteristic polynomial, 390 Clairaut’s theorem, 473 comparison theorem, 357 completing the square, 6 conic section, 415 conics circle, 26 ellipse, 26 horizontal hyperbola, 27 horizontal parabola, 25 standard form, 25 vertical hyperbola, 27 vertical parabola, 25 critical point, 169 cross product, 434 properties, 436 curl, 537 curvature, 530, 531 cylindrical coordinates, 445 convert to rectangular, 446 triple integral, 507 definite integral, 236 properties, 238 del (∇), 468 derivative, 117, 121, 130 acceleration, 128 chain rule, 137 differentiable, 125 exponential function, 143, 145 implicit differentiation, 147 intervals of increase and decrease, 197 leibniz notation, 121 logarithmic differentiation, 151 logarithmic function, 145 nth derivative, 127 of the inverse, 154 rate of change, 161 rules, 130 second derivative, 127 trigonometric functions, 136 velocity, 119, 128 difference quotient, 115 differentiable, 463 differential equation, 246, 375 Euler’s method, 386 first order, 375 general solution, 378 initial value problem, 246, 376, 381, 383 integrating factor, 384 method of undetermined coefficients, 393 particular solution, 378 second order, 389 second order linear, 393 separable, 377 separation of variables, 378 variation of parameters, 383 differentials, 184 direction vector, 440 directional derivative, 469 discriminant, 474 distance formula, 24, 420 divergence, 537 theorem, 552 599
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with Open Texts Calculus Early Tran
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Calculus: Early Transcendentals an
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Calculus: Early Transcendentals an
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Table of Contents Table of Contents
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v 5.4.3 Taylor Polynomials . . . .
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vii 13.2 Limits and Continuity . .
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Introduction The emphasis in this c
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4 Review In the table, the set of r
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6 Review 1.1.3 The Quadratic Formul
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8 Review 1.1.4 Inequalities, Interv
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10 Review Inequality Rules Add/subt
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12 Review Guidelines for Solving Ra
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14 Review Looking where the
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16 Review Example 1.17: Absolute Va
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18 Review (a) |x|≥2 (b) |x − 3|
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20 Review The most familiar form of
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22 Review Example 1.25: Equa
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24 Review |Δy|, as shown in figure
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26 Review • (h,k) is the vert
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28 Review Determining the Type of C
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30 Review To determine a, we substi
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32 Review (a) A(2,0),B(4,3) (b) A(
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34 Review • Secant (abbreviated b
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36 Review Reading from the unit cir
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38 Review Notice that we can now fi
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40 Review 1.3.4 Graphs of Trigonome
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42 Review Exercises for 1.3 Exercis
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44 Review Exercise 1.4.7 Simplify t
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46 Functions Example 2.2: Domain of
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48 Functions Example 2.5: Domain Fi
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50 Functions For horizontal and ver
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52 Functions Solution. The domain o
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54 Functions After the positive int
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56 Functions Example 2.11: Determin
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58 Functions Example 2.16: Finding
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60 Functions Since the function f (
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62 Functions Example 2.21: Solve Lo
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64 Functions We can do something si
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66 Functions Cancellation Rules sin
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68 Functions (a) sin −1 ( √ 3/2
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70 Functions Example 2.32: Computin
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72 Functions (a) f (x) (b) − f (x
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Chapter 3 Limits 3.1 The Limit The
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3.2. Precise Definition of a Limit
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3.2. Precise Definition of a Limit
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3.3. Computing Limits: Graphically
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3.4. Computing Limits: Algebraicall
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3.4. Computing Limits: Algebraicall
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3.5. Infinite Limits and Limits at
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5 + x −1 (n) lim x→∞ 1 + 2x
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3.6. A Trigonometric Limit 99 Figur
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3.6. A Trigonometric Limit 101 Solu
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3.7. Continuity 103 § § ¥
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3.7. Continuity 105 Definition 3.43
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3.7. Continuity 107 Definition 3.45
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3.7. Continuity 109 Solution. We wi
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3.7. Continuity 111 Therefore, our
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3.7. Continuity 113 Exercise 3.7.3
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116 Derivatives doesn’t meet the
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118 Derivatives of the slope of the
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120 Derivatives algebra to find a s
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122 Derivatives we often use f and
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124 Derivatives you are required to
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126 Derivatives We can summarize
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128 Derivatives 4.2.3 Velocities Su
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130 Derivatives Exercise 4.2.5 Find
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132 Derivatives Proof. For convenie
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134 Derivatives Exercises for Secti
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136 Derivatives since sinx cosx −
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138 Derivatives In practice, of cou
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140 Derivatives Exercises for Secti
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142 Derivatives Exercise 4.5.39 Fin
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144 Derivatives . . . . Figure 4.4:
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146 Derivatives = ( d dx x2 ln2) e
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148 Derivatives Example 4.38: Deriv
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150 Derivatives Example 4.42: Equat
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152 Derivatives Solution. We take l
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154 Derivatives Exercise 4.7.8 Find
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156 Derivatives 4.8.1 Derivatives o
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158 Derivatives Exercise 4.8.4 The
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Chapter 5 Applications of Derivativ
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5.1. Related Rates 163 Solution. He
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5.1. Related Rates 165 Example 5.6:
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5.2. Extrema of a Function 167 Exer
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5.2. Extrema of a Function 169 poin
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5.2. Extrema of a Function 171 Exer
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5.2. Extrema of a Function 173 3. E
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5.2. Extrema of a Function 175 A si
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5.3. The Mean Value Theorem 177 2.
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5.3. The Mean Value Theorem 179
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5.3. The Mean Value Theorem 181 Exe
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5.4. Linear and Higher Order Approx
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5.5. L’Hôpital’s Rule 191 "bou
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5.5. L’Hôpital’s Rule 193 Exam
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5.5. L’Hôpital’s Rule 195 Exer
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5.6. Curve Sketching 197 consistent
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5.6. Curve Sketching 199 Solution.
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5.6. Curve Sketching 201 the concav
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5.6. Curve Sketching 203 If there a
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5.6. Curve Sketching 205 Note that
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5.7. Optimization Problems 207 Guid
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5.7. Optimization Problems 209 is t
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5.7. Optimization Problems 211 (Alt
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5.7. Optimization Problems 213 Exer
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Chapter 6 Integration 6.1 Displacem
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6.1. Displacement and Area 217 Exam
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6.1. Displacement and Area 219 draw
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6.1. Displacement and Area 221 It i
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6.1. Displacement and Area 223
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6.1. Displacement and Area 225 We d
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6.1. Displacement and Area 227 x i
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6.1. Displacement and Area 229 Both
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6.1. Displacement and Area 231 does
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6.2. The Fundamental Theorem of Cal
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6.3. Indefinite Integrals 243 Exerc
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6.3. Indefinite Integrals 245 Solut
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6.3. Indefinite Integrals 247 Exerc
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250 Techniques of Integration This
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252 Techniques of Integration ♣ E
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254 Techniques of Integration u, th
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256 Techniques of Integration 7.2 P
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258 Techniques of Integration = u7
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260 Techniques of Integration and
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262 Techniques of Integration ∫ N
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264 Techniques of Integration To in
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266 Techniques of Integration Exerc
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268 Techniques of Integration Expre
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270 Techniques of Integration The t
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272 Techniques of Integration Examp
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276 Techniques of Integration = sec
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278 Techniques of Integration Find
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280 Techniques of Integration Solut
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282 Techniques of Integration So al
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292 Techniques of Integration There
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294 Techniques of Integration Now u
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296 Techniques of Integration The f
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298 Techniques of Integration (e)
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302 Applications of Integration For
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304 Applications of Integration Sup
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306 Applications of Integration Gui
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308 Applications of Integration The
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310 Applications of Integration . y
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312 Applications of Integration . F
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314 Applications of Integration so
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316 Applications of Integration Exe
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318 Applications of Integration In
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320 Applications of Integration Sol
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322 Applications of Integration mag
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324 Applications of Integration . 1
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326 Applications of Integration and
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328 Applications of Integration Exe
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330 Applications of Integration Unf
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332 Applications of Integration Fig
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334 Applications of Integration is
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Chapter 9 Sequences and Series Cons
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9.1. Sequences 339 log 2 (1 + ε) >
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9.1. Sequences 341 Example 9.8: Con
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9.1. Sequences 343 below it is boun
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9.2. Series 345 If {kx n } ∞ n=0
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9.2. Series 347 for some number s.
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9.3. The Integral Test 349 If all o
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9.3. The Integral Test 351 Proof. W
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9.4. Alternating Series 353 Exercis
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9.5. Comparison Tests 355 Exercises
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9.5. Comparison Tests 357 Solution.
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9.7. The Ratio and Root Tests 359 E
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9.7. The Ratio and Root Tests 361 P
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9.8. Power Series 363 Definition 9.
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Exercise 9.8.2 Find the radius of c
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f ′′ (x)= f ′′′ (x)= ∞
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9.10. Taylor Series 369 Solution. T
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9.11. Taylor’s Theorem 371 a posi
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9.11. Taylor’s Theorem 373 Note t
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Chapter 10 Differential Equations M
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10.1. First Order Differential Equa
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10.1. First Order Differential Equa
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10.2. First Order Homogeneous Linea
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10.3. First Order Linear Equations
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10.4. Approximation 385 Exercises f
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10.4. Approximation 387 y .. 0.5 .
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10.5. Second Order Homogeneous Equa
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10.5. Second Order Homogeneous Equa
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10.6. Second Order Linear Equations
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Chapter 11 Polar Coordinates, Param
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11.1. Polar Coordinates 403 Solutio
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11.2. Slopes in Polar Coordinates 4
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11.3. Areas in Polar Coordinates 40
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11.3. Areas in Polar Coordinates 40
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11.4. Parametric Equations 411 Figu
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11.5 Calculus with Parametric Equat
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11.6. Conics in Polar Coordinates 4
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11.6. Conics in Polar Coordinates 4
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Chapter 12 Three Dimensions 12.1 Th
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12.1. The Coordinate System 421 Now
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12.2. Vectors 423 12.2 Vectors A ve
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12.2. Vectors 425 3.54 0.77 . . . .
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12.3. The Dot Product 427 Exercise
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12.3. The Dot Product 429 Solution.
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12.3. The Dot Product 431 v .. v ..
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12.4. The Cross Product 433 Exercis
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12.4. The Cross Product 435 We know
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12.5. Lines and Planes 437 Exercise
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12.5. Lines and Planes 439 Solution
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12.5. Lines and Planes 441 Example
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12.5. Lines and Planes 443 Exercise
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12.6. Other Coordinate Systems 445
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452 Partial Differentiation the lin
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454 Partial Differentiation Exercis
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456 Partial Differentiation Fortuna
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460 Partial Differentiation paralle
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462 Partial Differentiation 3 z . 2
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466 Partial Differentiation lim ε
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468 Partial Differentiation 13.5 Di
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470 Partial Differentiation Example
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476 Partial Differentiation so we g
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478 Partial Differentiation 0.0 0.2
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480 Partial Differentiation and the
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482 Partial Differentiation xz = 2y
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484 Partial Differentiation x φ .
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486 Multiple Integration point mult
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488 Multiple Integration Figure 14.
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490 Multiple Integration 1 0 0 1 Fi
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492 Multiple Integration Exercise 1
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496 Multiple Integration This examp
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498 Multiple Integration Exercise 1
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500 Multiple Integration Finally, M
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504 Multiple Integration Example 14
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506 Multiple Integration ∫ 1 = 1
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508 Multiple Integration Solution.
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510 Multiple Integration ∫ ∫
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.. 512 Multiple Integration As befo
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514 Multiple Integration y . ......
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516 Multiple Integration ∫∫ Exe
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518 Vector Functions separate funct
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520 Vector Functions = 〈 f ′ (t
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522 Vector Functions Figure 15.6:
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524 Vector Functions 2.0 1.5 1.0 0.
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526 Vector Functions Exercise 15.2.
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528 Vector Functions (This integral
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530 Vector Functions To remove the
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532 Vector Functions Graphing this
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534 Vector Functions Example 15.20
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536 Vector Calculus which is the re
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538 Vector Calculus Since f x = 2e
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540 Vector Calculus Definition 16.4
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542 Vector Calculus Example 16.8: W
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544 Vector Calculus Proof. We write
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546 Vector Calculus Exercise 16.3.2
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Page 576 and 577: 560 Vector Calculus Exercise 16.6.7
Page 578 and 579: 562 Vector Calculus we might want t
Page 580 and 581: 564 Vector Calculus where e is an e
Page 582 and 583: 566 Vector Calculus ∫ b = ∫ = a
Page 585 and 586: Selected Exercise Answers 1.1.1 1.
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Page 599 and 600: 583 7.7.20 diverges 7.7.21 diverges
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Page 603 and 604: 587 9.8.2 R = e 10.1.2 y = arctant
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Page 611 and 612: 595 14.3.6 ¯x = 6/5, ȳ = 12/5 14.
Page 613: 597 15.5.8 〈−3sint,2cost + 1/10
Page 617 and 618: INDEX 601 at infinity, 87 indetermi
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Page 623 and 624: Integrals Involving u 2 − a 2 , a
Page 625 and 626: ∫ du 108. u √ a + bu = √ 1
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