Calculus- Early Transcendentals, 2021a

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82 Limits 10 8 6 4 2 K4 K2 0 2 4 6 K2 x (a) lim x→4 f (x) f (x) x→0 + (f) f (−2) f (x) (d) lim x→0 − (h) (e) lim (b) lim x→−3 (c) lim f (x) x→0 (g) lim x→2 − f (x) lim x→−2 − f (x) (i) lim f (x + 1) x→0 (j) f (0) (k) (l) lim f (x − 4) x→1− lim f (x − 2) x→0 + 3.4 Computing Limits: Algebraically Properties of limits We begin by deriving a handful of theorems to give us the tools to compute many limits without explicitly working with the precise definition of a limit.
3.4. Computing Limits: Algebraically 83 Theorem 3.8: Limit Properties Suppose that lim x→a f (x)=L and lim x→a g(x)=M, andk is some constant. Then • lim x→a kf(x)=k lim x→a f (x)=kL • lim x→a ( f (x)+g(x)) = lim x→a f (x)+lim x→a g(x)=L + M • lim x→a ( f (x) − g(x)) = lim x→a f (x) − lim x→a g(x)=L − M • lim x→a ( f (x)g(x)) = lim x→a f (x) · lim x→a g(x)=LM f (x) lim f (x) • lim x→a g(x) = x→a lim g(x) = L , if M is not 0 M x→a Roughly speaking, these rules say that to compute the limit of an algebraic expression, it is enough to compute the limits of the “innermost bits” and then combine these limits. This often means that it is possible to simply plug in a value for the variable, since lim x→a x = a. Example 3.9: Limit Properties x 2 − 3x + 5 Compute lim . x→1 x − 2 Solution. If we apply the theorem in all its gory detail, we get x 2 − 3x + 5 lim x→1 x − 2 = = = lim x→1 (x2 − 3x + 5) lim x→1 (x − 2) (lim x 2 ) − (lim 3x)+(lim x→1 x→1 (lim x) − (lim 2) x→1 x→1 (lim x→1 x) 2 − 3(lim x→1 x)+5 (lim x→1 x) − 2 = 12 − 3 · 1 + 5 1 − 2 = 1 − 3 + 5 = −3 −1 x→1 5) It is worth commenting on the trivial limit lim x→1 5. From one point of view this might seem meaningless, as the number 5 can’t “approach” any value, since it is simply a fixed number. However, 5 can, and ♣
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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
Page 48 and 49: 32 Review (a) A(2,0),B(4,3) (b) A(
Page 50 and 51: 34 Review • Secant (abbreviated b
Page 52 and 53: 36 Review Reading from the unit cir
Page 54 and 55: 38 Review Notice that we can now fi
Page 56 and 57: 40 Review 1.3.4 Graphs of Trigonome
Page 58 and 59: 42 Review Exercises for 1.3 Exercis
Page 60 and 61: 44 Review Exercise 1.4.7 Simplify t
Page 62 and 63: 46 Functions Example 2.2: Domain of
Page 64 and 65: 48 Functions Example 2.5: Domain Fi
Page 66 and 67: 50 Functions For horizontal and ver
Page 68 and 69: 52 Functions Solution. The domain o
Page 70 and 71: 54 Functions After the positive int
Page 72 and 73: 56 Functions Example 2.11: Determin
Page 74 and 75: 58 Functions Example 2.16: Finding
Page 76 and 77: 60 Functions Since the function f (
Page 78 and 79: 62 Functions Example 2.21: Solve Lo
Page 80 and 81: 64 Functions We can do something si
Page 82 and 83: 66 Functions Cancellation Rules sin
Page 84 and 85: 68 Functions (a) sin −1 ( √ 3/2
Page 86 and 87: 70 Functions Example 2.32: Computin
Page 88 and 89: 72 Functions (a) f (x) (b) − f (x
Page 91 and 92: Chapter 3 Limits 3.1 The Limit The
Page 93 and 94: 3.2. Precise Definition of a Limit
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Page 103 and 104: 3.5. Infinite Limits and Limits at
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Page 115 and 116: 3.6. A Trigonometric Limit 99 Figur
Page 117 and 118: 3.6. A Trigonometric Limit 101 Solu
Page 119 and 120: 3.7. Continuity 103 § § ¥
Page 121 and 122: 3.7. Continuity 105 Definition 3.43
Page 123 and 124: 3.7. Continuity 107 Definition 3.45
Page 125 and 126: 3.7. Continuity 109 Solution. We wi
Page 127 and 128: 3.7. Continuity 111 Therefore, our
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Page 132 and 133: 116 Derivatives doesn’t meet the
Page 134 and 135: 118 Derivatives of the slope of the
Page 136 and 137: 120 Derivatives algebra to find a s
Page 138 and 139: 122 Derivatives we often use f and
Page 140 and 141: 124 Derivatives you are required to
Page 142 and 143: 126 Derivatives We can summarize
Page 144 and 145: 128 Derivatives 4.2.3 Velocities Su
Page 146 and 147: 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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548 Vector Calculus In this case, n
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550 Vector Calculus We can now rewr
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552 Vector Calculus 16.5 The Diverg
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554 Vector Calculus The remaining f
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556 Vector Calculus Suppose we inst
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558 Vector Calculus circle of radiu
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560 Vector Calculus Exercise 16.6.7
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562 Vector Calculus we might want t
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564 Vector Calculus where e is an e
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566 Vector Calculus ∫ b = ∫ = a
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Selected Exercise Answers 1.1.1 1.
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571 2.8.4 1. [2,3) ∪ (3,∞) 2. (
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573 4.5.18 −3(4 − x) 2 4.5.19 6
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575 5.1.15 500/ √ 3 − 200 ≈ 8
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577 5.6.21 min at x = 1 5.6.22 none
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579 7.1.2 x 5 /5 + 2x 3 /3 + x +C 7
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581 7.4.19 1/2e x2 +C 7.4.20 x 3 e
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583 7.7.20 diverges 7.7.21 diverges
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585 8.6.5 ¯x = 45/28, ȳ = 93/70 8
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587 9.8.2 R = e 10.1.2 y = arctant
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589 10.6.8 Ae t + Be 3t +(1/2)te 3t
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591 12.5.2 4(x + 1)+5(y − 2) −
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593 13.6.5 f x = 3cos(3x)cos(2y), f
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595 14.3.6 ¯x = 6/5, ȳ = 12/5 14.
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597 15.5.8 〈−3sint,2cost + 1/10
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Index absolute extrema, 172, 206 ab
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INDEX 601 at infinity, 87 indetermi
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Table of Integrals Table of Integra
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∫ 41. sin n ucos m udu= − sinn
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Integrals Involving u 2 − a 2 , a
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∫ du 108. u √ a + bu = √ 1
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