Calculus- Early Transcendentals, 2021a

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492 Multiple Integration Exercise 14.1.6 Compute ∫ 1 ∫ x 2 ∫ √ π/2 Exercise 14.1.7 Compute Exercise 14.1.8 Compute Exercise 14.1.9 Compute: Exercise 14.1.10 Compute: Exercise 14.1.11 Compute: Exercise 14.1.12 Compute: Exercise 14.1.13 Compute: Exercise 14.1.14 Compute Exercise 14.1.15 Compute 0 0 0 y e x dydx. ∫ x 2 0 ∫ π/2 ∫ cosθ 0 0 ∫ 1 ∫ 1 0 ∫ 1 ∫ 1 0 ∫ 1 ∫ 1 0 ∫ 1 ∫ y 0 xcosydydx. r 2 (cosθ − r)drdθ. √ y √x 3 + 1dxdy. y 2 ysin(x2 )dxdy. x 2 x√ 1 + y 2 dydx. 0 ∫ 1 ∫ 3 0 3y ∫ 1 ∫ 1−x 2 −1 0 2 √ 1 − x 2 dxdy. e x2 dxdy. ∫ √ 2/2 ∫ √ 1−2x 2 0 x 2 − √ ydydx. − √ 1−2x 2 xdydx. ∫∫ Exercise 14.1.16 Evaluate x 2 dA over the region in the first quadrant bounded by the hyperbola xy = 16 and the lines y = x, y = 0, and x = 8. Exercise 14.1.17 Find the volume below z = 1 − y above the region −1 ≤ x ≤ 1, 0 ≤ y ≤ 1 − x 2 . Exercise 14.1.18 Find the volume bounded by z = x 2 + y 2 and z = 4. Exercise 14.1.19 Find the volume in the first octant bounded by y 2 = 4 − x and y = 2z. Exercise 14.1.20 Find the volume in the first octant bounded by y 2 = 4x, 2x + y = 4, z= y, and y = 0. Exercise 14.1.21 Find the volume in the first octant bounded by x+y+z = 9, 2x+3y = 18, and x+3y = 9. Exercise 14.1.22 Find the volume in the first octant bounded by x 2 + y 2 = a 2 and z = x + y.
14.1. Volume and Average Height 493 Exercise 14.1.23 Find the volume bounded by 4x 2 + y 2 = 4z and z = 2. Exercise 14.1.24 Find the volume bounded by z = x 2 + y 2 and z = y. Exercise 14.1.25 Find the volume under the surface z = xy above the triangle with vertices (1,1,0), (4,1,0), (1,2,0). Exercise 14.1.26 Find the volume enclosed by y = x 2 ,y= 4, z= x 2 ,z= 0. Exercise 14.1.27 A swimming pool is circular with a 40 meter diameter. The depth is constant along east-west lines and increases linearly from 2 meters at the south end to 7 meters at the north end. Find the volume of the pool. Exercise 14.1.28 Find the average value of f (x,y)=e y√ x + e y on the rectangle with vertices (0,0), (4,0), (4,1) and (0,1). Exercise 14.1.29 Figure 14.5 shows a temperature map of Colorado. Use the data to estimate the average temperature in the state using 4, 16 and 25 subdivisions. Give both an upper and lower estimate. Why do we like Colorado for this problem? What other state(s) might we like? Figure 14.5: Colorado temperatures. Exercise 14.1.30 Three cylinders of radius 1 intersect at right angles at the origin, as shown in Figure 14.6. Find the volume contained inside all three cylinders.
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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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Page 461 and 462: 12.6. Other Coordinate Systems 445
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Page 465: 12.6. Other Coordinate Systems 449
Page 468 and 469: 452 Partial Differentiation the lin
Page 470 and 471: 454 Partial Differentiation Exercis
Page 472 and 473: 456 Partial Differentiation Fortuna
Page 476 and 477: 460 Partial Differentiation paralle
Page 478 and 479: 462 Partial Differentiation 3 z . 2
Page 482 and 483: 466 Partial Differentiation lim ε
Page 484 and 485: 468 Partial Differentiation 13.5 Di
Page 486 and 487: 470 Partial Differentiation Example
Page 492 and 493: 476 Partial Differentiation so we g
Page 494 and 495: 478 Partial Differentiation 0.0 0.2
Page 496 and 497: 480 Partial Differentiation and the
Page 498 and 499: 482 Partial Differentiation xz = 2y
Page 500 and 501: 484 Partial Differentiation x φ .
Page 502 and 503: 486 Multiple Integration point mult
Page 504 and 505: 488 Multiple Integration Figure 14.
Page 506 and 507: 490 Multiple Integration 1 0 0 1 Fi
Page 512 and 513: 496 Multiple Integration This examp
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Page 516 and 517: 500 Multiple Integration Finally, M
Page 520 and 521: 504 Multiple Integration Example 14
Page 522 and 523: 506 Multiple Integration ∫ 1 = 1
Page 524 and 525: 508 Multiple Integration Solution.
Page 526 and 527: 510 Multiple Integration ∫ ∫
Page 528 and 529: .. 512 Multiple Integration As befo
Page 530 and 531: 514 Multiple Integration y . ......
Page 532 and 533: 516 Multiple Integration ∫∫ Exe
Page 534 and 535: 518 Vector Functions separate funct
Page 536 and 537: 520 Vector Functions = 〈 f ′ (t
Page 538 and 539: 522 Vector Functions Figure 15.6:
Page 540 and 541: 524 Vector Functions 2.0 1.5 1.0 0.
Page 542 and 543: 526 Vector Functions Exercise 15.2.
Page 544 and 545: 528 Vector Functions (This integral
Page 546 and 547: 530 Vector Functions To remove the
Page 548 and 549: 532 Vector Functions Graphing this
Page 550 and 551: 534 Vector Functions Example 15.20
Page 552 and 553: 536 Vector Calculus which is the re
Page 554 and 555: 538 Vector Calculus Since f x = 2e
Page 556 and 557: 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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