Calculus- Early Transcendentals, 2021a

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122 Derivatives we often use f and f (x) to mean the original function, we sometimes use df/dx and df(x)/dx to refer to the derivative. If the function f (x) is written out in full we often write the last of these something like this f ′ (x)= d dx √ 625 − x 2 with the function written to the side, instead of trying to fit it into the numerator. Example 4.3: Derivative of y = t 2 Find the derivative of y = f (t)=t 2 . Solution. We compute y ′ Δy = lim Δt→0 Δt (t + Δt) 2 −t 2 = lim Δt→0 Δt t 2 + 2tΔt + Δt 2 −t 2 = lim Δt→0 Δt 2tΔt + Δt 2 = lim Δt→0 Δt = lim 2t + Δt = 2t. Δt→0 ♣ Remember that Δt is a single quantity, not a “Δ” times a “t”, and so Δt 2 is (Δt) 2 not Δ(t 2 ). Doing the same example using the second formula for the derivative with h in place of Δt gives the following. Note that we compute f (t + h) by substituting t + h in place of t everywhere we see t in the expression f (t), while making no other changes (at least initially). For example, if f (t) =t + √ (t + 3) 2 −t then f (t + h)=(t + h)+ √ ((t + h)+3) 2 − (t + h)=t + h + √ (t + h + 3) 2 −t − h. Example 4.4: Derivative of y = t 2 Find the derivative of y = f (t)=t 2 .
4.2. The Derivative Function 123 Solution. We compute f ′ (t) = lim h→0 f (t + h) − f (t) h = lim h→0 (t + h) 2 −t 2 h = lim h→0 t 2 + 2th+ h 2 −t 2 h = lim h→0 2th+ h 2 h = lim h→0 2t + h = 2t. ♣ Example 4.5: Derivative Find the derivative of y = f (x)=1/x. Solution. The computation: y ′ = Δy lim Δx→0 Δx = 1 x+Δx lim x Δx→0 Δx = x x(x+Δx) lim x(x+Δx) Δx→0 Δx = lim Δx→0 = lim Δx→0 = lim Δx→0 = lim Δx→0 x−(x+Δx) x(x+Δx) Δx x − x − Δx x(x + Δx)Δx −Δx x(x + Δx)Δx −1 x(x + Δx) = −1 x 2 Note: If you happen to know some “derivative formulas” from an earlier course, for the time being you should pretend that you do not know them. In examples like the ones above and the exercises below, ♣
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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
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
Page 95 and 96: 3.2. Precise Definition of a Limit
Page 97 and 98: 3.3. Computing Limits: Graphically
Page 99 and 100: 3.4. Computing Limits: Algebraicall
Page 101 and 102: 3.4. Computing Limits: Algebraicall
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 144 and 145: 128 Derivatives 4.2.3 Velocities Su
Page 146 and 147: 130 Derivatives Exercise 4.2.5 Find
Page 148 and 149: 132 Derivatives Proof. For convenie
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Page 154 and 155: 138 Derivatives In practice, of cou
Page 156 and 157: 140 Derivatives Exercises for Secti
Page 158 and 159: 142 Derivatives Exercise 4.5.39 Fin
Page 160 and 161: 144 Derivatives . . . . Figure 4.4:
Page 162 and 163: 146 Derivatives = ( d dx x2 ln2) e
Page 164 and 165: 148 Derivatives Example 4.38: Deriv
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Page 170 and 171: 154 Derivatives Exercise 4.7.8 Find
Page 172 and 173: 156 Derivatives 4.8.1 Derivatives o
Page 174 and 175: 158 Derivatives Exercise 4.8.4 The
Page 177 and 178: Chapter 5 Applications of Derivativ
Page 179 and 180: 5.1. Related Rates 163 Solution. He
Page 181 and 182: 5.1. Related Rates 165 Example 5.6:
Page 183 and 184: 5.2. Extrema of a Function 167 Exer
Page 185 and 186: 5.2. Extrema of a Function 169 poin
Page 187 and 188: 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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