Calculus- Early Transcendentals, 2021a

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520 Vector Functions = 〈 f ′ (t),g ′ (t),h ′ (t)〉, Starting with a familiar expression for what appears to be a derivative, we find that we can make good computational sense out of it—but what does it actually mean? We know how to interpret r(t + Δt) and r(t)—they are vectors that point to locations in space; if t is time, we can think of these points as positions of a moving object at times that are Δt apart. We also know what Δr = r(t + Δt) − r(t) means—it is a vector that points from the head of r(t) to the head of r(t + Δt), assuming both have their tails at the origin. So when Δt is small, Δr is a tiny vector pointing from one point on the path of the object to a nearby point. As Δt gets close to 0, this vector points in a direction that is closer and closer to the direction in which the object is moving; geometrically, it approaches a vector tangent to the path of the object at a particular point. . . .. Δr .. r(t) r(t + Δt) Figure 15.3: Approximating the derivative. Unfortunately, the vector Δr approaches 0 in length; the vector 〈0,0,0〉 is not very informative. By dividing by Δt, when it is small, we effectively keep magnifying the length of Δr so that in the limit it doesn’t disappear. Thus the limiting vector 〈 f ′ (t),g ′ (t),h ′ (t)〉 will (usually) be a good, non-zero vector that is tangent to the curve. What about the length of this vector? It’s nice that we’ve kept it away from zero, but what does it measure, if anything? Consider the length of one of the vectors that approaches the tangent vector: ∣ r(t + Δt) − r(t) Δt ∣ = |r(t + Δt) − r(t)| |Δt| The numerator is the length of the vector that points from one position of the object to a “nearby” position; this length is approximately the distance travelled by the object between times t and t + Δt. Dividing this distance by the length of time it takes to travel that distance gives the average speed. As Δt approaches zero, this average speed approaches the actual, instantaneous speed of the object at time t. By performing an “obvious” calculation to get something that looks like the derivative of r(t),weget precisely what we would want from such a derivative: the vector r ′ (t) points in the direction of travel of the object and its length tells us the speed of travel. In the case that t is time, then, we call v(t)=r ′ (t) the velocity vector. Even if t is not time, r ′ (t) is useful—it is a vector tangent to the curve. Example 15.3 We have seen that r = 〈cost,sint,t〉 is a helix. Compute r ′ and |r ′ |. √ Solution. We compute r ′ = 〈−sint,cost,1〉, and|r ′ | = sin 2 t + cos 2 t + 1 = √ 2. Thinking of this as a description of a moving object, its speed is always √ 2; see Figure 15.4. ♣
15.2. <strong>Calculus</strong> with Vector Functions 521 Figure 15.4: A tangent vector on the helix. Example 15.4 The velocity vector for 〈cost,sint,cost〉 is 〈−sint,cost,−sint〉. As before, the first two coordinates mean that from above this curve looks like a circle. The z-coordinate is now also periodic, so that as the object moves around the curve its height oscillates up and down. In fact it turns out that the curve is a tilted ellipse, as shown in Figure 15.5. Figure 15.5: The ellipse r = 〈cost,sint,cost〉. Example 15.5 The velocity vector for 〈cost,sint,cos2t〉 is 〈−sint,cost,−2sin2t〉. Thez-coordinate is now oscillating twice as fast as in the previous example, so the graph is not surprising; see Figure 15.6.
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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
Page 486 and 487: 470 Partial Differentiation Example
Page 488 and 489: 472 Partial Differentiation Exercis
Page 490 and 491: 474 Partial Differentiation Exercis
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 508 and 509: 492 Multiple Integration Exercise 1
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 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
Page 558 and 559: 542 Vector Calculus Example 16.8: W
Page 560 and 561: 544 Vector Calculus Proof. We write
Page 562 and 563: 546 Vector Calculus Exercise 16.3.2
Page 564 and 565: 548 Vector Calculus In this case, n
Page 566 and 567: 550 Vector Calculus We can now rewr
Page 568 and 569: 552 Vector Calculus 16.5 The Diverg
Page 570 and 571: 554 Vector Calculus The remaining f
Page 572 and 573: 556 Vector Calculus Suppose we inst
Page 574 and 575: 558 Vector Calculus circle of radiu
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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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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