Calculus- Early Transcendentals, 2021a

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136 Derivatives since sinx cosx − 1 lim = 1and lim = 0. x→0 x x→0 x ♣ A formula for the derivative of the cosine function can be found in a similar fashion: d dx (cosx)=−sinx. Using the quotient rule we get formulas for the remaining trigonometric ratios. To summarize, here are the derivatives of the six trigonometric functions: d dx (sin(x)) = cos(x) d dx (tan(x)) = sec2 (x) d (csc(x)) = −csc(x)cot(x) dx d dx (cos(x)) = −sin(x) d dx (cot(x)) = −csc2 (x) d (sec(x)) = sec(x)tan(x) dx Example 4.29: Derivative of Product of Trigonometric Functions Find the derivative of f (x)=sinxtanx. Solution. Using the Product Rule we obtain f ′ (x)=cosxtanx + sinxsec 2 x. ♣ Exercises for Section 4.4 Exercise 4.4.1 Find the derivatives of the following functions. (a) sinxcosx (b) cotx (c) cscx − xtanx Exercise 4.4.2 Find the points on the curve y = x + 2cosx that have a horizontal tangent line. 4.5 The Chain Rule Let h(x) = √ 625 − x 2 . The rules stated previously do not allow us to find h ′ (x). However, h(x) is a composition of two functions. Let f (x)= √ x and g(x)=625 − x 2 .Thenweseethat h(x)=(f ◦ g)(x).
4.5. The Chain Rule 137 From our rules we know that f ′ (x)= 1 2 x−1/2 and g ′ (x)=−2x, thus it would be convenient to have a rule which allows us to differentiate f ◦ g in terms of f ′ and g ′ . This gives rise to the chain rule. The Chain Rule If g is differentiable at x and f is differentiable at g(x), then the composite function h = f ◦g [recall f ◦ g is defined as f (g(x))] is differentiable at x and h ′ (x) is given by: h ′ (x)= f ′ (g(x)) · g ′ (x). The chain rule has a particularly simple expression if we use the Leibniz notation for the derivative. The quantity f ′ (g(x)) is the derivative of f with x replaced by g; this can be written df/dg. As usual, g ′ (x)=dg/dx. Then the chain rule becomes df dx = df dg dg dx . This looks like trivial arithmetic, but it is not: dg/dx is not a fraction, that is, not literal division, but a single symbol that means g ′ (x). Nevertheless, it turns out that what looks like trivial arithmetic, and is therefore easy to remember, is really true. It will take a bit of practice to make the use of the chain rule come naturally—it is more complicated than the earlier differentiation rules we have seen. Example 4.30: Chain Rule Compute the derivative of √ 625 − x 2 . √ Solution. We already know that the answer is −x/ 625 − x 2 , computed directly from the limit. In the context of the chain rule, we have f (x) = √ x, g(x) =625 − x 2 . We know that f ′ (x) =(1/2)x −1/2 ,so f ′ (g(x)) = (1/2)(625−x 2 ) −1/2 . Note that this is a two step computation: first compute f ′ (x), then replace x by g(x). Sinceg ′ (x)=−2x we have f ′ (g(x))g ′ (x)= Example 4.31: Chain Rule √ Compute the derivative of 1/ 625 − x 2 . 1 2 √ 625 − x (−2x)= −x √ . 2 625 − x 2 ♣ Solution. This is a quotient with a constant numerator, so we could use the quotient rule, but it is simpler to use the chain rule. The function is (625−x 2 ) −1/2 , the composition of f (x)=x −1/2 and g(x)=625−x 2 . We compute f ′ (x)=(−1/2)x −3/2 using the power rule, and then f ′ (g(x))g ′ (x)= −1 2(625 − x 2 ) 3/2 (−2x)= x (625 − x 2 ) 3/2 .
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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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Page 115 and 116: 3.6. A Trigonometric Limit 99 Figur
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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 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
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
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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
Page 189 and 190: 5.2. Extrema of a Function 173 3. E
Page 191 and 192: 5.2. Extrema of a Function 175 A si
Page 193 and 194: 5.3. The Mean Value Theorem 177 2.
Page 195 and 196: 5.3. The Mean Value Theorem 179
Page 197 and 198: 5.3. The Mean Value Theorem 181 Exe
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5.4. Linear and Higher Order Approx
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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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