Calculus- Early Transcendentals, 2021a

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392 Differential Equations Exercise 10.5.2 Solve the initial value problem 2y ′′ + 18y = 0, y(0)=2,y ′ (0)=15. Exercise 10.5.3 Solve the initial value problem y ′′ + 6y ′ + 5y = 0, y(0)=1, y ′ (0)=0. Exercise 10.5.4 Solve the initial value problem y ′′ − y ′ − 12y = 0, y(0)=0, y ′ (0)=14. Exercise 10.5.5 Solve the initial value problem y ′′ + 12y ′ + 36y = 0, y(0)=5,y ′ (0)=−10. Exercise 10.5.6 Solve the initial value problem y ′′ − 8y ′ + 16y = 0, y(0)=−3, y ′ (0)=4. Exercise 10.5.7 Solve the initial value problem y ′′ + 5y = 0, y(0)=−2, y ′ (0)=5. Exercise 10.5.8 Solve the initial value problem y ′′ + y = 0,y(π/4)=0, y ′ (π/4)=2. Exercise 10.5.9 Solve the initial value problem y ′′ + 12y ′ + 37y = 0, y(0)=4,y ′ (0)=0. Exercise 10.5.10 Solve the initial value problem y ′′ + 6y ′ + 18y = 0, y(0)=0,y ′ (0)=6. Exercise 10.5.11 Solve the initial value problem y ′′ + 4y = 0, y(0)= √ 3,y ′ (0)=2. Exercise 10.5.12 Solve the initial value problem y ′′ + 100y = 0, y(0)=5,y ′ (0)=50. Exercise 10.5.13 Solve the initial value problem y ′′ + 4y ′ + 13y = 0, y(0)=1,y ′ (0)=1. Exercise 10.5.14 Solve the initial value problem y ′′ − 8y ′ + 25y = 0, y(0)=3,y ′ (0)=0. Exercise 10.5.15 A mass-spring system my ′′ + by ′ + kx has k = 29, b= 4, and m = 1. At time t = 0 the position is y(0)=2 and the velocity is y ′ (0)=1. Findy(t). Exercise 10.5.16 A mass-spring system my ′′ + by ′ + kx has k = 24, b= 12, and m = 3. At time t = 0 the position is y(0)=0 and the velocity is y ′ (0)=−1. Findy(t). Exercise 10.5.17 Consider the differential equation ay ′′ + by ′ = 0, with a and b both non-zero. Find the general solution by the method of this section. Now let g = y ′ ; the equation may be written as ag ′ +bg = 0, a first order linear homogeneous equation. Solve this for g, then use the relationship g = y ′ to find y. Exercise 10.5.18 Suppose that y(t) is a solution to ay ′′ + by ′ + cy = 0, y(t 0 )=0, y ′ (t 0 )=0. Show that y(t)=0. 10.6 Second Order Linear Equations - Method of Undetermined Coefficients Now we consider second order equations of the form ay ′′ + by ′ + cy = f (t), with a, b, andc constant. Of course, if a = 0 this is really a first order equation, so we assume a ≠ 0. Also, if c = 0 we can solve the
10.6. Second Order Linear Equations - Method of Undetermined Coefficients 393 related first order equation ah ′ +bh = f (t),andthensolveh = y ′ for y. So we will only examine examples in which c ≠ 0. Suppose that y 1 (t) and y 2 (t) are solutions to ay ′′ +by ′ +cy = f (t), and consider the function h = y 1 −y 2 . We substitute this function into the left hand side of the differential equation and simplify: a(y 1 − y 2 ) ′′ + b(y 1 − y 2 ) ′ + c(y 1 − y 2 )=ay ′′ 1 + by ′ 1 + cy 1 − (ay ′′ 2 + by ′ 2 + cy 2 )= f (t) − f (t)=0. So h is a solution to the homogeneous equation ay ′′ + by ′ + cy = 0. Since we know how to find all such h, then with just one particular solution y 2 we can express all possible solutions y 1 ,namely,y 1 = h + y 2 , where now h is the general solution to the homogeneous equation. Of course, this is exactly how we approached the first order linear equation. To make use of this observation we need a method to find a single solution y 2 . This turns out to be somewhat more difficult than the first order case, but if f (t) is of a certain simple form, we can find a solution using the method of undetermined coefficients, sometimes more whimsically called the method of judicious guessing. Example 10.24: Second Order Linear Equation Solve the differential equation y ′′ − y ′ − 6y = 18t 2 + 5. Solution. The general solution of the homogeneous equation is Ae 3t + Be −2t . We guess that a solution to the non-homogeneous equation might look like f (t) itself, namely, a quadratic y = at 2 + bt + c. Substituting this guess into the differential equation we get y ′′ − y ′ − 6y = 2a − (2at + b) − 6(at 2 + bt + c)=−6at 2 +(−2a − 6b)t +(2a − b − 6c). We want this to equal 18t 2 + 5, so we need −6a = 18 −2a − 6b = 0 2a − b − 6c = 5 This is a system of three equations in three unknowns and is not hard to solve: a = −3, b = 1, c = −2. Thus the general solution to the differential equation is Ae 3t + Be −2t − 3t 2 +t − 2. ♣ So the “judicious guess” is a function with the same form as f (t) but with undetermined (or better, yet to be determined) coefficients. This works whenever f (t) is a polynomial. Example 10.25: Mass-Spring System with No Damping Analyze the initial value problem my ′′ + ky = −mg, y(0)=2, y ′ (0)=50. Solution. The left hand side represents a mass-spring system with no damping, i.e., b = 0. Unlike the homogeneous case, we now consider the force due to gravity, −mg, assuming the spring is vertical at the surface of the earth, so that g = 980. To be specific, let us take m = 1andk = 100. The general solution to the homogeneous equation is Acos(10t)+Bsin(10t). For the solution to the non-homogeneous
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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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Page 365 and 366: 9.3. The Integral Test 349 If all o
Page 367 and 368: 9.3. The Integral Test 351 Proof. W
Page 369 and 370: 9.4. Alternating Series 353 Exercis
Page 371 and 372: 9.5. Comparison Tests 355 Exercises
Page 373 and 374: 9.5. Comparison Tests 357 Solution.
Page 375 and 376: 9.7. The Ratio and Root Tests 359 E
Page 377 and 378: 9.7. The Ratio and Root Tests 361 P
Page 379 and 380: 9.8. Power Series 363 Definition 9.
Page 381 and 382: Exercise 9.8.2 Find the radius of c
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Page 385 and 386: 9.10. Taylor Series 369 Solution. T
Page 387 and 388: 9.11. Taylor’s Theorem 371 a posi
Page 389 and 390: 9.11. Taylor’s Theorem 373 Note t
Page 391 and 392: Chapter 10 Differential Equations M
Page 393 and 394: 10.1. First Order Differential Equa
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Page 397 and 398: 10.2. First Order Homogeneous Linea
Page 399 and 400: 10.3. First Order Linear Equations
Page 401 and 402: 10.4. Approximation 385 Exercises f
Page 403 and 404: 10.4. Approximation 387 y .. 0.5 .
Page 405 and 406: 10.5. Second Order Homogeneous Equa
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Page 411 and 412: 10.6. Second Order Linear Equations
Page 417 and 418: Chapter 11 Polar Coordinates, Param
Page 419 and 420: 11.1. Polar Coordinates 403 Solutio
Page 421 and 422: 11.2. Slopes in Polar Coordinates 4
Page 423 and 424: 11.3. Areas in Polar Coordinates 40
Page 425 and 426: 11.3. Areas in Polar Coordinates 40
Page 427 and 428: 11.4. Parametric Equations 411 Figu
Page 429 and 430: 11.5 Calculus with Parametric Equat
Page 431 and 432: 11.6. Conics in Polar Coordinates 4
Page 433 and 434: 11.6. Conics in Polar Coordinates 4
Page 435 and 436: Chapter 12 Three Dimensions 12.1 Th
Page 437 and 438: 12.1. The Coordinate System 421 Now
Page 439 and 440: 12.2. Vectors 423 12.2 Vectors A ve
Page 441 and 442: 12.2. Vectors 425 3.54 0.77 . . . .
Page 443 and 444: 12.3. The Dot Product 427 Exercise
Page 445 and 446: 12.3. The Dot Product 429 Solution.
Page 447 and 448: 12.3. The Dot Product 431 v .. v ..
Page 449 and 450: 12.4. The Cross Product 433 Exercis
Page 451 and 452: 12.4. The Cross Product 435 We know
Page 453 and 454: 12.5. Lines and Planes 437 Exercise
Page 455 and 456: 12.5. Lines and Planes 439 Solution
Page 457 and 458: 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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