# Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.

8 Pages

### tcu11_09_02

Course: MATH 1240, Spring 2010
School: Westminster UT
Rating:

Word Count: 2471

#### Document Preview

9: 650 Chapter Further Applications of Integration 9.2 First-Order Linear Differential Equations The exponential growth&gt; decay equation dy&gt;dx = ky (Section 7.5) is a separable differential equation. It is also a special case of a differential equation having a linear form. Linear differential equations model a number of real-world phenomena, including electrical circuits and chemical mixture...

Register Now

#### Unformatted Document Excerpt

Coursehero >> Utah >> Westminster UT >> MATH 1240

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

Westminster UT - MATH - 1240
4100 AWL/Thomas_ch09p642-684 8/20/04 9:08 AM Page 6579.2 First-Order Linear Differential Equations657EXERCISES 9.2First-Order Linear EquationsSolve the differential equations in Exercises 114. dy dy + y = e x, x 7 0 + 2e x y = 1 1. x 2. e x dx dx 3.
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch09p642-684 8/20/04 9:08 AM Page 6599.3 Eulers Method6599.3Leonhard Euler (17031783)Eulers MethodIf we do not require or cannot immediately find an exact solution for an initial value problem y = s x, y d, ys x0 d = y0 we can often
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch09p642-684 8/20/04 9:08 AM Page 664664Chapter 9: Further Applications of IntegrationEXERCISES 9.3Calculating Euler ApproximationsIn Exercises 16, use Eulers method to calculate the first three approximations to the given initial val
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch09p642-684 8/20/04 9:08 AM Page 6659.4 Graphical Solutions of Autonomous Differential Equations6659.4Graphical Solutions of Autonomous Differential EquationsIn Chapter 4 we learned that the sign of the first derivative tells where t
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch09p642-684 8/20/04 9:08 AM Page 6719.4 Graphical Solutions of Autonomous Differential Equations671EXERCISES 9.4Phase Lines and Solution CurvesIn Exercises 18, a. Identify the equilibrium values. Which are stable and which are unstab
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch09p642-684 8/20/04 9:08 AM Page 6739.5Applications of First-Order Differential Equations6739.5Applications of First-Order Differential EquationsWe now look at three applications of the differential equations we have been studying.
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch09p642-684 8/20/04 9:08 AM Page 680680Chapter 9: Further Applications of IntegrationEXERCISES 9.51. Coasting bicycle A 66-kg cyclist on a 7-kg bicycle starts coasting on level ground at 9 m&gt; sec. The k in Equation (1) is about 3.9 kg
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch09p642-684 8/20/04 9:08 AM Page 683Chapter 9Additional and Advanced Exercises683Chapter 9Additional and Advanced Exercisesdy A = k sc - yd . V dt In this equation, y is the concentration of the substance inside the cell and dy&gt; dt
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch09p642-684 8/20/04 9:08 AM Page 682682Chapter 9: Further Applications of IntegrationChapter 9dy 1. = 2y cos2 2y dx 5. y = xe y 2x - 2 ey 9. y = xy 3. yy = sec y 2 sec2 xPractice Exercises24. x 25. dy + 2y = x 2 + 1, dx x 7 0, ys 1
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch09p642-684 8/20/04 9:08 AM Page 682682Chapter 9: Further Applications of IntegrationChapter 9Questions to Guide Your Review7. Describe the improved Eulers method for solving the initial value problem y = s x, y d, y s x0 d = y0 nume
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch09p642-684 8/20/04 9:08 AM Page 684684Chapter 9: Further Applications of IntegrationChapter 9Technology Application ProjectsMathematica / Maple ModuleDrug Dosages: Are They Effective? Are They Safe? Formulate and solve an initial v
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch10p685-745 8/25/04 2:34 PM Page 685Chapter10CONIC SECTIONS AND POLAR COORDINATESOVERVIEW In this chapter we give geometric definitions of parabolas, ellipses, and hyperbolas and derive their standard equations. These curves are calle
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch10p685-745 8/25/04 2:35 PM Page 69310.1 Conic Sections and Quadratic Equations693EXERCISES 10.1Identifying GraphsMatch the parabolas in Exercises 14 with the following equations: x 2 = 2y, x 2 = - 6y, y 2 = 8x, y 2 = - 4x .y x3.y
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch10p685-745 8/25/04 2:35 PM Page 69710.2 Classifying Conic Sections by Eccentricity69710.2Classifying Conic Sections by EccentricityWe now show how to associate with each conic section a number called the conic sections eccentricity.
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch10p685-745 8/25/04 2:35 PM Page 70110.2 Classifying Conic Sections by Eccentricity701EXERCISES 10.2EllipsesIn Exercises 18, find the eccentricity of the ellipse. Then find and graph the ellipses foci and directrices. 1. 16x 2 + 25y
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch10p685-745 8/25/04 2:35 PM Page 702702Chapter 10: Conic Sections and Polar Coordinates10.3Quadratic Equations and RotationsIn this section, we examine the Cartesian graph of any equation Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0,(1)in w
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch10p685-745 8/25/04 2:35 PM Page 70710.3 Quadratic Equations and Rotations707EXERCISES 10.3Using the DiscriminantUse the discriminant B - 4AC to decide whether the equations in Exercises 116 represent parabolas, ellipses, or hyperbol
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch10p685-745 8/25/04 2:35 PM Page 70910.4 Conics and Parametric Equations; The Cycloid70910.4Conics and Parametric Equations; The CycloidCurves in the Cartesian plane defined by parametric equations, and the calculation of their deriv
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch10p685-745 8/25/04 2:35 PM Page 712712Chapter 10: Conic Sections and Polar CoordinatesEXERCISES 10.4Parametric Equations for ConicsExercises 112 give parametric equations and parameter intervals for the motion of a particle in the x
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch10p685-745 8/25/04 2:35 PM Page 714714Chapter 10: Conic Sections and Polar Coordinates10.5r Origin (pole) OPolar CoordinatesIn this section, we study polar coordinates and their relation to Cartesian coordinates. While a point in t
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch10p685-745 8/25/04 2:35 PM Page 718718Chapter 10: Conic Sections and Polar CoordinatesEXERCISES 10.5Polar Coordinate Pairs1. Which polar coordinate pairs label the same point? d. s 2, 7p&gt; 3 d g. s - 3, 2p d a. s - 2, p&gt; 3 d a. (3, 0
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch10p685-745 8/25/04 2:35 PM Page 71910.6 Graphing in Polar Coordinates71910.6Graphing in Polar CoordinatesThis section describes techniques for graphing equations in polar coordinates.SymmetryFigure 10.43 illustrates the standard p
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch10p685-745 8/25/04 2:35 PM Page 724724Chapter 10: Conic Sections and Polar CoordinatesEXERCISES 10.6Symmetries and Polar GraphsIdentify the symmetries of the curves in Exercises 112. Then sketch the curves. 1. r = 1 + cos u 3. r = 1
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch10p685-745 8/25/04 2:35 PM Page 72510.7 Areas and Lengths in Polar Coordinates72510.7Areas and Lengths in Polar CoordinatesThis section shows how to calculate areas of plane regions, lengths of curves, and areas of surfaces of revol
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch10p685-745 8/25/04 2:35 PM Page 730730Chapter 10: Conic Sections and Polar CoordinatesEXERCISES 10.7Areas Inside Polar CurvesFind the areas of the regions in Exercises 16. 1. Inside the oval limaon r = 4 + 2 cos u 2. Inside the card
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch10p685-745 8/25/04 2:35 PM Page 732732Chapter 10: Conic Sections and Polar Coordinates10.8Conic Sections in Polar CoordinatesPolar coordinates are important in astronomy and astronautical engineering because the ellipses, parabolas,
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch10p685-745 8/25/04 2:35 PM Page 73710.8 Conic Sections in Polar Coordinates737EXERCISES 10.8LinesFind polar and Cartesian equations for the lines in Exercises 14. 1.y15.y Radius y 216.y Radius 1 22.0 5 6 0 x 0 2 3 4 x 0 xx3
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch10p685-745 8/25/04 2:35 PM Page 742742Chapter 10: Conic Sections and Polar CoordinatesChapter 10Additional and Advanced Exercisesb. Show that the line b 2xx1 - a 2yy1 - a 2b 2 = 0 is tangent to the hyperbola b 2x 2 - a 2y 2 - a 2b 2
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch10p685-745 8/25/04 2:35 PM Page 739Chapter 10Practice Exercises739Chapter 10Practice ExercisesFind the eccentricities of the ellipses and hyperbolas in Exercises 58. Sketch each conic section. Include the foci, vertices, and asympt
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch10p685-745 8/25/04 2:35 PM Page 739Chapter 10Questions to Guide Your Review739Chapter 10Questions to Guide Your Review9. What are some typical parametrizations for conic sections? 10. What is a cycloid? What are typical parametric
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch10p685-745 8/25/04 2:35 PM Page 745Chapter 10Technology Application Projects745Chapter 10Technology Application ProjectsMathematica / Maple ModuleRadar Tracking of a Moving Object Part I: Convert from polar to Cartesian coordinate
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch11p746-847 8/25/04 2:40 PM Page 772772Chapter 11: Infinite Sequences and Series11.3The Integral TestGiven a series g an , we have two questions: 1. 2. Does the series converge? If it converges, what is its sum?Much of the rest of t
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch11p746-847 8/25/04 2:41 PM Page 77511.3 The Integral Test775EXERCISES 11.3Determining Convergence or DivergenceWhich of the series in Exercises 130 converge, and which diverge? Give reasons for your answers. (When you check an answe
Westminster UT - MATH - 1240
12.2 Vectors85312.2VectorsSome of the things we measure are determined simply by their magnitudes. To record mass, length, or time, for example, we need only write down a number and name an appropriate unit of measure. We need more information to desc
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch12p848-905 8/25/04 2:43 PM Page 852852Chapter 12: Vectors and the Geometry of SpaceEXERCISES 12.1Sets, Equations, and InequalitiesIn Exercises 112, give a geometric description of the set of points in space whose coordinates satisfy
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch12p848-905 8/25/04 2:43 PM Page 848Chapter12VECTORS AND THE GEOMETRY OF SPACEOVERVIEW To apply calculus in many real-world situations and in higher mathematics, we need a mathematical description of three-dimensional space. In this c
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch12p848-905 8/25/04 2:43 PM Page 860860Chapter 12: Vectors and the Geometry of SpaceEXERCISES 12.2Vectors in the PlaneIn Exercises 18, let u = 83, - 29 and v = 8 - 2, 59 . Find the (a) component form and (b) magnitude (length) of the
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch12p848-905 8/25/04 2:43 PM Page 862862Chapter 12: Vectors and the Geometry of Space12.3FThe Dot ProductIf a force F is applied to a particle moving along a path, we often need to know the magnitude of the force in the direction of
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch12p848-905 8/25/04 2:43 PM Page 870870Chapter 12: Vectors and the Geometry of SpaceEXERCISES 12.3Dot Product and ProjectionsIn Exercises 18, find 1. v = 2i - 4j + 25k, d. the vector projv u . 2. v = s 3&gt; 5 di + s 4&gt; 5 dk, 3. v = 10i
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch12p848-905 8/25/04 2:43 PM Page 87312.4 The Cross Product87312.4The Cross ProductIn studying lines in the plane, when we needed to describe how a line was tilting, we used the notions of slope and angle of inclination. In space, we
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch12p848-905 8/25/04 2:43 PM Page 878878Chapter 12: Vectors and the Geometry of SpaceEXERCISES 12.4Cross Product CalculationsIn Exercises 18, find the length and direction (when defined) of u * v and v * u. 1. u = 2i - 2j - k, 2. u =
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch12p848-905 8/25/04 2:44 PM Page 880880Chapter 12: Vectors and the Geometry of Space12.5Lines and Planes in SpaceIn the calculus of functions of a single variable, we used our knowledge of lines to study curves in the plane. We inves
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch12p848-905 8/25/04 2:44 PM Page 88712.5 Lines and Planes in Space887EXERCISES 12.5Lines and Line SegmentsFind parametric equations for the lines in Exercises 112. 1. The line through the point Ps 3, - 4, - 1 d parallel to the vector
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch12p848-905 8/25/04 2:44 PM Page 88912.5 Lines and Planes in Space88912.6Cylinders and Quadric SurfacesUp to now, we have studied two special types of surfaces: spheres and planes. In this section, we extend our inventory to include
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch12p848-905 8/25/04 2:44 PM Page 89712.6 Cylinders and Quadric Surfaces897EXERCISES 12.6Matching Equations with SurfacesIn Exercises 112, match the equation with the surface it defines. Also, identify each surface by type (paraboloid
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch12p848-905 8/25/04 2:44 PM Page 902902Chapter 12: Vectors and the Geometry of SpaceChapter 12Additional and Advanced Exercises2. A helicopter rescue Two helicopters, H1 and H2 , are traveling together. At time t = 0 , they separate
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch12p848-905 8/25/04 2:44 PM Page 900900Chapter 12: Vectors and the Geometry of SpaceChapter 12Practice Exercises16. Find a vector 5 units long in the direction opposite to the direction of v = s 3&gt; 5 d i + s 4&gt; 5 d k. In Exercises 17
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch12p848-905 8/25/04 2:44 PM Page 899Chapter 12Questions to Guide Your Review899Chapter 12Questions to Guide Your Review3. How do you find a vectors magnitude and direction? 4. If a vector is multiplied by a positive scalar, how is t
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch12p848-905 8/25/04 2:44 PM Page 905Chapter 12Technology Application Projects905Chapter 12Technology Application ProjectsMathematica / Maple ModuleUsing Vectors to Represent Lines and Find Distances Parts I and II: Learn the advant
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch11p746-847 8/25/04 2:41 PM Page 786786Chapter 11: Infinite Sequences and Series 2n a 2n 1 + ln n = an n n + ln n = a n + 10 nnEXERCISES 11.5Determining Convergence or DivergenceWhich of the series in Exercises 126 converge, and whi
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch11p746-847 8/25/04 2:41 PM Page 78711.6Alternating Series, Absolute and Conditional Convergence78711.6Alternating Series, Absolute and Conditional ConvergenceA series in which the terms are alternately positive and negative is an a
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch11p746-847 8/25/04 2:41 PM Page 792792Chapter 11: Infinite Sequences and SeriesEXERCISES 11.6Determining Convergence or DivergenceWhich of the alternating series in Exercises 110 converge, and which diverge? Give reasons for your an
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch11p746-847 8/25/04 2:41 PM Page 794794Chapter 11: Infinite Sequences and Series11.7Power SeriesNow that we can test infinite series for convergence we can study the infinite polynomials mentioned at the beginning of this chapter. We
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch11p746-847 8/25/04 2:41 PM Page 804804Chapter 11: Infinite Sequences and SeriesEXERCISES 11.7Intervals of ConvergenceIn Exercises 132, (a) find the series radius and interval of convergence. For what values of x does the series conv
Westminster UT - MATH - 1240
11.8 Taylor and Maclaurin Series80511.8Taylor and Maclaurin SeriesThis section shows how functions that are infinitely differentiable generate power series called Taylor series. In many cases, these series can provide useful polynomial approximations
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch11p746-847 8/25/04 2:41 PM Page 810810Chapter 11: Infinite Sequences and SeriesEXERCISES 11.8Finding Taylor PolynomialsIn Exercises 18, find the Taylor polynomials of orders 0, 1, 2, and 3 generated by at a. 7. s x d = 2x, 1. s x d
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch11p746-847 8/25/04 2:41 PM Page 81111.9 Convergence of Taylor Series; Error Estimates81111.9Convergence of Taylor Series; Error EstimatesThis section addresses the two questions left unanswered by Section 11.8: 1. 2. When does a Tay
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch11p746-847 8/25/04 2:41 PM Page 81911.9 Convergence of Taylor Series; Error Estimates819EXERCISES 11.9Taylor Series by SubstitutionUse substitution (as in Example 4) to find the Taylor series at x = 0 of the functions in Exercises 1
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch11p746-847 8/25/04 2:41 PM Page 822822Chapter 11: Infinite Sequences and Series11.10Applications of Power SeriesThis section introduces the binomial series for estimating powers and roots and shows how series are sometimes used to a
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch11p746-847 8/25/04 2:41 PM Page 83111.10 Applications of Power Series831EXERCISES 11.10Binomial SeriesFind the first four terms of the binomial series for the functions in Exercises 110. 1. s 1 + x d1&gt;2 4. s 1 - 2x d 2. s 1 + x d1&gt;3
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch11p746-847 8/25/04 2:41 PM Page 83311.11 Fourier Series83311.11Fourier SeriesWe have seen how Taylor series can be used to approximate a function by polynomials. The Taylor polynomials give a close fit to near a particular point x =