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### tcu11_12_05ex

Course: MATH 1240, Spring 2010
School: Westminster UT
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Word Count: 1532

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AWL/Thomas_ch12p848-905 4100 8/25/04 2:44 PM Page 887 12.5 Lines and Planes in Space 887 EXERCISES 12.5 Lines and Line Segments Find parametric equations for the lines in Exercises 112. 1. The line through the point Ps 3, - 4, - 1 d parallel to the vector i+j+k 2. The line through Ps 1, 2, - 1 d and Qs - 1, 0, 1 d 3. The line through Ps - 2, 0, 3 d and Qs 3, 5, - 2 d 4. The line through P(1, 2, 0) and Qs 1, 1,...

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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 =
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch11p746-847 8/25/04 2:41 PM Page 838838Chapter 11: Infinite Sequences and SeriesEXERCISES 11.11Finding Fourier SeriesIn Exercises 18, find the Fourier series associated with the given functions. Sketch each function. 1. s x d = 1 0 x
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch11p746-847 8/25/04 2:41 PM Page 843Chapter 11Additional and Advanced Exercises843Chapter 11Additional and Advanced Exercises16. Find the sum of the infinite series 1+ + 17. Evaluateq n=0 n+1 n LConvergence or DivergenceWhich of
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch11p746-847 8/25/04 2:41 PM Page 840840Chapter 11: Infinite Sequences and SeriesChapter 11Practice ExercisesConvergent or Divergent SeriesWhich of the series in Exercises 2540 converge absolutely, which converge conditionally, and w
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch11p746-847 8/25/04 2:41 PM Page 839Chapter 11Questions to Guide Your Review839Chapter 11Questions to Guide Your Review15. When do p-series converge? Diverge? How do you know? Give examples of convergent and divergent p-series. 16.
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch11p746-847 8/25/04 2:41 PM Page 847Chapter 11Technology Application Projects847Chapter 11Technology Application ProjectsMathematica / Maple ModuleBouncing Ball The model predicts the height of a bouncing ball, and the time until i
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch13p906-964 8/25/04 2:47 PM Page 906Chapter13VECTOR-VALUED FUNCTIONS AND MOTION IN SPACEOVERVIEW When a body (or object) travels through space, the equations x = s t d, y = g s t d , and z = hs t d that give the bodys coordinates as f
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch13p906-964 8/25/04 2:48 PM Page 916916Chapter 13: Vector-Valued Functions and Motion in SpaceEXERCISES 13.1Motion in the xy-planeIn Exercises 14, r(t) is the position of a particle in the xy-plane at time t. Find an equation in x an
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch13p906-964 8/25/04 2:48 PM Page 920920Chapter 13: Vector-Valued Functions and Motion in Space13.2Modeling Projectile MotionWhen we shoot a projectile into the air we usually want to know beforehand how far it will go (will it reach
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch13p906-964 8/25/04 2:48 PM Page 92713.2 Modeling Projectile Motion927EXERCISES 13.2Projectile flights in the following exercises are to be treated as ideal unless stated otherwise. All launch angles are assumed to be measured from th
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch13p906-964 8/25/04 2:48 PM Page 93113.3Arc Length and the Unit Tangent Vector T93113.3Arc Length and the Unit Tangent Vector TImagine the motions you might experience traveling at high speeds along a path through the air or space.
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch13p906-964 8/25/04 2:48 PM Page 93513.3 Arc Length and the Unit Tangent Vector T935EXERCISES 13.3Finding Unit Tangent Vectors and Lengths of Curves11. rs t d = s 4 cos t di + s 4 sin t dj + 3t k, 13. rs t d = s e cos t di + s e sin
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch13p906-964 8/25/04 2:48 PM Page 936936Chapter 13: Vector-Valued Functions and Motion in Space13.4yCurvature and the Unit Normal Vector NIn this section we study how a curve turns or bends. We look first at curves in the coordinate
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch13p906-964 8/25/04 2:48 PM Page 942942Chapter 13: Vector-Valued Functions and Motion in SpaceEXERCISES 13.4Plane CurvesFind T, N, and k for the plane curves in Exercises 14. 1. rs t d = t i + s ln cos t dj, 2. rs t d = s ln sec t di
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch13p906-964 8/25/04 2:48 PM Page 94313.5 Torsion and the Unit Binormal Vector B94313.5zTorsion and the Unit Binormal Vector BIf you are traveling along a space curve, the Cartesian i, j, and k coordinate system for representing the
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch13p906-964 8/25/04 2:48 PM Page 94913.5 Torsion and the Unit Binormal Vector B949EXERCISES 13.5Finding Torsion and the Binormal VectorFor Exercises 18 you found T, N, and k in Section 13.4 (Exercises 916). Find now B and t for these
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch13p906-964 8/25/04 2:48 PM Page 950950Chapter 13: Vector-Valued Functions and Motion in Space13.6Planetary Motion and SatellitesIn this section, we derive Keplers laws of planetary motion from Newtons laws of motion and gravitation
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch13p906-964 8/25/04 2:48 PM Page 958958Chapter 13: Vector-Valued Functions and Motion in SpaceEXERCISES 13.6Reminder: When a calculation involves the gravitational constant G, express force in newtons, distance in meters, mass in kilo
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch13p906-964 8/25/04 2:48 PM Page 962962Chapter 13: Vector-Valued Functions and Motion in SpaceChapter 13ApplicationsAdditional and Advanced Exercises2. A straight river is 20 m wide. The velocity of the river at (x, y) is A boat lea
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch13p906-964 8/25/04 2:48 PM Page 960960Chapter 13: Vector-Valued Functions and Motion in SpaceChapter 13Practice Exercisesa. Sketch the curve traced by P during the interval 0 t 3 . b. Find v and a at t = 0, 1, 2 , and 3 and add thes
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch13p906-964 8/25/04 2:48 PM Page 959Chapter 13Questions to Guide Your Review959Chapter 13Questions to Guide Your Review6. How do you measure distance along a smooth curve in space from a preselected base point? Give an example. 7. W
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch13p906-964 8/25/04 2:48 PM Page 964964Chapter 13: Vector-Valued Functions and Motion in SpaceChapter 13Technology Application ProjectsMathematica / Maple ModuleRadar Tracking of a Moving Object Visualize position, velocity, and acc
Westminster UT - MATH - 1240
Chapter1414.1PARTIAL DERIVATIVESOVERVIEW In studying a real-world phenomenon, a quantity being investigated usually depends on two or more independent variables. So we need to extend the basic ideas of the calculus of functions of a single variable to
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch14p965-1066 8/25/04 2:52 PM Page 97314.1 Functions of Several Variables973EXERCISES 14.1Domain, Range, and Level CurvesIn Exercises 112, (a) find the functions domain, (b) find the functions range, (c) describe the functions level c
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch14p965-1066 8/25/04 2:53 PM Page 976976Chapter 14: Partial Derivatives14.2Limits and Continuity in Higher DimensionsThis section treats limits and continuity for multivariable functions. The definition of the limit of a function of
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch14p965-1066 8/25/04 2:53 PM Page 982982Chapter 14: Partial DerivativesEXERCISES 14.2Limits with Two VariablesFind the limits in Exercises 112. 3x 2 - y 2 + 5 lim 1. 2. sx, yd : s0,0d x 2 + y 2 + 2 3. 5. 7. 9. 11.sx, yd : s3,4dlim
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch14p965-1066 8/25/04 2:53 PM Page 994994Chapter 14: Partial DerivativesEXERCISES 14.3Calculating First-Order Partial DerivativesIn Exercises 122, find 0 &gt; 0 x and 0 &gt; 0 y . 1. s x, y d = 2x 2 - 3y - 4 7. s x, y d = 2x 2 + y 2 5. s x,
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch14p965-1066 8/25/04 2:53 PM Page 996996Chapter 14: Partial Derivatives14.4The Chain RuleThe Chain Rule for functions of a single variable studied in Section 3.5 said that when w = s x d was a differentiable function of x and x = gs
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch14p965-1066 8/25/04 2:53 PM Page 100314.4 The Chain Rule1003EXERCISES 14.4In Exercises 16, (a) express dw&gt; dt as a function of t, both by using the Chain Rule and by expressing w in terms of t and differentiating directly with respec
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch14p965-1066 8/25/04 2:53 PM Page 100514.5 Directional Derivatives and Gradient Vectors100514.5Directional Derivatives and Gradient VectorsIf you look at the map (Figure 14.23) showing contours on the West Point Area along the Hudson
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch14p965-1066 8/25/04 2:53 PM Page 101314.5 Directional Derivatives and Gradient Vectors1013A 22, 1 BEXERCISES 14.5Calculating Gradients at PointsIn Exercises 14, find the gradient of the function at the given point. Then sketch the
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch14p965-1066 8/25/04 2:53 PM Page 101514.6Tangent Planes and Differentials101514.6Tangent Planes and DifferentialsIn this section we define the tangent plane at a point on a smooth surface in space. We calculate an equation of the t
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch14p965-1066 8/25/04 2:53 PM Page 10241024Chapter 14: Partial DerivativesEXERCISES 14.6Tangent Planes and Normal Lines to SurfacesIn Exercises 18, find equations for the (a) tangent plane and 1. x 2 + y 2 + z 2 = 3, 2. x 2 + y 2 - z
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch14p965-1066 9/2/04 11:09 AM Page 102714.7 Extreme Values and Saddle Points102714.7zExtreme Values and Saddle PointsContinuous functions of two variables assume extreme values on closed, bounded domains (see Figures 14.36 and 14.37)
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch14p965-1066 8/25/04 2:53 PM Page 10341034Chapter 14: Partial DerivativesEXERCISES 14.7Finding Local ExtremaFind all the local maxima, local minima, and saddle points of the functions in Exercises 130. 1. s x, y d = x + xy + y + 3x -
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch14p965-1066 8/25/04 2:53 PM Page 10381038Chapter 14: Partial Derivatives14.8Joseph Louis Lagrange (17361813)Lagrange MultipliersSometimes we need to find the extreme values of a function whose domain is constrained to lie within so
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch14p965-1066 8/25/04 2:53 PM Page 104714.8 Lagrange Multipliers1047EXERCISES 14.8Two Independent Variables with One Constraint1. Extrema on an ellipse Find the points on the ellipse x 2 + 2y 2 = 1 where s x, y d = xy as its extreme v
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch14p965-1066 8/25/04 2:53 PM Page 104914.9 Partial Derivatives with Constrained Variables104914.9Partial Derivatives with Constrained VariablesIn finding partial derivatives of functions like w = s x, y d , we have assumed x and y to
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch14p965-1066 8/25/04 2:54 PM Page 105314.9 Partial Derivatives with Constrained Variables1053EXERCISES 14.9Finding Partial Derivatives with Constrained VariablesIn Exercises 13, begin by drawing a diagram that shows the relations amo
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch14p965-1066 8/25/04 2:54 PM Page 10541054Chapter 14: Partial Derivatives14.10Taylors Formula for Two VariablesThis section uses Taylors formula to derive the Second Derivative Test for local extreme values (Section 14.7) and the err
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch14p965-1066 8/25/04 2:54 PM Page 10581058Chapter 14: Partial DerivativesEXERCISES 14.10Finding Quadratic and Cubic ApproximationsIn Exercises 110, use Taylors formula for (x, y) at the origin to find quadratic and cubic approximatio
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch14p965-1066 8/25/04 2:54 PM Page 1063Chapter 14Additional and Advanced Exercises1063Chapter 14Partial DerivativesAdditional and Advanced Exercises(see the accompanying figure) is continuous at (0, 0). Find xys 0, 0 d and yxs 0, 0
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch14p965-1066 8/25/04 2:54 PM Page 10601060Chapter 14: Partial DerivativesChapter 14Practice Exercises23. Ps n, R, T, V d = 24. s r, l, T, w d = T 1 2rl A pw nRT (the ideal gas law) VDomain, Range, and Level CurvesIn Exercises 14, f
Westminster UT - MATH - 1240
4100 AWL/Thomas_ch14p965-1066 8/25/04 2:54 PM Page 1059Chapter 14 Questions to Guide Your Review1059Chapter 14Questions to Guide Your Review14. What is the derivative of a function (x, y) at a point P0 in the direction of a unit vector u? What rate d