Explanation Given: The vector-valued functions r ( t ) and u ( t ) whose limit exists as t → c . Formula used: The dot product of two vectors : ( a i + b j + c k ) ⋅ ( p i + q j + r k ) = a p + b q + c r . Proof: Let r ( t ) = a ( t ) i + b ( t ) j + c ( t ) k and u ( t ) = d ( t ) i + e ( t ) j + f ( t ) k . Then, r ( t ) ⋅ u ( t ) = [ a ( t ) i + b ( t ) j + c ( t ) k ] ⋅ [ d ( t ) i + e ( t ) j + f ( t ) k ] = a ( t ) d ( t ) + b ( t ) e ( t ) + c ( t ) f ( t ) Now, lim x → c r ( t ) ⋅ u ( t ) = lim x → c [ a ( t ) d ( t ) + b ( t ) e ( t ) +

11th Edition

ISBN: 9780357246412

Author: Ron Larson; Bruce H. Edwards

Publisher: Cengage Limited

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Chapter 12.1, Problem 84E

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To prove: The limit limx→c[r(t)⋅u(t)]=limx→cr(t)⋅limx→cu(t).

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Chapter 12.1, Problem 83E

Chapter 12.1, Problem 85E

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Chapter 12 Solutions

Calculus

Ch. 12.1 - CONCEPTS CHECK Vector-valued FunctionDescribe how...Ch. 12.1 - Prob. 2E Ch. 12.1 - Finding the Domain In exercises 310 find the...Ch. 12.1 - Finding the domain In exercises 310 find the...Ch. 12.1 - Prob. 5E Ch. 12.1 - Finding the domain In exercises 3-10 find the...Ch. 12.1 - Prob. 7E Ch. 12.1 - Prob. 8E Ch. 12.1 - Finding the domain In exercises 310 find the...Ch. 12.1 - Prob. 10E

Ch. 12.1 - Prob. 11E Ch. 12.1 - Evaluating a function In Exercises 11 and 12...Ch. 12.1 - Prob. 13E Ch. 12.1 - Prob. 14E Ch. 12.1 - Prob. 15E Ch. 12.1 - Writing a Vector-Valued FunctionIn Exercises 1316,...Ch. 12.1 - Prob. 17E Ch. 12.1 - Prob. 18E Ch. 12.1 - Prob. 19E Ch. 12.1 - Prob. 20E Ch. 12.1 - Prob. 21E Ch. 12.1 - Prob. 22E Ch. 12.1 - Prob. 23E Ch. 12.1 - Prob. 24E Ch. 12.1 - Prob. 25E Ch. 12.1 - Prob. 26E Ch. 12.1 - Prob. 27E Ch. 12.1 - Prob. 28E Ch. 12.1 - Prob. 29E Ch. 12.1 - Prob. 30E Ch. 12.1 - Prob. 31E Ch. 12.1 - Prob. 32E Ch. 12.1 - Prob. 33E Ch. 12.1 - Prob. 34E Ch. 12.1 - Prob. 35E Ch. 12.1 - Prob. 36E Ch. 12.1 - Prob. 37E Ch. 12.1 - Prob. 38E Ch. 12.1 - Prob. 39E Ch. 12.1 - Prob. 40E Ch. 12.1 - Prob. 41E Ch. 12.1 - Transformations of Vector-Valued Functions In...Ch. 12.1 - Prob. 43E Ch. 12.1 - Prob. 44E Ch. 12.1 - Prob. 45E Ch. 12.1 - Prob. 46E Ch. 12.1 - Prob. 47E Ch. 12.1 - Prob. 48E Ch. 12.1 - Representing a Graph by a Vector-Valued Function...Ch. 12.1 - Prob. 50E Ch. 12.1 - Prob. 51E Ch. 12.1 - Prob. 52E Ch. 12.1 - Prob. 53E Ch. 12.1 - Prob. 54E Ch. 12.1 - Representing a Graph by Vector-Valued Function In...Ch. 12.1 - Prob. 56E Ch. 12.1 - Prob. 57E Ch. 12.1 - Prob. 58E Ch. 12.1 - Prob. 59E Ch. 12.1 - Prob. 60E Ch. 12.1 - Prob. 61E Ch. 12.1 - Prob. 62E Ch. 12.1 - Prob. 63E Ch. 12.1 - Prob. 64E Ch. 12.1 - Prob. 65E Ch. 12.1 - Prob. 66E Ch. 12.1 - Prob. 67E Ch. 12.1 - Prob. 68E Ch. 12.1 - Prob. 69E Ch. 12.1 - Prob. 70E Ch. 12.1 - Prob. 71E Ch. 12.1 - Prob. 72E Ch. 12.1 - Prob. 73E Ch. 12.1 - Prob. 74E Ch. 12.1 - Prob. 75E Ch. 12.1 - Prob. 76E Ch. 12.1 - Prob. 77E Ch. 12.1 - Prob. 78E Ch. 12.1 - Prob. 79E Ch. 12.1 - Prob. 80E Ch. 12.1 - Prob. 81E Ch. 12.1 - HOW DO YOU SEE IT? The four figures below are...Ch. 12.1 - Prob. 83E Ch. 12.1 - Prob. 84E Ch. 12.1 - Prob. 85E Ch. 12.1 - Prob. 86E Ch. 12.1 - Prob. 87E Ch. 12.1 - Prob. 88E Ch. 12.1 - Prob. 89E Ch. 12.1 - Prob. 90E Ch. 12.2 - Prob. 1E Ch. 12.2 - Prob. 2E Ch. 12.2 - Prob. 3E Ch. 12.2 - Prob. 4E Ch. 12.2 - Prob. 5E Ch. 12.2 - Prob. 6E Ch. 12.2 - Prob. 7E Ch. 12.2 - Prob. 8E Ch. 12.2 - Prob. 9E Ch. 12.2 - Prob. 10E Ch. 12.2 - Prob. 11E Ch. 12.2 - Prob. 12E Ch. 12.2 - Prob. 13E Ch. 12.2 - Prob. 14E Ch. 12.2 - Prob. 15E Ch. 12.2 - Prob. 16E Ch. 12.2 - Prob. 17E Ch. 12.2 - Finding a Derivative In Exercises 11-18, find...Ch. 12.2 - Prob. 19E Ch. 12.2 - Prob. 20E Ch. 12.2 - Higher-Order DifferentiationIn Exercises 1922,...Ch. 12.2 - Prob. 22E Ch. 12.2 - Prob. 23E Ch. 12.2 - Prob. 24E Ch. 12.2 - Prob. 25E Ch. 12.2 - Prob. 26E Ch. 12.2 - Prob. 27E Ch. 12.2 - Prob. 28E Ch. 12.2 - Finding Intervals on Which a Curve Is Smooth In...Ch. 12.2 - Prob. 30E Ch. 12.2 - Prob. 31E Ch. 12.2 - Prob. 32E Ch. 12.2 - Prob. 33E Ch. 12.2 - Prob. 34E Ch. 12.2 - Using Properties of the Derivative In Exercises 35...Ch. 12.2 - Using Properties of the DerivativeIn Exercises 35...Ch. 12.2 - Using Two MethodsIn Exercises 37 and 38, find (a)...Ch. 12.2 - Prob. 38E Ch. 12.2 - Prob. 39E Ch. 12.2 - Prob. 40E Ch. 12.2 - Prob. 41E Ch. 12.2 - Prob. 42E Ch. 12.2 - Prob. 43E Ch. 12.2 - Prob. 44E Ch. 12.2 - Prob. 45E Ch. 12.2 - Finding an Indefinite Integral In Exercises 39-46,...Ch. 12.2 - Prob. 47E Ch. 12.2 - Evaluating a Definite Integral In Exercises 47-52,...Ch. 12.2 - Prob. 49E Ch. 12.2 - Prob. 50E Ch. 12.2 - Evaluating a Definite Integral In Exercises 47-52,...Ch. 12.2 - Prob. 52E Ch. 12.2 - Prob. 53E Ch. 12.2 - Prob. 54E Ch. 12.2 - Prob. 55E Ch. 12.2 - Prob. 56E Ch. 12.2 - Prob. 57E Ch. 12.2 - Finding an Antiderivative In Exercises 53-58, find...Ch. 12.2 - Prob. 59E Ch. 12.2 - Prob. 60E Ch. 12.2 - Prob. 61E Ch. 12.2 - Prob. 62E Ch. 12.2 - Prob. 63E Ch. 12.2 - Prob. 64E Ch. 12.2 - Prob. 65E Ch. 12.2 - Prob. 66E Ch. 12.2 - Prob. 67E Ch. 12.2 - Prob. 68E Ch. 12.2 - Particle MotionA particle moves in the xy-plane...Ch. 12.2 - Particle MotionA particle moves in the yz-plane...Ch. 12.2 - Prob. 71E Ch. 12.2 - Prob. 72E Ch. 12.2 - Prob. 73E Ch. 12.2 - True or False? In Exercises 73-76, determine...Ch. 12.2 - Prob. 75E Ch. 12.2 - Prob. 76E Ch. 12.3 - Prob. 1E Ch. 12.3 - Prob. 2E Ch. 12.3 - Prob. 3E Ch. 12.3 - Prob. 4E Ch. 12.3 - Prob. 5E Ch. 12.3 - Prob. 6E Ch. 12.3 - Prob. 7E Ch. 12.3 - Prob. 8E Ch. 12.3 - Prob. 9E Ch. 12.3 - Prob. 10E Ch. 12.3 - Finding Velocity and Acceleration Vectors in Space...Ch. 12.3 - Prob. 12E Ch. 12.3 - Prob. 13E Ch. 12.3 - Prob. 14E Ch. 12.3 - Prob. 15E Ch. 12.3 - Prob. 16E Ch. 12.3 - Prob. 17E Ch. 12.3 - Prob. 18E Ch. 12.3 - Prob. 19E Ch. 12.3 - Finding Velocity and Acceleration Vectors in Space...Ch. 12.3 - Prob. 21E Ch. 12.3 - Prob. 22E Ch. 12.3 - Prob. 23E Ch. 12.3 - Prob. 24E Ch. 12.3 - Prob. 25E Ch. 12.3 - Prob. 26E Ch. 12.3 - Prob. 27E Ch. 12.3 - Prob. 28E Ch. 12.3 - Prob. 29E Ch. 12.3 - A baseball player at second base throws a ball 90...Ch. 12.3 - Prob. 31E Ch. 12.3 - Prob. 32E Ch. 12.3 - Prob. 33E Ch. 12.3 - Prob. 34E Ch. 12.3 - Prob. 35E Ch. 12.3 - A bomber is flying horizontally at an altitude of...Ch. 12.3 - Prob. 37E Ch. 12.3 - Prob. 38E Ch. 12.3 - Prob. 39E Ch. 12.3 - Prob. 40E Ch. 12.3 - Projectile Motion In Exercises 41 and 42, use the...Ch. 12.3 - Prob. 42E Ch. 12.3 - Prob. 43E Ch. 12.3 - Prob. 44E Ch. 12.3 - Prob. 45E Ch. 12.3 - Prob. 46E Ch. 12.3 - Prob. 47E Ch. 12.3 - Prob. 48E Ch. 12.3 - Prob. 49E Ch. 12.3 - Prob. 50E Ch. 12.3 - Circular Motion In Exercises 51 and 52, use the...Ch. 12.3 - Prob. 52E Ch. 12.3 - Prob. 53E Ch. 12.3 - Prob. 54E Ch. 12.3 - Prob. 55E Ch. 12.3 - Particle Motion Consider a particle moving on an...Ch. 12.3 - Prob. 57E Ch. 12.3 - Prob. 58E Ch. 12.3 - Prob. 59E Ch. 12.3 - Prob. 60E Ch. 12.3 - Prob. 61E Ch. 12.3 - Prob. 62E Ch. 12.3 - Prob. 63E Ch. 12.4 - Prob. 1E Ch. 12.4 - Prob. 2E Ch. 12.4 - Prob. 3E Ch. 12.4 - Prob. 4E Ch. 12.4 - Finding the Unit Tangent Vector In Exercises 3-8,...Ch. 12.4 - Finding the Unit Tangent Vector In Exercises 3-8,...Ch. 12.4 - Finding the Unit Tangent Vector In Exercises 3-8,...Ch. 12.4 - Prob. 8E Ch. 12.4 - Prob. 9E Ch. 12.4 - Prob. 10E Ch. 12.4 - Prob. 11E Ch. 12.4 - Prob. 12E Ch. 12.4 - Prob. 13E Ch. 12.4 - Prob. 14E Ch. 12.4 - Prob. 15E Ch. 12.4 - Prob. 16E Ch. 12.4 - Prob. 17E Ch. 12.4 - Prob. 18E Ch. 12.4 - Prob. 19E Ch. 12.4 - Prob. 20E Ch. 12.4 - Prob. 21E Ch. 12.4 - Prob. 22E Ch. 12.4 - Prob. 23E Ch. 12.4 - Prob. 24E Ch. 12.4 - Prob. 25E Ch. 12.4 - Prob. 26E Ch. 12.4 - Prob. 27E Ch. 12.4 - Prob. 28E Ch. 12.4 - Prob. 29E Ch. 12.4 - Prob. 30E Ch. 12.4 - Prob. 31E Ch. 12.4 - Prob. 32E Ch. 12.4 - Prob. 33E Ch. 12.4 - Prob. 34E Ch. 12.4 - Prob. 35E Ch. 12.4 - Prob. 36E Ch. 12.4 - Prob. 37E Ch. 12.4 - Prob. 38E Ch. 12.4 - Prob. 39E Ch. 12.4 - Prob. 40E Ch. 12.4 - Prob. 41E Ch. 12.4 - Prob. 42E Ch. 12.4 - Prob. 43E Ch. 12.4 - Prob. 44E Ch. 12.4 - Cycloidal Motion The figure shows the path of a...Ch. 12.4 - Motion Along an Involute of a Circle The figure...Ch. 12.4 - Prob. 47E Ch. 12.4 - Prob. 48E Ch. 12.4 - Prob. 49E Ch. 12.4 - Prob. 50E Ch. 12.4 - Prob. 51E Ch. 12.4 - Prob. 52E Ch. 12.4 - Prob. 53E Ch. 12.4 - Prob. 54E Ch. 12.4 - Prob. 55E Ch. 12.4 - Prob. 56E Ch. 12.4 - Projectile Motion Find the tangential and normal...Ch. 12.4 - Prob. 58E Ch. 12.4 - Prob. 59E Ch. 12.4 - Prob. 60E Ch. 12.4 - Air Traffic ControlBecause of a storm, ground...Ch. 12.4 - Projectile Motion A plane flying at an altitude of...Ch. 12.4 - Prob. 63E Ch. 12.4 - Prob. 64E Ch. 12.4 - Prob. 65E Ch. 12.4 - Prob. 66E Ch. 12.4 - Prob. 67E Ch. 12.4 - Prob. 68E Ch. 12.4 - Prob. 69E Ch. 12.4 - Prob. 70E Ch. 12.4 - Prob. 71E Ch. 12.4 - Proof Prove that the principal unit normal vector...Ch. 12.4 - Prob. 73E Ch. 12.4 - Prob. 74E Ch. 12.4 - Prob. 75E Ch. 12.4 - Prob. 76E Ch. 12.5 - Prob. 1E Ch. 12.5 - Prob. 2E Ch. 12.5 - Prob. 3E Ch. 12.5 - Prob. 4E Ch. 12.5 - Prob. 5E Ch. 12.5 - Prob. 6E Ch. 12.5 - Prob. 7E Ch. 12.5 - Prob. 8E Ch. 12.5 - Projectile Motion The position of a baseball. is...Ch. 12.5 - Projectile Motion The position of a baseball is...Ch. 12.5 - Prob. 11E Ch. 12.5 - Prob. 12E Ch. 12.5 - Prob. 13E Ch. 12.5 - Prob. 14E Ch. 12.5 - Prob. 15E Ch. 12.5 - Prob. 16E Ch. 12.5 - Prob. 17E Ch. 12.5 - Investigation Consider the helix represented by...Ch. 12.5 - Prob. 19E Ch. 12.5 - Prob. 20E Ch. 12.5 - Prob. 21E Ch. 12.5 - Prob. 22E Ch. 12.5 - Prob. 23E Ch. 12.5 - Prob. 24E Ch. 12.5 - Prob. 25E Ch. 12.5 - Prob. 26E Ch. 12.5 - Prob. 27E Ch. 12.5 - Prob. 28E Ch. 12.5 - Finding CurvatureIn Exercises 2936, find the...Ch. 12.5 - Prob. 30E Ch. 12.5 - Prob. 31E Ch. 12.5 - Prob. 32E Ch. 12.5 - Prob. 33E Ch. 12.5 - Prob. 34E Ch. 12.5 - Prob. 35E Ch. 12.5 - Prob. 36E Ch. 12.5 - Prob. 37E Ch. 12.5 - Prob. 38E Ch. 12.5 - Prob. 39E Ch. 12.5 - Prob. 40E Ch. 12.5 - Prob. 41E Ch. 12.5 - Prob. 42E Ch. 12.5 - Prob. 43E Ch. 12.5 - Prob. 44E Ch. 12.5 - Prob. 45E Ch. 12.5 - Prob. 46E Ch. 12.5 - Prob. 47E Ch. 12.5 - Prob. 48E Ch. 12.5 - Prob. 49E Ch. 12.5 - Prob. 50E Ch. 12.5 - Prob. 51E Ch. 12.5 - Prob. 52E Ch. 12.5 - Prob. 53E Ch. 12.5 - Prob. 54E Ch. 12.5 - Prob. 55E Ch. 12.5 - Prob. 56E Ch. 12.5 - Prob. 57E Ch. 12.5 - Prob. 58E Ch. 12.5 - Prob. 59E Ch. 12.5 - Prob. 60E Ch. 12.5 - Prob. 61E Ch. 12.5 - Motion of a Particle A particle moves along the...Ch. 12.5 - Prob. 63E Ch. 12.5 - Prob. 64E Ch. 12.5 - Prob. 65E Ch. 12.5 - Speed The smaller the curvature of a bend in a...Ch. 12.5 - Prob. 67E Ch. 12.5 - Prob. 68E Ch. 12.5 - Prob. 69E Ch. 12.5 - Prob. 70E Ch. 12.5 - Curvature Given the polar curve r=ea,a0, use the...Ch. 12.5 - Prob. 72E Ch. 12.5 - Prob. 73E Ch. 12.5 - Prob. 74E Ch. 12.5 - Prob. 75E Ch. 12.5 - Prob. 76E Ch. 12.5 - Prob. 77E Ch. 12.5 - Prob. 78E Ch. 12.5 - Prob. 79E Ch. 12.5 - Prob. 80E Ch. 12.5 - Prob. 81E Ch. 12.5 - Prob. 82E Ch. 12.5 - True or False? In Exercises 83-86, determine...Ch. 12.5 - Prob. 84E Ch. 12.5 - Prob. 85E Ch. 12.5 - Prob. 86E Ch. 12.5 - Prob. 87E Ch. 12.5 - Prob. 88E Ch. 12.5 - Prob. 89E Ch. 12.5 - Prob. 90E Ch. 12.5 - Kepler's Laws In Exercises 87-94, you are asked to...Ch. 12.5 - Prob. 92E Ch. 12.5 - Prob. 93E Ch. 12.5 - Prove Keplers Third Law: The square of the period...Ch. 12 - Domain and Continuity In Exercises 1-4, (a) find...Ch. 12 - Prob. 2RE Ch. 12 - Prob. 3RE Ch. 12 - Prob. 4RE Ch. 12 - Prob. 5RE Ch. 12 - Prob. 6RE Ch. 12 - Prob. 7RE Ch. 12 - Writing a Vector-Valued Function In Exercises 7...Ch. 12 - Prob. 9RE Ch. 12 - Prob. 10RE Ch. 12 - Prob. 11RE Ch. 12 - Prob. 12RE Ch. 12 - Prob. 13RE Ch. 12 - Prob. 14RE Ch. 12 - Prob. 15RE Ch. 12 - Prob. 16RE Ch. 12 - Prob. 17RE Ch. 12 - Prob. 18RE Ch. 12 - Prob. 19RE Ch. 12 - Prob. 20RE Ch. 12 - Prob. 21RE Ch. 12 - Prob. 22RE Ch. 12 - Prob. 23RE Ch. 12 - Prob. 24RE Ch. 12 - Prob. 25RE Ch. 12 - Prob. 26RE Ch. 12 - Prob. 27RE Ch. 12 - Prob. 28RE Ch. 12 - Prob. 29RE Ch. 12 - Prob. 30RE Ch. 12 - Prob. 31RE Ch. 12 - Evaluating a Definite Integral In Exercises 31-34,...Ch. 12 - Prob. 33RE Ch. 12 - Prob. 34RE Ch. 12 - Prob. 35RE Ch. 12 - Prob. 36RE Ch. 12 - Prob. 37RE Ch. 12 - Prob. 38RE Ch. 12 - Prob. 39RE Ch. 12 - Prob. 40RE Ch. 12 - Prob. 41RE Ch. 12 - Prob. 42RE Ch. 12 - Prob. 43RE Ch. 12 - Prob. 44RE Ch. 12 - Prob. 45RE Ch. 12 - Prob. 46RE Ch. 12 - Prob. 47RE Ch. 12 - Prob. 48RE Ch. 12 - Prob. 49RE Ch. 12 - Prob. 50RE Ch. 12 - Prob. 51RE Ch. 12 - Prob. 52RE Ch. 12 - Prob. 53RE Ch. 12 - Prob. 54RE Ch. 12 - Prob. 55RE Ch. 12 - Prob. 56RE Ch. 12 - Prob. 57RE Ch. 12 - Prob. 58RE Ch. 12 - Prob. 59RE Ch. 12 - Prob. 60RE Ch. 12 - Prob. 61RE Ch. 12 - Prob. 62RE Ch. 12 - Prob. 63RE Ch. 12 - Prob. 64RE Ch. 12 - Prob. 65RE Ch. 12 - Prob. 66RE Ch. 12 - Prob. 67RE Ch. 12 - Prob. 68RE Ch. 12 - Prob. 69RE Ch. 12 - Prob. 70RE Ch. 12 - Prob. 71RE Ch. 12 - Prob. 72RE Ch. 12 - Prob. 73RE Ch. 12 - Prob. 1PS Ch. 12 - Prob. 2PS Ch. 12 - Prob. 3PS Ch. 12 - Prob. 4PS Ch. 12 - Cycloid Consider one arch of the cycloid...Ch. 12 - Prob. 6PS Ch. 12 - Prob. 7PS Ch. 12 - Prob. 8PS Ch. 12 - Prob. 9PS Ch. 12 - Prob. 10PS Ch. 12 - Prob. 11PS Ch. 12 - Prob. 12PS Ch. 12 - Prob. 13PS Ch. 12 - Ferris Wheel You want to toss an object to a...

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To prove: The limit lim x → c [ r ( t ) ⋅ u ( t ) ] = lim x → c r ( t ) ⋅ lim x → c u ( t ) .

Solution Summary: The author explains how to prove the limit undersetxto cmathrmlimr(t)cdot

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To prove: The limit limx→c[r(t)⋅u(t)]=limx→cr(t)⋅limx→cu(t).

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Chapter 12 Solutions