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Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781337552516

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Section
BuyFindarrow_forward

Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781337552516
Chapter 12.5, Problem 44E
Textbook Problem
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Finding Curvature in Rectangular Coordinates In Exercises 41-48, find the curvature and radius of curvature of the plane curve at the given value of r.

y = 2 9 x 2 , x = 0

To determine

To calculate: The curvature and radius of the curvature of the plane curve in the rectangular coordinates y=29x2,x=0 at the given value of x.

Explanation of Solution

Given:

The rectangular coordinate is y=29x2, x=0.

Formula Used:

The curvature K at the point (x,y) is K=|y|[1+(y)2]32 and radius of curvature r is r=1K.

Chain rule: derivative of f(g(x)) is f(g(x))=f(g(x))g(x).

Quotient rule: derivative of f(x)g(x)=f(x)g(x)f(x)g(x)g(x)2.

Calculation:

Consider function y=29x2,

Differentiating the function with respect to x by applying chain rule,

y=2(2x)29x2=2x9x2

Now, again differentiate with respect to x by using quotient rule,

y=29x2+2xx9x

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Chapter 12 Solutions

Calculus: Early Transcendental Functions
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Ch. 12.1 - Evaluating a function In Exercises 11 and 12...Ch. 12.1 - Evaluating a function In Exercises 11 and 12...Ch. 12.1 - Writing a Vector-Valued Function In Exercises...Ch. 12.1 - Writing a Vector-Valued Function In Exercises...Ch. 12.1 - Writing a Vector-Valued Function In Exercises...Ch. 12.1 - Writing a Vector-Valued Function In Exercises...Ch. 12.1 - Think About It In Exercises 17 and 18, find r( t)...Ch. 12.1 - Think About It In Exercises 17 and IS, find...Ch. 12.1 - Matching In Exercises 19-22. match the equation...Ch. 12.1 - Matching In Exercises 19-22. match the equation...Ch. 12.1 - Matching In Exercises 19-22. match the equation...Ch. 12.1 - Matching In Exercises 19-22, match the equation...Ch. 12.1 - Think about it In Exercises 23-30 sketch the plane...Ch. 12.1 - Think about it In Exercises 23-30 sketch the plane...Ch. 12.1 - Think about it In Exercises 23-30 sketch the plane...Ch. 12.1 - Think about it In Exercises 23-30 sketch the plane...Ch. 12.1 - Think about it In Exercises 23-30 sketch the plane...Ch. 12.1 - Think about it In Exercises 23-30 sketch the plane...Ch. 12.1 - Think about it In Exercises 23-30 sketch the plane...Ch. 12.1 - Think about it In Exercises 23-30 sketch the plane...Ch. 12.1 - Sketching a space curve In Exercises 31-38 sketch...Ch. 12.1 - Sketching a space curve In Exercises 31-38 sketch...Ch. 12.1 - Sketching a space curve In Exercises 31-38 sketch...Ch. 12.1 - Sketching a space curve In Exercises 31-38 sketch...Ch. 12.1 - Sketching a space curve In Exercises 31-38 sketch...Ch. 12.1 - Sketching a space curve In Exercises 31-38 sketch...Ch. 12.1 - Sketching a space curve In Exercises 31-38 sketch...Ch. 12.1 - Sketching a space curve In Exercises 31-38 sketch...Ch. 12.1 - Identifying Common Curve In Exercises 39 and 40...Ch. 12.1 - Identifying Common Curve In Exercises 39 and 40...Ch. 12.1 - Transformation of a vector valued valued in...Ch. 12.1 - Transformations of Vector-Valued Functions In...Ch. 12.1 - Writing a Transformation In exercises 43-46...Ch. 12.1 - Writing a Transformation In exercises 43-46...Ch. 12.1 - Writing a Transformation In exercises 43-46...Ch. 12.1 - Writing a Transformation In exercises 43-46...Ch. 12.1 - Representing a graph by vector valued function In...Ch. 12.1 - Representing a graph by vector valued function In...Ch. 12.1 - Representing a graph by vector valued function In...Ch. 12.1 - Representing a graph by vector valued function In...Ch. 12.1 - Representing a graph by vector valued function In...Ch. 12.1 - Representing a graph by vector valued function In...Ch. 12.1 - Representing a graph by vector valued function In...Ch. 12.1 - Representing a graph by vector valued function In...Ch. 12.1 - Representing a Graph by Vector-Valued FunctionIn...Ch. 12.1 - Representing a Graph by Vector-Valued FunctionIn...Ch. 12.1 - Representing a Graph by Vector-Valued FunctionIn...Ch. 12.1 - Representing a Graph by Vector-Valued FunctionIn...Ch. 12.1 - Representing a Graph by Vector-Valued FunctionIn...Ch. 12.1 - Representing a Graph by Vector-Valued FunctionIn...Ch. 12.1 - Representing a Graph by Vector-Valued FunctionIn...Ch. 12.1 - Representing a Graph by Vector-Valued Function In...Ch. 12.1 - Sketching a Curve Show that the vector-valued...Ch. 12.1 - Sketching a Curve Show that the vector-valued...Ch. 12.1 - Finding a Limit In Exercises 65-70, find the limit...Ch. 12.1 - Finding a Limit In Exercises 65-70, find the limit...Ch. 12.1 - Finding a Limit In Exercises 65-70, find the limit...Ch. 12.1 - Finding a Limit In Exercises 65-70, find the limit...Ch. 12.1 - Finding a Limit In Exercises 65-70, find the limit...Ch. 12.1 - Finding a Limit In Exercises 65-70, find the limit...Ch. 12.1 - Continuity of a Vector-Valued Function In...Ch. 12.1 - Continuity of a Vector-Valued Function In...Ch. 12.1 - Continuity of a Vector-Valued Function In...Ch. 12.1 - Continuity of a Vector-Valued Function In...Ch. 12.1 - Continuity of a Vector-Valued Function In...Ch. 12.1 - Continuity of a Vector-Valued Function In...Ch. 12.1 - EXPLORING CONCEPTS Think About It Consider...Ch. 12.1 - EXPLORING CONCEPTS Think About It The curve...Ch. 12.1 - EXPLORING CONCEPTS Continuity of a Vector-Valued...Ch. 12.1 - Comparing Functions Which of the following...Ch. 12.1 - Staircase The outer bottom edge of a staircase is...Ch. 12.1 - HOW DO YOU SEE IT? 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What...Ch. 12.5 - Curvature Given a twice-differentiable function...Ch. 12.5 - Investigation Consider the function f(x)=x4x2. (a)...Ch. 12.5 - Motion of a Particle A particle moves along the...Ch. 12.5 - Investigation Find all a and b such that the two...Ch. 12.5 - HOW DO YOU SEE IT? Using the graph of the ellipse,...Ch. 12.5 - Sphere and Paraboloid A sphere of radius 4 is...Ch. 12.5 - The smaller the curvature of a bend in a road, the...Ch. 12.5 - Center of Curvature Let C be a curve given by...Ch. 12.5 - Center of Curvature Use the result of Exercise 67...Ch. 12.5 - Curvature A curve C is given by the polar equation...Ch. 12.5 - Curvature Use the result of Exercise 69 to find...Ch. 12.5 - Curvature Given the polar curve r=ea,a0, use the...Ch. 12.5 - Curvature at the Pole Show. that the formula for...Ch. 12.5 - Curvature at the Pole In Exercises 73 and 74, use...Ch. 12.5 - Curvature at the Pole In Exercises 73 and 74, use...Ch. 12.5 - Proof For a smooth curve given by the parametric...Ch. 12.5 - Horizontal Asymptotes Use the result of Exercise...Ch. 12.5 - Curvature of a Cycloid Use the result of Exercise...Ch. 12.5 - Tangential and Normal Components of Acceleration...Ch. 12.5 - Frictional Force A 5500-pound vehicle is driven at...Ch. 12.5 - Frictional Force A 6400-pound vehicle is driven at...Ch. 12.5 - Curvature Verify that the curvature at any point...Ch. 12.5 - Formulas for Curvature Use the definition of...Ch. 12.5 - True or False? In Exercises 83-86, determine...Ch. 12.5 - True or False? In Exercises 83-86, determine...Ch. 12.5 - True or False? In Exercises 83-86, determine...Ch. 12.5 - True or False? In Exercises 83-86, determine...Ch. 12.5 - Keplers Laws In Exercises 87-94, you are asked to...Ch. 12.5 - Keplers Laws In Exercises 87-94, you are asked to...Ch. 12.5 - Keplers Laws In Exercises 87-94, you are asked to...Ch. 12.5 - Show that rGMLrr=e is a constant vector.Ch. 12.5 - Prove Keplers First Law: Each planet moves in an...Ch. 12.5 - Assume that the elliptical orbit r=ed1+ecos is in...Ch. 12.5 - Prove Keplers Second Law: Each ray from the sun to...Ch. 12.5 - Prove Keplers Third Law: The square of the period...Ch. 12 - Domain and Continuity In Exercises 1-4, (a) And...Ch. 12 - Domain and Continuity In Exercises 1-4, (a) And...Ch. 12 - Domain and Continuity In Exercises 1-4, (a) And...Ch. 12 - Domain and Continuity In Exercises 1-4, (a) And...Ch. 12 - Evaluating a Function In Exercises 5 and 6....Ch. 12 - Evaluating a Function In Exercises 5 and 6....Ch. 12 - Writing a Vector-Valued Function In Exercises 7...Ch. 12 - Writing a Vector-Valued Function In Exercises 7...Ch. 12 - Sketching a Curve In Exercises 9-12, sketch the...Ch. 12 - 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Finding the Unit Tangent Vector In Exercises 43...Ch. 12 - Finding a Tangent Line In Exercises 45 and 46,...Ch. 12 - Finding a Tangent Line In Exercises 45 and 46,...Ch. 12 - Finding the Principal Unit Normal Vector In...Ch. 12 - Finding the Principal Unit Normal Vector In...Ch. 12 - Finding the Principal Unit Normal Vector In...Ch. 12 - Finding the Principal Unit Normal Vector In...Ch. 12 - Finding Tangential and Normal Components of...Ch. 12 - Finding Tangential and Normal Components of...Ch. 12 - Finding Tangential and Normal Components of...Ch. 12 - Finding Tangential and Normal Components of...Ch. 12 - r(t)=2ti3tj,[0,5] Finding the Arc Length of a...Ch. 12 - Finding the Arc Length of a Plane Curve In...Ch. 12 - Finding the Arc Length of a Plane Curve In...Ch. 12 - Finding the Arc Length of a Plane Curve In...Ch. 12 - Finding the Arc Length of a Curve in Space In...Ch. 12 - Finding the Arc Length of a Curve in Space In...Ch. 12 - Finding the Arc Length of a Curve in Space In...Ch. 12 - 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Cardioid Consider the cardioid r=1cos,02 as shown...Ch. 12 - Proof If r(t) is a nonzero differentiable function...Ch. 12 - Satellite A communications satellite moves in a...Ch. 12 - Binormal Vector In Exercises 9-11, use the...Ch. 12 - Binormal Vector In Exercises 9-11, use the...Ch. 12 - Binormal Vector In Exercises 9-11, use the...Ch. 12 - Exit Ramp A highway has an exit ramp that begins...Ch. 12 - Arc Length and Curvature Consider the...Ch. 12 - Ferris Wheel You want to toss an object to a...

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