Explanation Given: The function y = x and the point x = 4 . Calculation: Find the first and the second derivative of the function y = x . The first and the second derivative of the function y = x are y ′ = 1 2 x and y ″ = − 1 4 x 3 / 2 respectively. So, the curvature of the function can be found as shown below. k ( x ) = | y ″ | ( 1 + ( y ′ ) 2 ) 3 / 2 = | − 1 4 x 3 / 2 | ( 1 + ( 1 2 x ) 2 ) 3 / 2 So, at the point x = 4 , the curvature of the function is: k ( 4 ) = | − 1 32 | ( 1 + ( 1 2 4 ) 2 ) 3 / 2 = 1 / 32 ( 17 16 ) 3 / 2 = 1 / 32 17 17 / 64 = 2 17 17 Therefore, radius of curvature of the function at the point x = 4 is r ( 4 ) = 1 k ( 4 ) = 17 17 2 . The parametric equation of the function is r ( t ) = 〈 t , t 〉 . The tangent vector is given as r ′ ( t ) = 〈 1 , 1 2 t 〉 . The vector perpendicular to r ′ ( t ) will be of the form ± 〈 1 2 t , − 1 〉 . So, the unit normal vector is N ( t ) = ± 〈 1 2 t , − 1 〉 ( 1 2 t ) 2 + 1 The normal vector is pointing towards negativey -axis, so it must be N ( t ) = 〈 1 2 t , − 1 〉 1 4 t + 1

4th Edition

ISBN: 9781319050740

Author: Jon Rogawski, Colin Adams, Robert Franzosa

Publisher: W. H. Freeman

See similar textbooks

Concept explainers

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the se…

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integratio…

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects th…

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.

Videos

Question

Chapter 13, Problem 37CRE

To determine

The equation of the osculating circle of the curve y=x at the point x=4 or state that the osculating circle does not exist.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 13, Problem 36CRE

Chapter 13, Problem 38CRE

Students have asked these similar questions

Please focus on problem ii.

Please focus on problem x.

Please focus on vii.

Chapter 13 Solutions

Calculus: Early Transcendentals

Ch. 13.1 - Prob. 1PQ Ch. 13.1 - Prob. 2PQ Ch. 13.1 - Prob. 3PQ Ch. 13.1 - Prob. 4PQ Ch. 13.1 - Prob. 5PQ Ch. 13.1 - Prob. 6PQ Ch. 13.1 - Prob. 1E Ch. 13.1 - Prob. 2E Ch. 13.1 - Prob. 3E Ch. 13.1 - Prob. 4E

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

The equation of the osculating circle of the curve y = x at the point x = 4 or state that the osculating circle does not exist.

Solution Summary: The author states that the osculating circle of the curve y=sqrtx is not present.

Concept explainers

Videos

The equation of the osculating circle of the curve y=x at the point x=4 or state that the osculating circle does not exist.

Want to see the full answer?

Chapter 13 Solutions