Explanation Given information: The curve is given as a function of x such that, y = x 2 − x 3 Formula used: The curvature of a curve is given by, κ ( x ) = | f ' ' ( x ) | ( 1 + f ' ( x ) 2 ) 3 / 2 The normal vector N , N ( x ) = T ' ( x ) ‖ T ' ( x ) ‖ where T ( x ) is the unit tangent vector. The vector for the center of the osculating circle. r ( x 0 ) + 1 κ ( x ) N Let the vector function be, r ( x ) = 〈 x , x 2 − x 3 〉 The first derivative of y is, y ' = 2 x − 3 x 2 and, The second derivative od y is, y ' ' = 2 − 6 x The curvature at x = 1 can be found by the formula, κ ( x ) = | y ' ' ( x ) | ( 1 + y ' ( x ) 2 ) 3 / 2 = | 2 − 6 x | ( 1 + ( 2 x − 3 x 2 ) 2 ) 3 / 2 κ ( x = 1 ) = | 2 − 6 ( 1 ) | ( 1 + ( 2 ( 1 ) − 3 ( 1 ) 2 ) 2 ) 3 / 2 = | 2 − 6 | ( 1 + ( 2 − 3 ) 2 ) 3 / 2 = | − 4 | ( 1 + ( − 1 ) 2 ) 3 / 2 = 4 ( 2 ) 3 / 2 = 4 2 2 = 2 2 κ ( 1 ) = 2 The derivative of the vector function is, r ' ( x ) = 〈 1 ,<

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

See similar textbooks

Concept explainers

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in ter…

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which …

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the …

Videos

Question

Chapter 14, Problem 36CRE

To determine

To find: The equation of the osculating circle of the given curve at x=1 .

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 14, Problem 35CRE

Chapter 14, Problem 37CRE

Chapter 14 Solutions

CALCULUS (CLOTH)

Ch. 14.1 - Prob. 1PQ Ch. 14.1 - Prob. 2PQ Ch. 14.1 - Prob. 3PQ Ch. 14.1 - Prob. 4PQ Ch. 14.1 - Prob. 5PQ Ch. 14.1 - Prob. 6PQ Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - 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 given curve at x = 1 .

Solution Summary: The author explains that the equation of the osculating circle at x=1 can be found by the formula.

Concept explainers

Videos

To find: The equation of the osculating circle of the given curve at x=1 .

Want to see the full answer?

Chapter 14 Solutions