   Chapter 13, Problem 14RE

Chapter
Section
Textbook Problem

# Find an equation of the osculating circle of the curve y = x4 − x2 at the origin. Graph both the curve and its osculating circle.

To determine

To find: An equation of the osculating circle of the curve y=x4x2 at the origin.

Explanation

Given data:

Equation of the curve is y=x4x2 .

The equation of the osculating circle is to be determined at the point of origin.

Formula used:

Write the expression for equation of osculating circle.

(xh)2+(yk)2=ρ2 (1)

Here,

(h,k) is the center of the osculating circle and

ρ is the radius of the osculating circle.

Write the expression for radius of osculating circle (ρ) .

ρ=1k(x)|at the specified point (2)

Here,

k(x)|at the specified point is the curvature at the specified point.

Write the expression to find the curvature k(x) for a curve.

k(x)=|y||1+(y)2|3 (3)

Here,

y is the derivative of y and

y is the derivative of y .

Write the required differential formula to find y and y as follows.

ddxxn=nxn1

Calculation of y :

Differentiate the expression y=x4x2 with respect to x as follows.

ddx(y)=ddx(x4x2)

Rewrite and compute the expression as follows.

y=ddx(x4)+ddx(x2)=4x41+(2x21)=4x32x

Calculation of y :

Differentiate the expression (4x32x) with respect to x as follows.

ddx(y)=ddx(4x32x)

Rewrite and compute the expression as follows.

y=ddx(4x3)+ddx(2x)=4(3x31)+(2)(1)=12x22

Calculation of curvature k(x) :

Substitute (4x32x) for y and (12x22) for y in equation (3),

k(x)=|12x22||1+

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