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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

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BuyFindarrow_forward

Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem
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Center of Curvature Let C be a curve given by y = f ( x ) .

Let K be the curvature ( K 0 ) at the point P ( x 0 , y 0 ) and let

z = 1 + f ' ( x 0 ) 2 f " ( x 0 )

Show that the coordinates ( α , β ) of the center of curvature at P are ( ( α , β ) = ( x 0 f ' ( x 0 ) z 0 , y 0 = z ) .

To determine

To prove: The co-ordinates (α,β) of the center of curvature at P are (α,β)=(x0f(x0)z,y0+z ).

Explanation

Given:

The curve C is given by y=f(x), K is the curvature at point P(x0,y0) and z=1+f(x0)2f(x0).

Proof:

P(x0,y0) Is the point on curve y=f(x).

Let (α,β) be the center of curvature.

The radius of curvature is 1K.

Since,

y=f(x)

So, slope of normal line at (x0,y0) is 1f(x0).

Equation of normal line,

yy0 = 1f(x0)(xx0)

Since, (α,β) is on normal line.

Therefore,

f(x0)(βy0)=αx0 ...... (1)

And, (x0,y0) lies on the circle.

Therefore,

(x0α)2+(y0β)2=(1k)2=[(1+f(x0)2)32|f(x0)|]2 .....

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Sect-12.1 P-11ESect-12.1 P-12ESect-12.1 P-13ESect-12.1 P-14ESect-12.1 P-15ESect-12.1 P-16ESect-12.1 P-17ESect-12.1 P-18ESect-12.1 P-19ESect-12.1 P-20ESect-12.1 P-21ESect-12.1 P-22ESect-12.1 P-23ESect-12.1 P-24ESect-12.1 P-25ESect-12.1 P-26ESect-12.1 P-27ESect-12.1 P-28ESect-12.1 P-29ESect-12.1 P-30ESect-12.1 P-31ESect-12.1 P-32ESect-12.1 P-33ESect-12.1 P-34ESect-12.1 P-35ESect-12.1 P-36ESect-12.1 P-37ESect-12.1 P-38ESect-12.1 P-39ESect-12.1 P-42ESect-12.1 P-41ESect-12.1 P-40ESect-12.1 P-43ESect-12.1 P-44ESect-12.1 P-45ESect-12.1 P-46ESect-12.1 P-47ESect-12.1 P-48ESect-12.1 P-49ESect-12.1 P-50ESect-12.1 P-51ESect-12.1 P-52ESect-12.1 P-53ESect-12.1 P-54ESect-12.1 P-55ESect-12.1 P-56ESect-12.1 P-57ESect-12.1 P-58ESect-12.1 P-59ESect-12.1 P-60ESect-12.1 P-61ESect-12.1 P-62ESect-12.1 P-63ESect-12.1 P-64ESect-12.1 P-65ESect-12.1 P-66ESect-12.1 P-67ESect-12.1 P-68ESect-12.1 P-69ESect-12.1 P-70ESect-12.1 P-71ESect-12.1 P-72ESect-12.1 P-73ESect-12.1 P-74ESect-12.1 P-75ESect-12.1 P-76ESect-12.1 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P-93ESect-12.5 P-94ESect-12.5 P-95ESect-12.5 P-96ECh-12 P-1RECh-12 P-2RECh-12 P-3RECh-12 P-4RECh-12 P-5RECh-12 P-6RECh-12 P-7RECh-12 P-8RECh-12 P-9RECh-12 P-10RECh-12 P-11RECh-12 P-12RECh-12 P-13RECh-12 P-14RECh-12 P-15RECh-12 P-16RECh-12 P-17RECh-12 P-18RECh-12 P-19RECh-12 P-20RECh-12 P-21RECh-12 P-22RECh-12 P-23RECh-12 P-24RECh-12 P-25RECh-12 P-26RECh-12 P-27RECh-12 P-28RECh-12 P-29RECh-12 P-30RECh-12 P-31RECh-12 P-32RECh-12 P-33RECh-12 P-34RECh-12 P-35RECh-12 P-36RECh-12 P-37RECh-12 P-38RECh-12 P-39RECh-12 P-40RECh-12 P-41RECh-12 P-42RECh-12 P-43RECh-12 P-44RECh-12 P-45RECh-12 P-46RECh-12 P-47RECh-12 P-48RECh-12 P-49RECh-12 P-50RECh-12 P-51RECh-12 P-52RECh-12 P-53RECh-12 P-54RECh-12 P-55RECh-12 P-56RECh-12 P-57RECh-12 P-58RECh-12 P-59RECh-12 P-60RECh-12 P-61RECh-12 P-62RECh-12 P-63RECh-12 P-64RECh-12 P-65RECh-12 P-66RECh-12 P-67RECh-12 P-68RECh-12 P-69RECh-12 P-70RECh-12 P-71RECh-12 P-1PSCh-12 P-2PSCh-12 P-3PSCh-12 P-4PSCh-12 P-5PSCh-12 P-6PSCh-12 P-7PSCh-12 P-8PSCh-12 P-9PSCh-12 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