   Chapter 12.5, Problem 51E

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# Maximum Curvature In Exercises 49-54, (a) find the point on the curve at which the curvature is a maximum and (b) find the limit of the curvature as x → ∞ . y = x 2 / 3

(a)

To determine

To calculate: The point on the curve y=x23 at which the curvature is a maximum.

Explanation

Given:

The provided function is: y=x23.

Formula used:

The formula of curvature

k=|y|[1+(y)2]32

Calculation:

Consider the provided function:

y=x23

Differentiate the function with respect to x.

y=23x13=29x43

Now, substitute the values in formula.

k=|(29)x43[1+(49)x23]32|=|29×27x431[9x23+4]32|=|6x13[9x23+4]32| ……(1)

Now, differentiate k with respect to x

(b)

To determine

To calculate: The limit of the curvature as x→∞.

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