   Chapter 12.5, Problem 63E

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# Investigation Consider the function f ( x ) = x 4 − x 2 .(a) Use a computer algebra system to find the curvature K of the curve as a function of x.(b) Use the result of part (a) to find the circles of curvature to the graph of f when x = 0 and x = 1 . Use a computer algebra system to graph the function and the two circles of curvature.(c) Graph the function K(x) and compare it with the graph of f(x). For example, do the extrema of f and K occur at the same critical numbers? Explain your reasoning.

(a)

To determine

To calculate: The curvature K of the curve f(x)=x4x2 as a function of x by the use of computer algebra system.

Explanation

Given:

The equation of the curve f(x)=x4x2.

Formula used:

The formula for the curvature K of curve is K=|f(x)|[1+(f(x))2]32.

Calculation:

Consider the provided curve f(x)=x4x2

(b)

To determine

To calculate: The circle of curvature to the graph of f when x=0 and x=1 also graph the function and two circles by the use of computer algebra system.

(c)

To determine

To graph: The function of the curvature K(x) and compare it with the graph of the function f(x).

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