Answer:
To find the radius of curvature (\(R\)) for the curve \(y = x\) at a given point \(x = 1\), we can use the formula:
\[ R = \frac{[1 + (y')^2]^{3/2}}{y''} \]
Here, \(y'\) is the first derivative of \(y\) with respect to \(x\), and \(y''\) is the second derivative.
For the curve \(y = x\), the first derivative \(y'\) is \(1\), and the second derivative \(y''\) is \(0\).
Substituting these values into the formula:
\[ R = \frac{[1 + 1^2]^{3/2}}{0} \]
Since the denominator is \(0\), it indicates that the radius of curvature is infinite for the curve \(y = x\) at \(x = 1\).
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Answer:
To find the radius of curvature (\(R\)) for the curve \(y = x\) at a given point \(x = 1\), we can use the formula:
\[ R = \frac{[1 + (y')^2]^{3/2}}{y''} \]
Here, \(y'\) is the first derivative of \(y\) with respect to \(x\), and \(y''\) is the second derivative.
For the curve \(y = x\), the first derivative \(y'\) is \(1\), and the second derivative \(y''\) is \(0\).
Substituting these values into the formula:
\[ R = \frac{[1 + 1^2]^{3/2}}{0} \]
Since the denominator is \(0\), it indicates that the radius of curvature is infinite for the curve \(y = x\) at \(x = 1\).