Find the radius of curvature for the curve y = e^x at the point (0,1).

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Step-by-Step Solution

Step 1: Find the first and second derivatives

To find the radius of curvature, we first need to calculate the first and second derivatives of the given function y = e^x$. The derivative of e^x is e^x itself, so both the first derivative \frac{dy}{dx} and the second derivative \frac{d^2y}{dx^2} are equal to e^x$.

Step 2: Evaluate derivatives at the given point

Next, we evaluate the first and second derivatives at the given point (0,1)$. Substituting x=0 into both derivatives, we find that \frac{dy}{dx} = e^0 = 1 and \frac{d^2y}{dx^2} = e^0 = 1$.

Step 3: Apply the formula for radius of curvature

The formula for the radius of curvature \rho for a curve y = f(x) is given by \rho = $\frac${$\left[1 + \left(\frac{dy}{dx}\right)^2\right]$^{3/2}}{$\left$|$\frac{d^2y}{dx^2}$\right$|}$. This formula relates the first and second derivatives to the curvature of the curve.

Step 4: Substitute values and calculate

Now, we substitute the calculated values of \frac{dy}{dx} = 1 and \frac{d^2y}{dx^2} = 1 into the formula for \rho$. This simplifies to (1+1)^{3/2}$, which is 2^{3/2}$. Finally, 2^{3/2} can be written as 2\sqrt{2}$.