The formula for the radius of curvature for a curve y = f(x) is given by [1 + ((dy)/(dx))²] ³/2 | (d²y)/(dx²) |. This formula relates the first and second derivatives to the curvature of the curve.

The formula for the radius of curvature for a curve y = f(x) is given by [1 + ((dy)/(dx))²] ³/2 | (d²y)/(dx²) |. This formula relates the first and second derivatives to the curvature of the curve.

Now, we substitute the calculated values of (dy)/(dx) = 1 and (d²y)/(dx²) = 1 into the formula for This simplifies to (1+1)³/2, which is 2³/2. Finally, 2³/2 can be written as 2√2.

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

Answer: The radius of curvature for the curve $y = e^x$ at the point (0,1) is 2 $\sqrt{2}$ .

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}}{ $\left. \right.$ | $\frac{d^2y}{dx^2}$ $\left. \right.$ | . 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 $2\sqrt{2}$ .

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