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=ex The derivative of e xise x itself, so both the first derivative dxdy and the second derivative dx2d2y 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)
.Substitutingx=0
into both derivatives, we find that
dxdy=e0=1anddx2d2y=e0=1
.
Step 3: Apply the formula for radius of curvature
The formula for the radius of curvature ρ for a curve y=f(x) is given by ρ \frac {[1+(dxdy)2] ^{3/2}}{\left. \right. ∣ dx2d2y |$
. 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 dxdy=1 and dx2d2y=1 into the formula for ρ This simplifies to (1+1)3/2, which is 23/2 .Finally,23/2 can be written as 22 $
.