Final answer:
To find the curvature of the plane curve y = 2e^(x/3) at x = 3, we need to find the second derivative of y, evaluate it at x = 3, and use the formula for curvature.
Step-by-step explanation:
To find the curvature -κ of the plane curve y = 2e^(x/3) at x = 3, we first need to find the second derivative of y with respect to x.
The second derivative can be found by taking the derivative of the first derivative. In this case, the first derivative of y is y' = (2/3)e^(x/3), and the second derivative is y'' = (2/9)e^(x/3).
Next, we can evaluate the second derivative at x = 3 to find y''. Substituting x = 3 in the equation y'' = (2/9)e^(x/3), we get y'' = (2/9)e^(1) = (2/9)e.
The curvature -κ of the plane curve at x = 3 is equal to the absolute value of y'' divided by the square of (1 + (y')^2) raised to the power of 3/2. Substituting the values, we have -κ = |(2/9)e| / (1 + [(2/3)e^(x/3)]²)^(3/2).