Final answer:
The curvature κ of the plane curve y = 2x² - 4x + 5 at x = 3 is κ = 6.
Explanation:
The curvature of a curve is the measure of the rate of change of its direction at a point on the curve. We can calculate the curvature of a curve at a certain point by using the formula κ=|y''|/[1+(y')²]³/². To find the curvature of the plane curve y = 2x² - 4x + 5 at x = 3, we first need to find the derivatives of y with respect to x.
The first derivative of y with respect to x is y'= 4x - 4 and the second derivative of y with respect to x is y''= 4. Substituting the values of y' and y'' in the curvature formula, we get κ=|4|/[1+(4x - 4)²]³/². At x = 3, κ = |4|/[1+(12 - 4)²]³/² = 4/[1+64]³/² = 4/1728 = 6. Therefore, the curvature of the plane curve y = 2x² - 4x + 5 at x = 3 is κ = 6.