Question

Find the curvature κ of the plane curve y=2x³/x at x=2.
A) 2
B) 4
C) 1
D) 3

Answer 1

To calculate the curvature κ of the plane curve y=2x² at x=2, we find the derivatives y'=4x and y''=4, and then we use the provided formula. The curvature at x=2 is 4 / 65^(3/2), which is not an integer and does not match the given answer choices, suggesting a possible error.

Step-by-step explanation:

To find the curvature κ of the plane curve y = 2x³/x at x=2, we first simplify the given function to y = 2x². The next step is to compute the first and second derivatives of y with respect to x. The first derivative y' is 4x and the second derivative y'' is 4.

Then we use the formula for the curvature of a curve at a point x, which is given by:

κ = |y''| / (1 + (y')²)^(3/2)

At x = 2, we have:

κ = |4| / (1 + (4*2)²)^(3/2)

Substituting all known values, this simplifies to:

κ = 4 / (1 + 64)^(3/2) = 4 / 65^(3/2)

When evaluated, the curvature κ at x=2 is 4 / 65^(3/2), which none of the given answer choices A) 2 B) 4 C) 1 D) 3 matches. Therefore, it appears there is an error in the given answer choices or in the calculation. It should be noted that the curvature value obtained is not an integer.