Question

The formula k(x) = |f’’(x)|/ [1+(f’(x))^2]^3/2 expresses the curvature of a twice-differentiable plane curve as a function of x. use this formula to find the curvature function of the following curve: f(x)= In(2 sec x) for -π/2

Answer 1

Explanation:

We can use the given formula to find the curvature function k(x) for the curve f(x) = ln(2sec(x)) as follows:

First, we find the first and second derivatives of f(x):

f'(x) = 2tan(x)

f''(x) = 2sec^2(x)

Next, we substitute these derivatives into the formula for curvature:

k(x) = |f''(x)| / [1 + (f'(x))^2]^(3/2)

k(x) = |2sec^2(x)| / [1 + (2tan(x))^2]^(3/2)

k(x) = 2sec^2(x) / [1 + 4tan^2(x)]^(3/2)

Now, we can use the identity sec^2(x) = 1 + tan^2(x) to simplify the formula for k(x):

k(x) = 2(1 + tan^2(x)) / [1 + 4tan^2(x)]^(3/2)

k(x) = 2 / [1 + 4tan^2(x)]^(1/2) * (1 + tan^2(x)) / [1 + tan^2(x)]

k(x) = 2 / [1 + 4tan^2(x)]^(1/2) * sec^2(x)

Therefore, the curvature function for the curve f(x) = ln(2sec(x)) is:

k(x) = 2 / [1 + 4tan^2(x)]^(1/2) * sec^2(x)

for -π/2 < x < π/2.