Answer to Question #129635 in Calculus for abhi

Question

Find the curvature of r(t) at the point (1, 1, 1). Given that r(t) = (t, t^2, t^3).

Answer 1

Let's find the value for t where r(t) = P

t=1

curvate=|r'(t) x r"(t)|/|r'(t)|³ (x mean vector product of r'(t) and r"(t))

r'(t)=(1,2t, 3t²)

r''(t)=(0,2,6t)

r'(t) x r"(t)=(6t² , -6t, 2)

|r'(t) x r"(t)|=(36t⁴+36t²+4)^1/2

|r'(t)|=(1+4t²+9t⁴)^1/2

|r'(t)|³=(1+4t²+9t⁴)^3/2

curvate=(36t⁴+36t²+4)^1/2/(1+4t²+9t⁴)^3/2

Let's find curvate in t=1

curvate=(36+36+4)^1/2/(1+4+9)^3/2=76^1/2/14^3/2