Question

If the curve C is parametrized by ~r(t) = 1, t^2 , t^3 , what is the rate of change of the arc length function when t = 1?

Answer 1

The rate of change of the arc length function can be found by taking the derivative of the position vector ~r(t) = 1, t^2 , t^3 with respect to the parameter t. The derivative ~r'(t) is computed and its magnitude is used to find the rate of change of the arc length function. Substituting t = 1 into the rate of change function gives the desired result.

Step-by-step explanation:

The rate of change of the arc length function can be found by taking the derivative of the position vector ~r(t) = 1, t^2 , t^3 with respect to the parameter t. Let's denote the arc length function as S(t). Using the formula for arc length, we have:

S(t) = ∫ ||~r'(t)|| dt, where ~r'(t) is the derivative of ~r(t).

Computing the derivative ~r'(t) = 0, 2t, 3t^2, and taking its magnitude ||~r'(t)|| = √(0^2 + (2t)^2 + (3t^2)^2) = √(4t^2 + 9t^4), we can find the rate of change of the arc length function S'(t):

S'(t) = ||~r'(t)||.

Substitute t = 1 into S'(t) to find the rate of change of the arc length function when t = 1.