Question

Use the theorem given below to find the curvature of r(t)=t^5j+t^4k. Theorem: The curvature of the curve given by the vector function r is kappa (t)=frac |r'(t)* r''(t)||r'(t)|^3. K(t)=□ ×

Answer 1

To find the curvature of the curve defined by the vector function r(t) = t^5j + t^4k, use the curvature formula kappa(t) = |r'(t) x r''(t)| / |r'(t)|^3.

Step-by-step explanation:

To find the curvature of the curve defined by the vector function r(t) = t^5j + t^4k, we can use the theorem given: kappa(t) = |r'(t) x r''(t)| / |r'(t)|^3.

First, find the first derivative of r(t) by differentiating each component with respect to t. r'(t) = 0j + 4t^3k.

Next, find the second derivative of r(t) by differentiating each component of r'(t) with respect to t. r''(t) = 0j + 12t^2k.

Substitute these values into the curvature formula and simplify to find the curvature: kappa(t) = |(0j + 4t^3k) x (0j + 12t^2k)| / |(0j + 4t^3k)|^3.

The question asks to find the curvature of the vector function r(t) = t^5j + t^4k using the given theorem for curvature, which is κ (t) = |r'(t) × r''(t)| / |r'(t)|^3. To solve this, we need to calculate the first derivative r'(t) and the second derivative r''(t) of the vector function, and then evaluate the cross product and magnitudes necessary to apply the formula.

Calculate the first derivative, r'(t).

Calculate the second derivative, r''(t).

Find the cross product r'(t) × r''(t).

Compute the magnitudes |r'(t)| and |r'(t) × r''(t)|.

Apply the formula for curvature to find κ(t).