Final answer:
To calculate the curvature of the curve at a given point, first find the first and second derivatives of the position vector, then use the curvature formula involving these derivatives and the vector cross product.
Step-by-step explanation:
To find the curvature of the curve r(t) = 9t, t², t³ at the point (9, 1, 1), you need to follow a sequence of steps involving differentiation and the use of formulas relating to the curvature of a parametric curve. Firstly, compute the first and second derivatives of the given position vector r(t) with respect to time t. The curvature k at a given point on the curve can then be found using the formula k = |r'(t) × r''(t)| / |r'(t)|³. By substituting the values of r'(t) and r''(t) into this formula, we can calculate the curvature at the specific point of interest.
The given expression in the reference does not match the curvature calculation, but serves to remind us that we must first find the velocity (derivative of position) and acceleration (derivative of velocity) vectors before finding the radius of curvature. The radius of curvature can then help determine the curvature itself.