Final answer:
To find r'(t) · r''(t) for r(t) = -4t⁵i + 2tj + 3t²k, we first need to find the derivatives of r(t) and then multiply the corresponding components and add them together.
Step-by-step explanation:
To find r'(t) · r''(t) for r(t) = -4t⁵i + 2tj + 3t²k, we first need to find the derivatives of r(t). The derivative of r(t) is r'(t) = -20t⁴i + 2j + 6tk. The second derivative of r(t) is r''(t) = -80t³i + 6k. To find the dot product r'(t) · r''(t), we multiply the corresponding components and add them together:
-20t⁴ * -80t³ + 2 * 0 + 6t * 6 = 1600t⁷ + 36t. So, r'(t) · r''(t) = 1600t⁷ + 36t.