Question

Find r(t) if r'(t) = 8t^7i + 9t^8j + tk and r(1) = i + j.

Answer 1

To find r(t), integrate r'(t) component by component: r_x(t) = t^8 + C₁, r_y(t) = t^9 + C₂, and r_z(t) = (t^2)/2 + C₃.

Step-by-step explanation:

To find r(t), we can integrate r'(t) with respect to t. Since r'(t) = 8t^7i + 9t^8j + tk, integrating each component separately will give us the position function r(t). Integrating the x-component, we get r_x(t) = 8 * (t^8)/8 + C₁ = t^8 + C₁. Integrating the y-component, we get r_y(t) = 9 * (t^9)/9 + C₂ = t^9 + C₂. Integrating the z-component, we get r_z(t) = (t^2)/2 + C₃.