Question

If r(t) = 6t, 2t², 4t³ , find r'(t), t(1), r''(t), and r'(t) × r ''(t).

Answer 1

To solve the student's question, we found the first and second derivatives of the vector function r(t), evaluated them at t=1, and calculated their cross product.

Step-by-step explanation:

The student's question involves finding the first and second derivatives of the vector function r(t), evaluating the function and its first derivative at t=1, and finding the product of the first and second derivatives.

The vector function given is r(t) = <6t, 2t², 4t³>.

1. To find r'(t), take the derivative of each component of the vector:
   r'(t) = <6, 4t, 12t²>.
2. To evaluate the function at t=1, we simply substitute 1 into the function:
   r(1) = <6(1), 2(1)², 4(1)³> = <6, 2, 4>.
3. The second derivative, r''(t), is the derivative of r'(t):
   r''(t) = <0, 4, 24t>.
4. Finally, to find r'(t) × r''(t), calculate the cross product of the two derivatives:

r'(t) × r''(t) = <6, 4t, 12t²> × <0, 4, 24t> = <(4t)(24t) - (12t²)(4), -(6)(24t), (6)(4) - (4t)(0)> = <24t² - 48t², -144t, 24>

Therefore, r'(t) × r''(t) = <-24t², -144t, 24>.