Question

Let ????(t)=⟨t2,1−t,4t⟩r(t)=⟨t2,1−t,4t⟩. Calculate the derivative of ????(t)⋅????(t)r(t)⋅a(t) at t=2t=2, assuming that ????(2)=⟨2,5,−3⟩a(2)=⟨2,5,−3⟩ and ????′(2)=⟨4,−3,9⟩

Answer 1

The derivative is
`( d (r(t) \cdot a(t)))/(dt) = 82`

Explanation:

From the question we are told that

`r(t) = (t^2 ,1 - t , 4t)`

`a(2) = (2, 5, -3)` and
`a'(2) = (4,-3 , 9)`

At t = 2

`r(t) = (2^2 ,1 - 2 , 4(2))`

`r(t) = (4 ,-1 , 8 )`

Now the derivative of r(t) is

`r'(t) = (2t, -1 ,4)`

At t = 2

`r'(t) = (2(2), -1 ,4)`

`r'(t) = (4, -1 ,4)`

Now the derivative of
`r(t) \cdot a(t)` At t = 2 is

`= r'(2) a(2) + a'(2)r(2)`

`= (4,-1,4)(2,5,-3) + (4,-3,9)(4,-1,8)`

`= (8 - 5 -12) + (16+3+72)`

`= -9 + 91`

`( d (r(t) \cdot a(t)))/(dt) = 82`