Question

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

Answer 1

`\mathbf r(t)=\langle t^2,1-t,4t\rangle`

`\implies\mathbf r'(t)=\langle 2t,-1,4\rangle`


`(\mathrm d(\mathbf r(t)\cdot\mathbf a(t)))/(\mathrm dt)\bigg|_(t=5)=\mathbf r'(5)\cdot\mathbf a(5)+\mathbf r(5)\cdot\mathbf a'(5)`

`=\langle10,-1,4\rangle\cdot\langle-4,4,-5\rangle+\langle25,-4,20\rangle\cdot\langle-5,9,3\rangle`

`=(-40-4-20)+(-125-36+60)`

`=-165`