Suppose $r(t)$ and $s(t)$ are vector functions with $r(2) = \langle 1,2,−1 \rangle$, $r'(t) =\langle 3,0,4\rangle$, and $s(t) =\langle t,t^2,t^3 \rangle$.
(a) Find the value of $f'(2)$, when $f(t)=r(t)·s(t)$
(b) Find the value of $u'(2)$, when $u(t)=r(t)\times s(t)$.
To find the value of $f(2)$ and $u(2)$, I need $r(t)$ which I do not know how to find. My guess is that since $r'(t)=\langle 3,0,4 \rangle$ then $r(t)$ could equal $\langle 3x,0,4z \rangle$.
You don't need $r(t)$ for any $t\ne2$. By the product rule,$$f^\prime(t)=r^\prime(t)\cdot s(t)+r(t)\cdot s^\prime(t)\to f^\prime(2)=\langle3,\,0,\,4\rangle\cdot\langle2,\,4,\,8\rangle+\langle1,\,2,\,-1\rangle\cdot\langle2,\,4,\,12\rangle.$$Similarly,$$u^\prime(2)=\langle3,\,0,\,4\rangle\times\langle2,\,4,\,8\rangle+\langle1,\,2,\,-1\rangle\times\langle2,\,4,\,12\rangle.$$I'l leave the rest of the arithmetic to you.