Answer
$$r'(t) = \lt e^t, 2t\gt, \quad T(t)= \frac{ \lt e^t, 2t\gt}{\sqrt{e^{2t}+4t^2}}, \quad T(1)=\frac{ \lt e, 2 \gt}{\sqrt{e^4+4}}.$$
Work Step by Step
Since $ r(t) = \lt e^t, t^2\gt$, then $ r'(t) = \lt e^t, 2t\gt$ and $\|r'(t)\|=\sqrt{e^{2t}+4t^2} $.
Hence, we have
$$T(t)=\frac{r'(t)}{\|r'(t)\|}=\frac{ \lt e^t, 2t\gt}{\sqrt{e^{2t}+4t^2}}, \quad T(1)=\frac{ \lt e, 2 \gt}{\sqrt{e^4+4}}.$$
