Total growth via Net Change Theorem

We need an antiderivative of $2t^{1/2}$. Using the power rule for integration, $\int t^{1/2}\,dt = \frac{t^{3/2}}{3/2} = \frac{2}{3}t^{3/2}$, so $\int 2t^{1/2}\,dt = \frac{4}{3}t^{3/2} + C$. This gives us $H(t) = \frac{4}{3}t^{3/2}$ as an antiderivative of $H'(t)$.

By the Net Change Theorem, the total growth equals $H(9) - H(0) = \frac{4}{3}(9)^{3/2} - \frac{4}{3}(0)^{3/2}$. Since $9^{3/2} = (\sqrt{9})^3 = 3^3 = 27$, this becomes $\frac{4}{3}(27) - 0 = 36$. So the plant grew $36$ cm over the first 9 weeks.