Chapter 12 - Vector-Valued Functions - 12.3 Change Of Parameter; Arc Length - Exercises Set 12.3 - Page 866: 18

False

Consider \[ \mathbf{r}(t) = t^2\mathbf{i} + t^2\mathbf{j} \] Then \[ \mathbf{r}'(t) = 2t\mathbf{i} + 2t\mathbf{j} \] It is clear that \(2t\) and \(2t\) are continuous, but \(\mathbf{r}(t) = 0\) at \(t = 0\). Hence, \(\mathbf{r}(t)\) is not a smooth function. Result : False