Chapter 12.3, Problem 38ES

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(a) Sketch the graph of $\text{r}\left(t\right)=t\text{i}+t^2\text{j}.$ Show that $\text{r}\left(t\right)$ is a smooth vector-valued function but

the change of parameter $t={\tau }^3$ produces a vector-valued function that is not smooth, yet

has the same graph as $\text{r}\left(t\right).$

(b) Examine how the two vector-valued functions are traced, and see if you can explain what

causes the problem.