Chapter 12.1, Problem 42E

Transformations of Vector-Valued Functions In Exercises 41 and 42, use a computer algebra system tograph the vector-valued function r( t). For each u( r), make aconjecture about the transformation (if any) of the graph of r( f). Use a computer algebra system to verify your conjecture.

$r(t)=ti+t^{2}j+\frac{1}{2}{t}^{3}k$

(a) $u(t)=(-t)i+{(-t)}^{2}j+\frac{1}{2}{(-t)}^{3}k$

(b) $u(t)=t^{2}i+tj+\frac{1}{2}{t}^{3}k$

(c) $u(t)=ti+t^{2}j+\left(\frac{1}{2}{t}^3+4\right)k$

(d) $u(t)=ti+t^{2}j+\frac{1}{8}{t}^{3}k$