# Thread: Calculus of Vector-Valued Func.

1. ## Calculus of Vector-Valued Func.

1.) Given: r(t)=tcos(t)i+e(t^2)j+ln(t)k,

what is r'(t)?

2.) Suppose r(t) is a vector-valued func.,

What geometric/graphical info does r'(a) tell us?

3.) Let r(t)=cos(t)i+sin(t)j and s(t)=sin(5t)i+cos(5t)j.

(a) What does the graphs of r(t) and s(t) look like?

(b) Suppose the graphs of two vector-valued func. r(t) and s(t) are the same, then must r'(0)=s'(0)? (Explain whether this is a new result, or was it also true for functions f(x) and g(x)?)

2. Hello, fifthrapiers!

Here's some help . . .

1) Given: .$\displaystyle \vec{r}(t)\;=\;t\cos(t)\vec{i} + e^{t^2}\vec{j} + \ln(t)\vec{k}$. . Find $\displaystyle \vec{r'}(t)$
$\displaystyle \vec{r'}(t)\;=\;(\cos t - t\sin t)\vec{i} + \left(2te^{t^2}\right)\vec{j} + \left(\frac{1}{t}\right)\vec{k}$

3) Let $\displaystyle r(t)\:=\:\cos(t)i + \sin(t)j$ and $\displaystyle s(t)\:=\:\sin(5t)i + \cos(5t)j$

(a) What does the graphs of $\displaystyle r(t)$ and $\displaystyle s(t)$ look like?
Both are unit circles, centered at the origin.

(b) Suppose the graphs of two vector-valued func. $\displaystyle r(t)$ and $\displaystyle s(t)$ are the same,
. . .then must $\displaystyle r'(0) = s'(0)$? . no
Consider: .$\displaystyle \begin{array}{cc}\vec{r}(t) \;= & \cos(t)\vec{i} + \sin(t)\vec{j} \\ \vec{s}(t) \:= & \cos(t + \pi)\vec{i} + \sin(t + \pi)\vec{j} \end{array}$

When $\displaystyle t = 0:$

. . $\displaystyle \begin{array}{ccc}r(0) = \langle 1,\,0\rangle & \text{ and } & r'(0) = \langle 0,\,1\rangle\!:\;\uparrow \\ \\ s(0) = \langle \text{-}1,0\rangle & \text{ and } &s'(0) = \langle 0,\,\text{-}1\rangle\!: \;\downarrow\end{array}$ . . . different derivatives

Explain whether this is a new result,
or was it also true for functions $\displaystyle f(x)$ and $\displaystyle g(x)$ ?

This is a new result.

With rectangular functions, $\displaystyle y = f(x)$ and $\displaystyle y = g(x)$,
. . if their graphs are identical, then the functions are identical.
. . Hence, their derivatives are equal.

With parametric functions, a graph can be generated in a number of ways.
. . As seen in part (b), their derivatives need not be equal.