13.1 Videos Guide

13.1a

• Forms of vector functions

  • $\mathbf{r}(t)=\left\langle f(t),\ g(t),\ h(t)\right\rangle =f(t)\mathbf{i}+g(t)\mathbf{j}+h(t)\mathbf{k}$

  • Parametric equations: $x=f(t)$, $y=g(t)$, $z=h(t)$

• Limit of a vector function

  • $\lim_{x\rightarrow a}{\mathbf{r}(t)=\left\langle \lim_{x\rightarrow a}f(t),\ \lim_{x\rightarrow a}g(t),\ \lim_{x\rightarrow a}h(t)\right\rangle }$

Exercises:

• Find the domain of the vector function $\mathbf{r}(t)=\cos t\ \mathbf{i}+\ln t\ \mathbf{j}+\frac{1}{t-2}\ \mathbf{k}$.

• Find the limit $\lim_{t\rightarrow1}\left(\frac{t^{2}-t}{t-1}\ \mathbf{i}+\sqrt{t+8}\ \mathbf{j}+\frac{\sin{\pi t}}{\ln t}\ \mathbf{k}\right)$.

13.1b

• Sketch the curve with the given vector equation. Indicate with an arrow the direction in which $t$ increases.

  • $\mathbf{r}(t)=2\cos t\mathbf{i}+2\sin t\mathbf{j}+\mathbf{k}$

  • $\mathbf{r}(t)=\left\langle \sin{\pi t},\ t,\cos{\pi t}\right\rangle$

13.1c

• Find a vector function that represents the curve of intersection of the cylinder
  $x^{2}+y^{2}=4$ and the surface $z=xy$.

• Using graphing utilities