Ch 13: Vector Functions

Question 1

Definition If $r(t)=<f(t),g(t),h(t)>$, then $\lim_{t	o a}{r(t)}=$

Accepted Answer

$<\lim_{t	o a}{f(t)},\lim_{t	o a}{g(t)},\lim_{t	o a}{h(t)}>$

Question 2

A vector function r is continuous at a if

Accepted Answer

$\lim_{t \rightarrow a} r(t) = r(a)$

Question 3

Theorem If $	extbf{r}(t) = <f(t),g(t),h(t)>$, where f, g, and h are differentiable functions, then $	extbf{r}'(t)=$

Accepted Answer

$<f'(t),g'(t),h'(t)>$

Question 4

Differentiation Rules $\frac{d}{dt}[	extbf{u}(t)+	extbf{v}(t)]=$

Accepted Answer

$	extbf{u}'(t)+	extbf{v}'(t)$

Question 5

Differentiation Rules $\frac{d}{dt}[c	extbf{u}(t)]=$

Accepted Answer

$c	extbf{u}'(t)$

Question 6

Differentiation Rules $\frac{d}{dt}[f(t)	extbf{u}(t)]=$

Accepted Answer

$f'(t)	extbf{u}(t)+f(t)	extbf{u}'(t)$

Question 7

Differentiation Rules $\frac{d}{dt}[	extbf{u}(t)\cdot	extbf{v}(t)]=$

Accepted Answer

$	extbf{u}'(t)\cdot	extbf{v}(t)+	extbf{u}(t)\cdot	extbf{v}'(t)$

Question 8

Differentiation Rules $\frac{d}{dt}[	extbf{u}(t)	imes	extbf{v}(t)]=$

Accepted Answer

$	extbf{u}'(t)	imes	extbf{v}(t)+	extbf{u}(t)	imes	extbf{v}'(t)$

Question 9

Differentiation Rules $\frac{d}{dt}[	extbf{u}(f(t))]=$

Accepted Answer

$f'(t)	extbf{u}'(f(t))$

Question 10

If $|	extbf{r}(t) = c|$ (a constant), then $	extbf{r}'(t)$ is

Accepted Answer

orthogonal to $	extbf{r}(t)$ for all t

Question 11

Arc Length Formula

Accepted Answer

$L = \int_a^b{|	extbf{r}'(t)|} dt$

Question 12

Definition Arc Length Function

Accepted Answer

$s(t)=\int_a^t{|	extbf{r}'(t)|}dt$

Question 13

Definition Curvature of C is how quickly the curve changes direction at a given point

Accepted Answer

$\kappa(t) = \frac{	extbf{T}'(t)}{|	extbf{r}'(t)|}$

Question 14

Theorem The curvature of the curve given by the vector function r is

Accepted Answer

$\kappa(t)=\frac {|	extbf{r}'(t) 	imes 	extbf{r}''(t)|} { |	extbf{r}'(t)|^3} $

Question 15

Unit Tangent Vector

Accepted Answer

$	extbf{T}(t)=\frac{	extbf{r}'(t)}{|	extbf{r}'(t)|}$

Question 16

Unit Normal Vector

Accepted Answer

$N(t)=\frac {|	extbf{T}'(t)}{|T'(t)|} $

Question 17

Definition Binormal Vector is perpendicular to both T and N

Accepted Answer

$	extbf{B}(t)= 	extbf{T}(t)	imes	extbf{N}(t)$

Question 18

Definition The radius of the circle of curviture of ocilating plane is

Accepted Answer

$\frac{1}{\kappa}$

Question 19

Definition The torsion of a curve is

Accepted Answer

$	au(t)=-\frac{	extbf{B}'(t)\cdot	extbf{N}(t)}{|	extbf{r}'(t)|}$

Question 20

Theorem The torsion of the curve given by the vector function r is

Accepted Answer

$	au(t)=\frac{[	extbf{r}'(t)	imes	extbf{r}''(t)]	extbf{r}'''(t)}{|	extbf{r}'(t)	imes	extbf{r}''(t)|^2}$

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