This document discusses vector-valued functions and some key concepts related to them. It provides examples of parametric curves and vector functions, how to sketch their graphs, and calculations related to them. Key topics covered include the domain of a vector function, determining the equation of a line from points it passes through, sketching parametric curves, properties of vector operations and functions, differentiation and integration of vector functions, arc length of curves, unit tangent/normal/binormal vectors, curvature, and radius of curvature.
3. DOMAIN
Example 1
Determine the domain of the fo11owing function
cos ,1n 4 , 1
So1ution:
The first component is defined for a11 's.
The second component is on1y defined for 4.
The third component is on1y defined for
t t t t
t
t
r
1.
Putting a11 of these together gives the fo11owing domain.
1,4
This is the 1argest possib1e interva1 for which a11 three
components are defined.
t
26. Notes: Smooth Curve
The graph of the vector function defined by
r(t) is smooth on any interval of t where is
continuous and .
The graph is piecewise smooth on an interval
that can be subdivided into a finite number of
subintervals on which r is smooth.
r
0t r
27. rahimahj@ump.edu.my
Find the arc length of the parametric curve
4
3
0;2,sin,cos)( 33
tztytxa
10;2,,)(
ttzeyexb tt
Find the arc length of the graph of r(t)
42;6
2
1
)()( 23
ttttta kjir
20;2sin3cos3)()( tttttb kjir
4
3
: LAns
1
:
eeLAns
58: LAns
132: LAns
Example 8
Example 9
28.
29. If r(t) is a vector function that defines a smooth graph,
then at each point a unit tangent vector is
t
t
t
r
T
r
UNIT TANGENT VECTOR
3
a) Find the derivative of 1 sin 2
b) Find the unit tangent vector at the point where 0.
t
t t te t
t
r i j k
Example 10
43. Find the position vector R(t), given the
velocity V(t) and the initial position R(0) for
2 2
; 0 4t
t t e t V i j k R i j k
Example 15