James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Chapter 13
Concept Check Answers
Cut here and keep for reference
1. What is a vector function? How do you find its derivative and
its integral?
A vector function is a function whose domain is a set of real
numbers and whose range is a set of vectors. To find the
derivative or integral, we can differentiate or integrate each
component function of the vector function.
2. What is the connection between vector functions and space
curves?
A continuous vector function r defines a space curve that is
traced out by the tip of the moving position vector rstd.
3. How do you find the tangent vector to a smooth curve at a
point? How do you find the tangent line? The unit tangent
vector?
The tangent vector to a smooth curve at a point P with position
vector rstd is the vector r9std. The tangent line at P is the
line through P parallel to the tangent vector r9std. The unit
tangent vector is Tstd −
r9std
| r9std | .
4. If u and v are differentiable vector functions, c is a scalar,
and f is a real-valued function, write the rules for differentiating
the following vector functions.
(a) ustd 1 vstd
(b) custd
(c) f std ustd
(d) ustd vstd
(e) ustd 3 vstd
(f) us f stdd
d
fustd 1 vstdg − u9std 1 v9std
dt
d
fcustdg − cu9std
dt
d
f f std ustdg − f 9std ustd 1 f std u9std
dt
d
fustd vstdg − u9std vstd 1 ustd v9std
dt
d
fustd 3 vstdg − u9std 3 vstd 1 ustd 3 v9std
dt
d
fus f stddg − f 9stdu9s f stdd
dt
5. How do you find the length of a space curve given by a vector
function rstd?
If rstd − k f std, tstd, hstdl, a < t < b, and the curve is
traversed exactly once as t increases from a to b, then the
length is
L − y b
a
| r9std | dt − y b
a sf f 9stdg2 1 ft9stdg 2 1 fh9stdg 2 dt
6. (a) What is the definition of curvature?
dT
The curvature of a curve is − Z
ds
Z where T is the
unit tangent vector.
(b) Write a formula for curvature in terms of r9std and T9std.
std − | T9std |
| r9std |
(c) Write a formula for curvature in terms of r9std and r0std.
std − | r9std 3 r0std |
| r9std | 3
(d) Write a formula for the curvature of a plane curve with
equation y − f sxd.
sxd −
| f 0sxd |
f1 1 s f 9sxdd 2 g 3y2
7. (a) Write formulas for the unit normal and binormal vectors
of a smooth space curve rstd.
Unit normal vector: Nstd −
T9std
| T9std |
Binormal vector: Bstd − Tstd 3 Nstd
(b) What is the normal plane of a curve at a point? What is
the osculating plane? What is the osculating circle?
The normal plane of a curve at a point P is the plane
determined by the normal and binormal vectors N and
B at P. The tangent vector T is orthogonal to the normal
plane.
The osculating plane at P is the plane determined by the
vectors T and N. It is the plane that comes closest to containing
the part of the curve near P.
The osculating circle at P is the circle that lies in the
osculating plane of C at P, has the same tangent as C at
P, lies on the concave side of C (toward which N points),
and has radius − 1y (the reciprocal of the curvature).
It is the circle that best describes how C behaves near P; it
shares the same tangent, normal, and curvature at P.
(continued)
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