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

Chapter 14
Concept Check Answers (continued)
8. Define the linearization of f at sa, bd. What is the corresponding
linear approximation? What is the geometric interpretation
of the linear approximation?
The linearization of f at sa, bd is the linear function whose
graph is the tangent plane to the surface z − f sx, yd at the
point sa, b, f sa, bdd:
Lsx, yd − f sa, bd 1 f xsa, bdsx 2 ad 1 f ysa, bdsy 2 bd
The linear approximation of f at sa, bd is
f sx, yd < f sa, bd 1 f xsa, bdsx 2 ad 1 f ysa, bdsy 2 bd
Geometrically, the linear approximation says that function
values f sx, yd can be approximated by values Lsx, yd from the
tangent plane to the graph of f at sa, b, f sa, bdd when sx, yd is
near sa, bd.
9. (a) What does it mean to say that f is differentiable at sa, bd?
If z − f sx, yd, then f is differentiable at sa, bd if Dz can be
expressed in the form
Dz − f xsa, bd Dx 1 f ysa, bd Dy 1 « 1 Dx 1 « 2 Dy
where « 1 and « 2 l 0 as sDx, Dyd l s0, 0d. In other
words, a differentiable function is one for which the linear
approximation as stated above is a good approximation
when sx, yd is near sa, bd.
(b) How do you usually verify that f is differentiable?
If the partial derivatives f x and f y exist near sa, bd and are
continuous at sa, bd, then f is differentiable at sa, bd.
10. If z − f sx, yd, what are the differentials dx, dy, and dz?
The differentials dx and dy are independent variables that
can be given any values. If f is differentiable, the differential
dz is then defined by
dz − f xsx, yd dx 1 f ysx, yd dy
11. State the Chain Rule for the case where z − f sx, yd and x and
y are functions of one variable. What if x and y are functions
of two variables?
Suppose that z − f sx, yd is a differentiable function of x and
y, where x − tstd and y − hstd are both differentiable functions
of t. Then z is a differentiable function of t and
dz
dt − −f
−x
dx
dt 1 −f
−y
If z − f sx, yd is a differentiable function of x and y, where
x − tss, td and y − hss, td are differentiable functions of s and
t, then
−z
−s − −z
−x
−x
−s 1 −z
−y
−y
−s
dy
dt
−z
−t − −z
−x
−x
−t 1 −z
−y
−y
−t
12. If z is defined implicitly as a function of x and y by an equation
of the form Fsx, y, zd − 0, how do you find −zy−x and
−zy−y?
If F is differentiable and −Fy−z ± 0, then
−z
−x − 2
−F
−x
−F
−z
−z
−y − 2
−F
−y
−F
−z
13. (a) Write an expression as a limit for the directional derivative
of f at sx 0, y 0d in the direction of a unit vector
u − k a, b l. How do you interpret it as a rate? How do
you interpret it geometrically?
The directional derivative of f at sx 0, y 0d in the direction
of u is
D u f sx 0, y 0d − lim
h l 0
if this limit exists.
f sx 0 1 ha, y 0 1 hbd 2 f sx 0, y 0d
h
We can interpret it as the rate of change of f (with respect
to distance) at sx 0, y 0d in the direction of u.
Geometrically, if P is the point sx 0, y 0, f sx 0, y 0dd on the
graph of f and C is the curve of intersection of the graph
of f with the vertical plane that passes through P in the
direction of u, then D u f sx 0, y 0d is the slope of the tangent
line to C at P.
(b) If f is differentiable, write an expression for D u f sx 0, y 0d
in terms of f x and f y.
D u f sx 0, y 0d − f xsx 0, y 0d a 1 f ysx 0, y 0d b
14. (a) Define the gradient vector =f for a function f of two or
three variables.
If f is a function of two variables, then
=f sx, yd − k f xsx, yd, f ysx, yd l − −f
−x i 1 −f
−y j
For a function f of three variables,
=f sx, y, zd − k f xsx, y, zd, f ysx, y, zd, f zsx, y, zd l
− −f
−x i 1 −f
−y j 1 −f
−z k
(b) Express D u f in terms of =f .
or
D u f sx, yd − =f sx, yd u
D u f sx, y, zd − =f sx, y, zd u
(continued)
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.