3) Suppose f (x, y) is a differentiable function. a) If (a, b) is a critical point of f (y) (in the sense that or (a, b)-0,(a,b)-0 will (a, b) be a critical point of g(x, y) = (f(x, y)) ? b) if (a, b) is a critical point of 9 (x, y) = (f(x, y))², in the sense that (a,b) = 0,(a, b) = 0, will (a, b) be a critical point of f(x,y)? hint: how are the partial derivatives of f(x,y) and 9 (x,y) related?
3) Suppose f (x, y) is a differentiable function. a) If (a, b) is a critical point of f (y) (in the sense that or (a, b)-0,(a,b)-0 will (a, b) be a critical point of g(x, y) = (f(x, y)) ? b) if (a, b) is a critical point of 9 (x, y) = (f(x, y))², in the sense that (a,b) = 0,(a, b) = 0, will (a, b) be a critical point of f(x,y)? hint: how are the partial derivatives of f(x,y) and 9 (x,y) related?
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.5: Graphs Of Functions
Problem 36E
Related questions
Question
Please help me with questions 3 and 4 of this homework. Thanks
![Description
f(x) == What is
1 a) Consider the function
its domain? b) Now consider the function
g(x, y) =
What is its domain? c) What is
the domain of the function h (x, y) = √√x+y
? d) What is the domain of the function
k (x, y) =
√2+8?
2 Suppose that I (r, y, z) represents the
temperature at the point (x, y, z), measured in
8T
Kelvin. What are the units of ax, if "x" is
measured in miles? What about the units of
²T
82² ?
3) Suppose f(",) is a differentiable function.
a) If (a, b) is a critical point of f(x, y) (in the
af
sense that
(a,b)-0,(a,b) - 0
Will
(a, b) be a critical point of
g(x, y) = (f (r. y))²? b) if (a, b) is a critical
point of 9 (x, y) = (f (x, y))², in the sense
(a, b) = 0,(a, b) = 0
89
that de
will (a, b)
be a critical point of f(x,y)? hint: how are the
partial derivatives of f(x,y) and g(x, y)
related?
9
4) Suppose that the level curves of a function
z = f(x, y) consists of straight lines. Must the
graph of be a plane?
5) In thermodynamics, the ideal gas law states
that
PVNKT
Here
N,k are constants
• P is the pressure of the gas
. V is the volume of the container
. T is the temperature.
Notice that you can play with this equation in
several ways. For example:
• You can think of P as a function of V and T
P =
NKT
via the equation
V. So you can
OP OP
find the partial derivatives T¹ V
. You can think of T as a function of P and V
T -
PV
=
via the equation
Nk. So you can find
8T
OT T
the partial derivatives
P¹ av
• You can think of V as a function of P and T
V =
NKT
via the equation
P. So you can
av
av
find the partial derivatives OP T
a) What do you think
OP OT BV
ST OV DP
will be equal to, based on a "naive" manipulation
of the symbols? Similar to how "naively one
dy
dy da
dzd in the case of the chain
writes d
rule?
b) What do you actually find if you do the
product of these partial derivatives? [hint: there
is a video on our Canvas sites:
Thermodynamics and Partial Derivatives, a
Cautionary Tale, where I go over this.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F41a1af84-f79e-495e-9bf6-e155c46aa5b8%2F40275b71-ced7-4f6d-8f90-1d6e795147bf%2Fveimth_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Description
f(x) == What is
1 a) Consider the function
its domain? b) Now consider the function
g(x, y) =
What is its domain? c) What is
the domain of the function h (x, y) = √√x+y
? d) What is the domain of the function
k (x, y) =
√2+8?
2 Suppose that I (r, y, z) represents the
temperature at the point (x, y, z), measured in
8T
Kelvin. What are the units of ax, if "x" is
measured in miles? What about the units of
²T
82² ?
3) Suppose f(",) is a differentiable function.
a) If (a, b) is a critical point of f(x, y) (in the
af
sense that
(a,b)-0,(a,b) - 0
Will
(a, b) be a critical point of
g(x, y) = (f (r. y))²? b) if (a, b) is a critical
point of 9 (x, y) = (f (x, y))², in the sense
(a, b) = 0,(a, b) = 0
89
that de
will (a, b)
be a critical point of f(x,y)? hint: how are the
partial derivatives of f(x,y) and g(x, y)
related?
9
4) Suppose that the level curves of a function
z = f(x, y) consists of straight lines. Must the
graph of be a plane?
5) In thermodynamics, the ideal gas law states
that
PVNKT
Here
N,k are constants
• P is the pressure of the gas
. V is the volume of the container
. T is the temperature.
Notice that you can play with this equation in
several ways. For example:
• You can think of P as a function of V and T
P =
NKT
via the equation
V. So you can
OP OP
find the partial derivatives T¹ V
. You can think of T as a function of P and V
T -
PV
=
via the equation
Nk. So you can find
8T
OT T
the partial derivatives
P¹ av
• You can think of V as a function of P and T
V =
NKT
via the equation
P. So you can
av
av
find the partial derivatives OP T
a) What do you think
OP OT BV
ST OV DP
will be equal to, based on a "naive" manipulation
of the symbols? Similar to how "naively one
dy
dy da
dzd in the case of the chain
writes d
rule?
b) What do you actually find if you do the
product of these partial derivatives? [hint: there
is a video on our Canvas sites:
Thermodynamics and Partial Derivatives, a
Cautionary Tale, where I go over this.]
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