Another useful tool for understanding the
graph of a function of two variables is called a vertical trace. Level curves are always graphed in the
but as their name implies, vertical traces are graphed in the
- or
Definition
Consider a function
with domain
A
vertical trace of the function can be either the set of points that solves the equation
for a given constant
or
for a given constant
Finding vertical traces
Find vertical traces for the function
corresponding to
and
First set
in the equation
This describes a cosine graph in the plane
The other values of
appear in the following table.
Vertical traces parallel to the
For the function
Vertical Trace for
In a similar fashion, we can substitute the
in the equation
to obtain the traces in the
as listed in the following table.
Vertical traces parallel to the
For the function
Vertical Trace for
The three traces in the
are cosine functions; the three traces in the
are sine functions. These curves appear in the intersections of the surface with the planes
and
as shown in the following figure.
Functions of two variables can produce some striking-looking surfaces. The following figure shows two examples.
Functions of more than two variables
So far, we have examined only functions of two variables. However, it is useful to take a brief look at functions of more than two variables. Two such examples are
and
In the first function,
represents a point in space, and the function
maps each point in space to a fourth quantity, such as temperature or wind speed. In the second function,
can represent a point in the plane, and
can represent time. The function might map a point in the plane to a third quantity (for example, pressure) at a given time
The method for finding the domain of a function of more than two variables is analogous to the method for functions of one or two variables.
Domains for functions of three variables
Find the domain of each of the following functions:
For the function
to be defined (and be a real value), two conditions must hold:
The denominator cannot be zero.
The radicand cannot be negative.
Combining these conditions leads to the inequality
Moving the variables to the other side and reversing the inequality gives the domain as
which describes a ball of radius
centered at the origin. (
Note : The surface of the ball is not included in this domain.)
For the function
to be defined (and be a real value), two conditions must hold:
The radicand cannot be negative.
The denominator cannot be zero.
Since the radicand cannot be negative, this implies
and therefore that
Since the denominator cannot be zero,
or
Which can be rewritten as
which are the equations of two lines passing through the origin. Therefore, the domain of
is
is it possible to leave every good at the same level
Joseph
I don't think so. because check it, if the demand for chicken increases, people will no longer consume fish like they used to causing a fall in the demand for fish
Anuolu
is not really possible to let the value of a goods to be same at the same time.....
Salome
Suppose the inflation rate is 6%, does it mean that all the goods you purchase will cost
6% more than previous year? Provide with reasoning.
Not necessarily. To measure the inflation rate economists normally use an averaged price index of a basket of certain goods. So if you purchase goods included in the basket, you will notice that you pay 6% more, otherwise not necessarily.
Good day
How do I calculate this question: C= 100+5yd G= 2000 T= 2000 I(planned)=200.
Suppose the actual output is 3000. What is the level of planned expenditures at this level of output?
I am Camara from Guinea west Africa... happy to meet you guys here
Sekou
ma management ho
Amisha
ahile becheclor ho
Amisha
hjr ktm bta ho
ani k kaam grnu hunxa tw
Amisha
belatari
Amisha
1st year ho
Amisha
nd u
Amisha
ahh
Amisha
kaha biratnagar
Amisha
ys
Amisha
kina k vo
Amisha
money as unit of account means what?
Kalombe
A unit of account is something that can be used to value goods and services and make calculations
Jim
all of you please speak in English I can't understand you're language
Muhammad
I want to know how can we define macroeconomics in one line
Muhammad
it must be .9 or 0.9
no Mpc is greater than 1
Y=100+.9Y+50
Y-.9Y=150
0.1Y/0.1=150/0.1
Y=1500
Kalombe
Mercy is it clear?😋
Kalombe
hi can someone help me on this question
If a negative shocks shifts the IS curve to the left, what type of policy do you suggest so as to stabilize the level of output?
discuss your answer using appropriate graph.