Recall that for any base
the function
is an exponential function with domain
and range
If
is increasing over
If
is decreasing over
For the natural exponential function
Therefore,
is increasing on
and the range is
The exponential function
approaches
as
and approaches
as
as shown in
[link] and
[link] .
End behavior of the natural exponential function
Recall that the natural logarithm function
is the inverse of the natural exponential function
Therefore, the domain of
is
and the range is
The graph of
is the reflection of the graph of
about the line
Therefore,
as
and
as
as shown in
[link] and
[link] .
End behavior of the natural logarithm function
Determining end behavior for a transcendental function
Find the limits as
and
for
and describe the end behavior of
To find the limit as
divide the numerator and denominator by
We conclude that
and the graph of
approaches the horizontal asymptote
as
To find the limit as
use the fact that
as
to conclude that
and therefore the graph of approaches the horizontal asymptote
as
We now have enough analytical tools to draw graphs of a wide variety of algebraic and transcendental functions. Before showing how to graph specific functions, let’s look at a general strategy to use when graphing any function.
Problem-solving strategy: drawing the graph of a function
Given a function
use the following steps to sketch a graph of
Determine the domain of the function.
Locate the
- and
-intercepts.
Evaluate
and
to determine the end behavior. If either of these limits is a finite number
then
is a horizontal asymptote. If either of these limits is
or
determine whether
has an oblique asymptote. If
is a rational function such that
where the degree of the numerator is greater than the degree of the denominator, then
can be written as
where the degree of
is less than the degree of
The values of
approach the values of
as
If
is a linear function, it is known as an
oblique asymptote .
Determine whether
has any vertical asymptotes.
Calculate
Find all critical points and determine the intervals where
is increasing and where
is decreasing. Determine whether
has any local extrema.
Calculate
Determine the intervals where
is concave up and where
is concave down. Use this information to determine whether
has any inflection points. The second derivative can also be used as an alternate means to determine or verify that
has a local extremum at a critical point.
Questions & Answers
differentiate between demand and supply
giving examples
In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,
When MP₁ becomes negative, TP start to decline.
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab
Kelo
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)
Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded
Ezea
ok
Shukri
how do you save a country economic situation when it's falling apart
Economic growth as an increase in the production and consumption of goods and services within an economy.but
Economic development as a broader concept that encompasses not only economic growth but also social & human well being.
Shukri
production function means
Jabir
What do you think is more important to focus on when considering inequality ?
sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏
Asui
it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs
Awais
thank you so much 👍 sir
Asui
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p
Cornelius
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities,
Cornelius
Suppose a consumer consuming two commodities X and Y has
The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50.
A,Calculate quantities of x and y which maximize utility.
B,Calculate value of Lagrange multiplier.
C,Calculate quantities of X and Y consumed with a given price.
D,alculate optimum level of output .
the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema
suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product