Identify all discontinuities for the following functions as either a jump or a removable discontinuity.
Notice that the function is defined everywhere except at
Thus,
does not exist, Condition 2 is not satisfied. Since Condition 1 is satisfied, the limit as
approaches 5 is 8, and Condition 2 is not satisfied.This means there is a removable discontinuity at
Condition 2 is satisfied because
Notice that the function is a
piecewise function , and for each piece, the function is defined everywhere on its domain. Let’s examine Condition 1 by determining the left- and right-hand limits as
approaches 2.
Left-hand limit:
The left-hand limit exists.
Right-hand limit:
The right-hand limit exists. But
So,
does not exist, and Condition 2 fails: There is no removable discontinuity. However, since both left- and right-hand limits exist but are not equal, the conditions are satisfied for a jump discontinuity at
Recognizing continuous and discontinuous real-number functions
Many of the functions we have encountered in earlier chapters are continuous everywhere. They never have a hole in them, and they never jump from one value to the next. For all of these functions, the limit of
as
approaches
is the same as the value of
when
So
There are some functions that are continuous everywhere and some that are only continuous where they are defined on their domain because they are not defined for all real numbers.
Examples of continuous functions
The following functions are continuous everywhere:
Polynomial functions
Ex:
Exponential functions
Ex:
Sine functions
Ex:
Cosine functions
Ex:
The following functions are continuous everywhere they are defined on their domain:
Logarithmic functions
Ex:
,
Tangent functions
Ex:
is an integer
Rational functions
Ex:
Given a function
determine if the function is continuous at
Check Condition 1:
exists.
Check Condition 2:
exists at
Check Condition 3:
If all three conditions are satisfied, the function is continuous at
If any one of the conditions is not satisfied, the function is not continuous at
Determining whether a piecewise function is continuous at a given number
Determine whether the function
is continuous at
To determine if the function
is continuous at
we will determine if the three conditions of continuity are satisfied at
.
Condition 1: Does
exist?
Condition 2: Does
exist?
To the left of
to the right of
We need to evaluate the left- and right-hand limits as
approaches 1.
Left-hand limit:
Right-hand limit:
Because
does not exist.
There is no need to proceed further. Condition 2 fails at
If any of the conditions of continuity are not satisfied at
the function
is not continuous at
Condition 1: Does
exist?
Condition 2: Does
exist?
To the left of
to the right of
We need to evaluate the left- and right-hand limits as
approaches
Left-hand limit:
Right-hand limit:
Because
exists,
Condition 3: Is
Because all three conditions of continuity are satisfied at
the function
is continuous at
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