Given the polar equation for a conic, identify the type of conic, the directrix, and the eccentricity.
Multiply the numerator and denominator by the reciprocal of the constant in the denominator to rewrite the equation in standard form.
Identify the eccentricity
as the coefficient of the trigonometric function in the denominator.
Compare
with 1 to determine the shape of the conic.
Determine the directrix as
if cosine is in the denominator and
if sine is in the denominator. Set
equal to the numerator in standard form to solve for
or
Identifying a conic given the polar form
For each of the following equations, identify the conic with focus at the origin, the
directrix , and the
eccentricity .
For each of the three conics, we will rewrite the equation in standard form. Standard form has a 1 as the constant in the denominator. Therefore, in all three parts, the first step will be to multiply the numerator and denominator by the reciprocal of the constant of the original equation,
where
is that constant.
Multiply the numerator and denominator by
Because
is in the denominator, the directrix is
Comparing to standard form, note that
Therefore, from the numerator,
Since
the conic is an
ellipse . The eccentricity is
and the directrix is
Multiply the numerator and denominator by
Because
is in the denominator, the directrix is
Comparing to standard form,
Therefore, from the numerator,
Since
the conic is a
hyperbola . The eccentricity is
and the directrix is
Multiply the numerator and denominator by
Because sine is in the denominator, the directrix is
Comparing to standard form,
Therefore, from the numerator,
Because
the conic is a
parabola . The eccentricity is
and the directrix is
When graphing in Cartesian coordinates, each conic section has a unique equation. This is not the case when graphing in polar coordinates. We must use the eccentricity of a conic section to determine which type of curve to graph, and then determine its specific characteristics. The first step is to rewrite the conic in standard form as we have done in the previous example. In other words, we need to rewrite the equation so that the denominator begins with 1. This enables us to determine
and, therefore, the shape of the curve. The next step is to substitute values for
and solve for
to plot a few key points. Setting
equal to
and
provides the vertices so we can create a rough sketch of the graph.
Graphing a parabola in polar form
Graph
First, we rewrite the conic in standard form by multiplying the numerator and denominator by the reciprocal of 3, which is
Because
we will graph a
parabola with a focus at the origin. The function has a
and there is an addition sign in the denominator, so the directrix is
The directrix is
Plotting a few key points as in
[link] will enable us to see the vertices. See
[link] .
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