We then substitute this into the equation of the graph (i.e.
) to determine the
-coordinate of the turning point:
This corresponds to the point that we have previously calculated.
Calculate the turning points of the graph of the function
.
Using the rules of differentiation we get:
Therefore, the turning points are at
and
.
The turning points of the graph of
are (2,-11) and (1,-10).
We are now ready to sketch graphs of functions.
Method:
Sketching Graphs:
Suppose we are given that
, then there are
five steps to be followed to sketch the graph of the function:
If
, then the graph is increasing from left to right, and may have a maximum and a minimum. As
increases, so does
. If
, then the graph decreasing is from left to right, and has first a minimum and then a maximum.
decreases as
increases.
Determine the value of the
-intercept by substituting
into
Determine the
-intercepts by factorising
and solving for
. First try to eliminate constant common factors, and to group like terms together so that the expression is expressed as economically as possible. Use the factor theorem if necessary.
Find the turning points of the function by working out the derivative
and setting it to zero, and solving for
.
Determine the
-coordinates of the turning points by substituting the
values obtained in the previous step, into the expression for
.
Use the information you're given to plot the points and get a rough idea of the gradients between points. Then fill in the missing parts of the function in a smooth, continuous curve.
Draw the graph of
The
-intercept is obtained by setting
.
The turning point is at (0,2).
The
-intercepts are found by setting
.
Using the quadratic formula and looking at
we can see that this would be negative and so this function does not have real roots. Therefore, the graph of
does not have any
-intercepts.
Work out the derivative
and set it to zero to for the
coordinate of the turning point.
coordinate of turning point is given by calculating
.
The turning point is at
Sketch the graph of
.
Find the turning points by setting
.
If we use the rules of differentiation we get
The
-coordinates of the turning points are:
and
.
The
-coordinates of the turning points are calculated as:
Therefore the turning points are:
and
.
We find the
-intercepts by finding the value for
.
We find the
-intercepts by finding the points for which the function
.
Use the factor theorem to confirm that
is a factor. If
, then
is a factor.
Therefore,
is a factor.
If we divide
by
we are left with:
This has factors
Therefore:
The
-intercepts are:
Sketching graphs
Given
:
Show that
is a factor of
and hence fatorise
fully.
Find the coordinates of the intercepts with the axes and the turning points and sketch the graph
Sketch the graph of
showing all the relative turning points and intercepts with the axes.
Sketch the graph of
, showing all intercepts with the axes and turning points.
Find the equation of the tangent to
at
.
Local minimum, local maximum and point of inflextion
If the derivative (
) is zero at a point, the gradient of the tangent at that point is zero. It means that a turning point occurs as seen in the previous example.
From the drawing the point (1;0) represents a
local minimum and the point (3;4) the
local maximum .
A graph has a horizontal
point of inflexion where the derivative is zero but the sign of the sign of the gradient does not change. That means the graph always increases or always decreases.
From this drawing, the point (3;1) is a horizontal point of inflexion, because the sign of the derivative does not change from positive to negative.
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