Assignment 7: Maximum of a profit function.

Assignment 7: Maximum of a profit function.

Based on the assignment, we got the:
TR=1400Q -7.5Q^2
TC=Q^3-6Q^2+140Q+750

Revenue= TR-TC
So, we can get the function of profit is :
Profit=1260Q-1.5Q^2-Q^3-750

MR=1400-15Q
MC=3Q^2-12Q+140

We can draw the graph of TR,TC,PROFIT ,MR and MC.

When MC=MR , the company achieve the most profit.

According to the graph of MC and MR, we can see that when Q=20, MC=MR.
When Q=20 the profit is biggest. Profit=15850

Assignmet 6: Tipping (braking) point of a product life cycle.

Assignmet 6: Tipping (braking) point of a product life cycle.

TC=Q^3-12Q^2+60Q

According to the TC function, we can get the MC and AC function.
MC=3Q^2-24Q+60
AC=Q^2-12Q+60

When we have those product function, we can draw the graph.

When MC=AC, we can get Q=6 and MC=AC=24, then when Q=6 TC=144.
If Q<6, the TC graph is degressive increasing.
If Q>6, the TC graph is progressive increasing.
So the Point (6,144) is the tipping point of  TC.
We can get the tangent could assume be y=24x.

y=24+25 is the point (1.49) tangent of TC. It paralla with y=24x.


After point(6.144). the MC>AC, the productor could consider keep on product, because the cost is progressive increasing.

Assignment 5: Smooth function

Assignment 5: Smooth function

A smooth function is a function that has continuous derivatives up to some desired order over some domain.

y=-2|x-1|+3
We can separate this function to three parts:
y=-2(x-1)+3=-2x+5  when x>1
y=-2(1-1)+3=3         when x=1
y=2(x-1)+3=2x+1     when x<1

We can see from the first graph, the max of this function is 3.

But when x<1 , the function have a positive slope.

when x>1, the function have a negative slope.

when x=1, the function's slope is exist.

Because the when X=1, the slope is not exit, so this function is a non-smooth function.



Assignment 4: Non smooth function

Assignment 4: Non smooth function

According to above data, we got the first graph.
The red grapgs are composed of four linear functions:
1. y=x+2
2. y=-2x+5
3. y=1
4. y=7x-20

The blue graph is a quadratic function:
5. y=-1/2x^2+4x

After the  derivation, we can find out the slope of each functions:
1. y'=1
2. y'=-2
3. y'=0
4. y'=7
5. y'=-x+4

HW 2: Function y = f(x) - Independent and dependent variable (concave or convex S, D).

HW 3: Slope of an Economical Curve (theory of derivation, slope of linear function, slope of quadratic function, degresive or progresive increase)

HW 3: Slope of an Economical Curve (theory of derivation, slope of linear function, slope of quadratic function, degresive or progresive increase）

In this homework I choose the economical curve is the total utility curve.

A curve illustrating the relation between the total utility obtained from consuming a good and the quantity of the good consumed. The shape of the total utility curve, increasing at a decreasing rate, reflects the law of diminishing marginal utility. The reason for this is that slope of the total utility curve is marginal utility, meaning the total utility curve can be use to derive the marginal utility curve.
 

Q	Total Utility	Marginal Utility	P
0	0
1	11	10	5
2	20	8	4
3	27	6	3
4	32	4	2
5	35	2	1
6	36	0	0
7	35	-2

In this case, the total utility have the binomial expression: y=-x^2+12x.

According to the Q=2, we find a linear y=9x+2 which connect with the point(2,20) in the second graph.

If we find countless linear to connect with total utility graph, we can find the slope of the total utility.

The marginal utility actually is the slope of total utility, the linear expression of marginal utility is y=-2x+12.

We can see that when marginal utility equal to 0, that total utility achieve to the max.



HW 1: Approximation of discrete values by continuous diferenciable function (MS Excel)

HW 1: Approximation of discrete values by continuous diferenciable function (MS Excel)

In this homework, we set up the datas about virtual company's supply.
The Price is the independent variable and the Quality is the dependent variable.

P	1	2	3	4	5	6	7	8	9	10
Q	8	10	14	16	18	22	24	26	25	23

Like the first graph, it's based on the datas from the P and Q. However, in the economic graph, the independent variable P is in Y axis, and the dependent variable Q is in X axis.

The first graph is use the trend line to describe the trend of the Q, but here we use the binominal expression.

So, it's easy to find out that after the Q=6. the price start to decreasing.

Second graph has the same data with first graph, the only different is that they has different trend line. The second trend line is the linear expression. In the linear expression, it's hard to find out the real trend of P.