MATH221 Business Mathematics II
Group Project
Part 1:
For the function
- Find the equations of the two tangent lines at the points x = 0 and x = 1, respectively.
- Find the intersection point of the two tangent lines, if any.
Solution:
To simulate the function f(x) in MATLAB following program is executed to plot and determine the tangent.
![MATH221 business mathematics 2 image 1 MATH221 business mathematics 2 image 1](https://www.assignmenthelp.net/webimg/yah/images/math221-business-mathematics-2-image-1.jpg)
![MATH221 business mathematics 2 image 2 MATH221 business mathematics 2 image 2](https://www.assignmenthelp.net/webimg/yah/images/math221-business-mathematics-2-image-2.jpg)
a)
![MATH221 business mathematics 2 image 3 MATH221 business mathematics 2 image 3](https://www.assignmenthelp.net/webimg/yah/images/math221-business-mathematics-2-image-3.jpg)
The equation of tangent line is,
y = 1 – x, for x = 0,
y = x, for x = 1
- b) The intersection point of the two tangent lines is ( ½ , ½)
![MATH221 business mathematics 2 image 4 MATH221 business mathematics 2 image 4](https://www.assignmenthelp.net/webimg/yah/images/math221-business-mathematics-2-image-4.jpg)
Part 2:
Consider the following demand, supply and total cost functions:
Demand function:
Supply function:
- Determine the price and quantity at the equilibrium.
- Calculate the consumer surplus.
- Calculate the producer surplus.
Solution:
![MATH221 business mathematics 2 image 5 MATH221 business mathematics 2 image 5](https://www.assignmenthelp.net/webimg/yah/images/math221-business-mathematics-2-image-5.jpg)
![MATH221 business mathematics 2 image 6 MATH221 business mathematics 2 image 6](https://www.assignmenthelp.net/webimg/yah/images/math221-business-mathematics-2-image-6.jpg)
- Determine the price and quantity at the equilibrium.
at equilibrium Demand function = Supply function.
Price is 9 units and quantity is 14 units.
- Calculate the consumer surplus.
We have to find the area under demand curve to get Consumer Surplus for this we need to integrate Demand curev w r t Quantity from q = 0 to Equilibrium
Quantity and then subtract area (Price)x(Quantity) at equilibrium
The consumer surplus is 881.
- Calculate the producer surplus.
We have to find the area under demand curve to get Producer Surplus for this we need to integrate Supply curve w r t Quantity from q = 0 to Equilibrium
Quantity and then subtract it from area (Price)x(Quantity) at equilibrium
The Producer surplus is 881.
Part 3
If the function is subject to the constrain
- Use Lagrangian multipliers method to find the critical points of the function .
- Plot the function in the 3-D graph in MATLAB.
- Using Matlab function “fmincon”, find the maximum and minimum of the function .
Solution:
![MATH221 business mathematics 2 image 7 MATH221 business mathematics 2 image 7](https://www.assignmenthelp.net/webimg/yah/images/math221-business-mathematics-2-image-7.jpg)
![MATH221 business mathematics 2 image 8 MATH221 business mathematics 2 image 8](https://www.assignmenthelp.net/webimg/yah/images/math221-business-mathematics-2-image-8.jpg)
- Use Lagrangian multipliers method to find the critical points of the function .
![MATH221 business mathematics 2 image 9 MATH221 business mathematics 2 image 9](https://www.assignmenthelp.net/webimg/yah/images/math221-business-mathematics-2-image-9.jpg)
The critical point is -3 for the function.
- Plot the function in the 3-D graph in MATLAB.
- Using Matlab function “fmincon”, find the maximum and minimum of the function .’
The maximum is -6.0 and minimum is -8.0 of the function.
Part 4
Suppose that a restaurant has certain fixed costs per month of $5000. The fixed costs could be interpreted as rent, insurance etc. The marginal cost function of the restaurant is given by:
dc/dq = [0.8 (0.5q2 - 25q ) + 0.4]
wherec is the total cost in dollars of producing q units of good per week.
- Find the cost of producing units, units and units per week.
- What do you notice? Explain your results.
Solution:
![MATH221 business mathematics 2 image 12 MATH221 business mathematics 2 image 12](https://www.assignmenthelp.net/webimg/yah/images/math221-business-mathematics-2-image-12.jpg)
- Find the cost of producing units, units and units per week.
![MATH221 business mathematics 2 image 13 MATH221 business mathematics 2 image 13](https://www.assignmenthelp.net/webimg/yah/images/math221-business-mathematics-2-image-13.jpg)
The total cost of producing q1 is 1.7208 x 1011, q2 is 6.8410 x 1011 and q3 is 1.7542 x 1011
- What do you notice? Explain your results.
![MATH221 business mathematics 2 image 14 MATH221 business mathematics 2 image 14](https://www.assignmenthelp.net/webimg/yah/images/math221-business-mathematics-2-image-14.jpg)
Total cost is increasing with the increase in the quantity. As it is observed from the graph of the function plotted between price and quantity.
Group |
Part 1 |
Part 2 |
Part 3 |
Part 4 |
2 |
f(x)= x2-x+1 |
D: p=100 e-0.05q S: p= 25 e0.05q | f(x,y)= 4x+3y g(x,y)=x2+y2=100 | dc/dq=[0.8(0.5q^2-25q)+0.4] s q1=12000;q2=19000;q3=26000 |