ECS3145 Homework II
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Homework II
2023-09-29
This homework covers material from module 1 &2 (linear algebra and matrices), modules 3 & 4 (continuity and derivatives). You will explore the following functions in R:
• matrices
• functions
• optimize
• plotting
Question 1
The demand for products x1 and x2 follows the following function:
200 = −3x1 + 7x2
The supply for products x1 and x2 follows the following function:
400 = 5x1 − 10x2
1 - a) Enter matrices of demand and supply for products x1 and x2 . Print the two matrices.
1 - b) How many solutions does this system have? Please use “Solve” to obtain solution to this linear system.
1 - c) Illustrate the solution(s) or no-solution on a plot. You have to include axes titles for x and y and the two equations have to be represented by different colors.
## [1] "#1-a"
## [1] "#1-b"
## [1] "#1-c"
Question 2
The production function for product x1 is the following:
x1 = 1000 − x2 − 4 * x3
The production function for product x2 is the following:
x2 = 1000 − x1 − 2 * x3
The production function for product x3 is the following:
x3 = 500 − 3 * x1
2 - a) Enter the three equations as functions using “functions”. Please print those functions. 2 - b) Enter matrices of the production system. Please print the matrices.
2 - c) How many solutions does this system have? If it exists more than one solution, what is/are the solutions?
## [1] "#2-a"
## [1] "#2-b"
## [1] "#2-c"
Question 3
Find the derivatives of x1 of the following functions:
3 - a)
y = 2 * x1(4)
3 - b)
y = 100 − 2 * x1 + 3 * x2
## [1] "#3 - a"
## [1] "#3 - b"
Question 4
You have to create a plot for each of the equations below. Plots have to include a plot title, axes titles, one or more red dot indicating the extremums (maximum or minimum), as well as the coordinates of the extremums.
f (x) = cos(x) − (x − 5)2 + 1000
over [0, 10]
4-b) Find the local minimum
f (x) = x5 − 5 * x4 − x + 10
over [0, 8]
## [1] "#4 - a"
## [1] "#4 - b"
2023-10-17