Question 1 [35 marks] A foundry that specializes in producing custom blended alloys has received an order for 1 000 kg of an alloy containing at least 5% chromium and not more than 50% iron. Four types of scrap which can be easily acquired can be blended to produce the order. The cost and metal characteristics of the four scrap types are given below: Scrap type Item 1 2 3 4 Chromium 5% 4% - 8% Iron 40% 80% 60% 32% Cost per kg R6 R5 R4 R7 The purchasing manager has formulated the following LP model: Minimise COST = 6M1 + 5M2 + 4M3 + 7M4 subject to 0,05M1 + 0,04M2 + 0,08M4 ≥ 50 (CHRM) 0,40M1 + 0,80M2 + 0,60M3 + 0,32M4 ≤ 500 (IRON) M1 + M2 + M3 + M4 = 1000 (MASS) and all variables ≥ 0, where Mi = number of kg of scrap type i purchased, i=1,2,3,4. (a) Solve this model using LINDO or SOLVER. (b) Write down the foundry's optimal purchasing plan and cost. 4 PBA4804 OCTOBER/NOVEMBER 2019 PORTFOLIO EXAMINATION [TURN OVER] Based on your LINDO or SOLVER solution answer the following questions by using only the initial printout of the optimal solution. (This means that you may not change the relevant parameters in the model and do reruns.) (c) How good a deal would the purchasing manager need to get on scrap type 1 before he would be willing to buy it for this order? (d) Upon further investigation, the purchasing manager finds that scrap type 2 is now being sold at R5,40 per kg. Will the purchasing plan change? By how much will the cost of purchasing the metals increase? (e) The customer is willing to raise the ceiling on the iron content in order to negotiate a reduction in the price he pays for the order. How should the purchasing manager react to this? (f) The customer now specifies that the alloy must contain at least 6% chromium. Can the purchasing manager comply with this new specification? Will the price charged for the order change?

Show that g(x) = 1/2sin(x-1) has a unique fixed point on the interval [-1,1]

Estimate the number of iterations required to achieve 10^-2 accuracy

Find all homomorphism

(1) f : z -> z_5

(2) f: z_5 -> z_5

(3) f: z_3 -> S_3

1) what are Cauchy’s Integral Theorem and Cauchy's integral formula

2) Explain about the consequences and applications of these theorems.

3) Explain about different types of singularities and the difference between Taylor series and Laurent series.

Given the plot of y=f(x) below, find the plot of y=f−1(x).

A coordinate plane has a horizontal x-axis labeled from negative 7 to 7 in increments of 1 and a vertical y-axis labeled from negative 7 to 7 in increments of 1. A curve starts at the point left-parenthesis negative 1 comma 0 right-parenthesis, rises at an increasing rate from left to right and passes through left-parenthesis 1 comma 1 right-parenthesis and left-parenthesis 4 comma 6 right-parenthesis.

Select the correct answer below:

A coordinate plane has a horizontal x-axis labeled from negative 7 to 7 in increments of 1 and a vertical y-axis labeled from negative 7 to 7 in increments of 1. A curve starts at the point left-parenthesis 0 comma negative 1 right-parenthesis, rises at a decreasing rate from left to right and passes through left-parenthesis 1 comma 1 right-parenthesis and left-parenthesis 4 comma 3 right-parenthesis.

A coordinate plane has a horizontal x-axis labeled from negative 7 to 7 in increments of 1 and a vertical y-axis labeled from negative 7 to 7 in increments of 1. A curve starts at the point left-parenthesis 0 comma 1 right-parenthesis, rises at a decreasing rate from left to right and passes through left-parenthesis 1 comma 3 right-parenthesis and left-parenthesis 4 comma 5 right-parenthesis.

A coordinate plane has a horizontal x-axis labeled from negative 7 to 7 in increments of 1 and a vertical y-axis labeled from negative 7 to 7 in increments of 1. A curve starts at the point left-parenthesis 1 comma 0 right-parenthesis, falls at an increasing rate from right to left and passes through left-parenthesis negative 1 comma negative 1 right-parenthesis and left-parenthesis negative 3 comma negative 4 right-parenthesis.

A coordinate plane has a horizontal x-axis labeled from negative 7 to 7 in increments of 1 and a vertical y-axis labeled from negative 7 to 7 in increments of 1. From left to right, a curve falls shallowly in quadrant 2, passing through the points left-parenthesis negative 4 comma 6 right-parenthesis and left-parenthesis negative 2 comma 2 right-parenthesis, crosses the positive y axis at 0.5, and ends at left-parenthesis 1 comma 0 right-parenthesis.

Can someone explain Bessel's Equations in Layman's terms

If a function f(x) is odd about a point, say (a,0), on the x-axis what exactly does this mean? How would you relate f(x values to left of a) to f(x values to right of a)?

Similarly, if a function f(x) is even about a point, say (a,0), on the x-axis what exactly does this mean? How would you relate f(x values to left of a) to f(x values to right of a)?

I understand what is meant by odd and even functions about the origin, I just want to make sure I understand how a function can be odd or even about a point on the x axis. Can you also have odd and even functions about points on the y-axis? How would you express these?

Find a Formula for the degree 2 Taylor polynomial T₂(x,y) at (a,b)=(pi/2,0). Do not simplify your formula. Use a 3d graphing tool to verify T2(x,y) does a good job of approximating f(x,y) near (a,b)

3. What are the necessary and sufficient conditions for a bipartite graph to have a perfect matching? Justify your answer.

4. Illustrate Lemma 3.1.21 using the Peterson graph.

8. This problem is about the definition of periodic function. We assume you already know intuitively what periodic means, and now we want a formal definition. For simplicity, we will restrict ourselves to functions with domain R. A naive (but incorrect) definition of periodic function with period T is

   f (x + T ) = f (x)

   Without accompanying words, this is not a good definition because it does not introduce the variables x and T and it does not explain their role. For which values of x and T does the above definition have to be valid?

   Here is an attempt at a definition, with various ways to complete it:Definition. Let f be a function with domain (−∞, ∞). We say that

   f is periodic when...
   (a) Foreveryx∈(−∞,∞)andforeveryT >0,f(x+T)=f(x).
   (b) For every x ∈ (−∞,∞) there exists T > 0 such that f(x+T) = f(x). (c) There exists T > 0 such that x ∈ (−∞, ∞) =⇒ f (x + T ) = f (x). (d) There exists T > 0 such that for every x ∈ (−∞,∞), f(x+T) = f(x). (e) For every T > 0 there exists x ∈ (−∞,∞) such that f(x+T) = f(x).

9. One or more of the above are valid ways to complete the definition of periodic function. Identify which ones are correct and which ones are wrong. For any property which is wrong, show it by giving an example of a function which satisfies the property but is not periodic, or an example of a function which is periodic but does not satisfy the property. It is okay to give your examples as equations or graphs.

Prove that a subharmonic function remains subharmonic if the independent variable is subjected to a conformal mapping.

A company needs to purchase several new machines to meet its future production needs. It can purchase three different types of machines A,B,andC. Each machine A costs $80,000 and requires 2,000 square feet of floor space. Each machine B costs$50,000 and requires 3,000 square feet of floor space. Each machine C costs $40,000 and requires 5,000 square feet of floor space. The machines can produce 200, 250 and 350 units per day respectively. The plant can only afford $500,000 for all the machines and has at most 20,000 square feet of room for the machines. The company wants to buy as many machines as possible to maximize daily production.

Let                       Xi= number of machines to be purchased

MAX:                         200X1+250X2+300X3

Subject to:            2X1+3X2+5X3≤20

80X1+50X2+40X3≤500

X1,X2,X3≥0

How many of each machine should they purchase?

A floor-refinishing company charges $1.83 per square foot to strip and refinish a tile floor for up to 1000 square feet. There is an additional charge of $350 for any job over 150 square feet.

a) Create a piecewise model to express the cost, C, of refinishing a floor as a function of the number of square feet, s, to be refinished.
b) Graph the function. Be sure to label your axis and use an appropriate scale.
c) Give the domain and range.

Will Rogers spun a lasso in a vertical circle. The diameter of the loop was 6 ft, and the loop spun 50 times each minutes. If the lowest point on the rope was 6 inches above the ground, write an equation to describe the height of this point above the ground after t seconds.

Please write nicely.