1. Develop a well-structured MATLAB function to compute the Maclaurin series expansion

   for the cosine function and name the function cosMac. The function should have the following features:

   1. Iterate until the relative error (variable name “relErr” ) falls below a stopping criterion OR exceeds a maximum number of iterations (variable name“maxIter”) for a given value of x.

   2. Include a default value of relErr = 1e-6 if the user does not enter the value (use nargin function).

   3. Include a default value for maxIter = 100 if the user does not enter the value (use nargin function).

   4. Return the estimate of cos(x), the approximate relative error, the number of iterations, and the true relative error (that you can calculate based on the built- in cosine function).

Use the Newton’s method to find the root for e^x+1 = 2 + x, over [−2, 2]. Can you find a way to numerically determine whether the convergence is roughly quadratic using error produced at each iteration? Include your answers as Matlab code comments

Describe a setting where you could use exponential functions to make investment decisions.

What kind of information could exponential functions tell you that would be valuable?

show that for any n the matrix ring M_n(F) is simple over a field F.

show your work. Do not use quotient rings!

A number of guests gather around a round table for a dinner. Between every adjacent pair of guests, there is a plate for tips. When everyone has finished eating, each person places half their tip in the plate to their left and half in the plate to their right. Suppose you can only see the amount of tips in each plate after everyone has left. Can you deduce the amount that each individual tipped?

(a) Suppose six guests sit around a table and there are six plates of tips. If we know the amount of tip in each plate, P1 to P6, can we determine each individual’s tip amount, G1 to G6 (G1+G6 = P1, G1+G2 = P2, G2+G3 = P3, ... , G5+G6 = P6)? If yes, explain why by examining the relationship between the plate values, P1 to P6, and guest tips, G1 to G6. If not, give two different assignments of G1 to G6 that will result in the same P1 to P6.

(b) Now lets consider five guests at the table, G1 to G5, and we can see the amount of tips in the five plates, P1 to P5 ((G1+G5 = P1, G1+G2 = P2, G2+G3 = P3, ... , G4+G5 = P5)). In this new setting can you figure out each guests tip values, G1 to G5?

(c) If n is the total number of guests sitting around a table, for which values of n can you figure out everyone’s tip? You do not have to rigorously prove your answer. (Hint: consider what is different about parts a and b.)

Suppose that we have access to an unlimited number of 5 and 11 cent stamps. Prove, using simple induction, that we can use these stamps to make any amount of postage that is at least 40 cents.

1. Calculate the weighted averages for two additional used cars using the weights given in the table below.

Make sure that you fill in each answer area before checking "How Did I Do?".

1. Weighted average for Car 4:  
2. Weighted average for Car 5:   

• In your weighted average equations in the problem above, how many total weights are used?
  
  
  
• For each category, what percent of the total weights are represented by the category weights? Enter your answers in the table.
  

  	New Weights as a Percentage of Total Weights
  Reliability	%
  Gas Mileage	%
  Interior Features/Comfort	%
  Cargo Space	%

  
  
• In the tables below, multiply each category’s percentage weight by its rating, and add up the four amounts.
  

  	Percentage Weights (written as a decimal)		Car 4 Ratings
  Reliability		×	7	=
  Gas Mileage		×	5	=
  Interior Features/Comfort		×	8	=
  Cargo Space		×	6	=
  Total

  
  
  

  	Percentage Weights (written as a decimal)		Car 5 Ratings
  Reliability		×	6	=
  Gas Mileage		×	7	=
  Interior Features/Comfort		×	7	=
  Cargo Space		×	8	=
  Total

  
  
• Write an equation for the weighted average using the percentage weights as decimals. Use the following variables to represent the individual rating in each category: R= Reliability; G= Gas Mileage; I= Interior Features/Comfort; C= Cargo Space.
  
  Weighted average =

Find a particular solution to y′′+4y′+4y=(e^−2x)/x^4 using variation of parameters

yp= ?

Construct a derivation (for the following argument) from the premise to the conclusion, only using these :(&I, &E, ∨I, ∨E, ≡ I, ≡ E ,⊃ I, ⊃ E,∼ I, ∼ E)

(a) Premise 1: W ⊃ X

     Premise 2: X ⊃ Y

     Premise 3: Y ⊃ Z

     Premise 4: W

    Conclusion: W & Z

(b)Premise 1: (A & B) ≡ C

    Premise 2: A ≡ B

   Conclusion: A ≡ C

(c)Premise 1: B ⊃ D

Premise 2: C ⊃ D

Conclusion: (B ∨ C) ⊃ D

(d) Premise 1: B ⊃ C

Conclusion: ∼ C ⊃∼ B

Let a,b be any relatively prime positive numbers. (The case b = 1 is allowed). We have the rational number a/b. Without appealing to anything but Euclid’s lemma (no use of ε(p, m)) show that if p is prime then p does not equal (a/b)^2. That is √p is irrational. Hint: if p = (a/b)^2 then we have p(b^2) = a^2. Derive p|b and then show p|a which is a contradiction.

What is the relation between “Green’s Theorem” and “Stokes’s Theorem”?

Explain about the transformations defined in these theorems.

What is the most important application and consequence of “Stokes’s Theorem”?  

Explain Independence of path?

Use laplace Transform in solving the ff.:

After cooking for 45 minutes, when a cake is removed from an oven its temperature is measured at 300°F. 3 minutes later its temperature is 200°F. The oven is pre heated, and so at t=0, the cake mixture is at the room temperature of 70°F. The temperature of the oven increases linearly until t=4 minutes, when the desired temperature of 300°F is attained; thereafter the oven temperature is constant 300°F for t is greater than or equal to 4 minutes.

Devised a mathematical model for the temperature of a cake while it is inside the oven and after it is taken out of the oven.

View S₃ as a subset of S₅ in the obvious way. For σ, τ ∈ S₅, define σ ∼ τ if στ ^-1 ∈ S₃.

(a) Prove that ∼ is an equivalence relation on S5.

(b) Find the equivalence class of (4, 5).

(c) Find the equivalence class of (1, 2, 3, 4, 5).

(d) Determine the total number of equivalence classes

In this project you are given the task of estimating the number of trees that will maximize the fruit production per acre of a local orchard. By researching other orchards, you discover that at 400 trees per acre, each tree will produce an average of 30 pounds of fruit. For each tree that is added to the acre, the average production per tree goes down 0.05 pounds

1. Use the information to build a table of values for this situation. Let t be the number of trees (above 400) planted per acre and find the total amount of fruit each acre will produce. Find at least eight sets of values. (Hint: You should add 20 trees at a time to obtain your values.)

2. Create a scatterplot using the values you found in the table on the calculator or computer and print it out.

3. Find a QUADRATIC function for the total production per acre when t trees over 400 are planted per acre.

4. Find the total production per acre when 400 trees are planted.

5. Estimate how many trees per acre should be planted per acre to produce 12,400 pounds of fruit.

6. Find the number of trees per acre that will maximize fruit production.

7. Find a reasonable domain and range for your model.