Please solve all of the questions, questions 1, 2, and 3. Please show all work and all steps.

1.) Find x(t) = Σ a_kt^k such that tx'' = x

2.) Find x(t) = Σ_k>=0 a_kt^k such that x'' = tx + 1 and x(0) = 0, x'(0) = 1

3.) Using the Frobenius method, solve t²x'' - 3tx' + (4-t)x = 0

1. Use induction to prove that Summation with n terms where i=1 and Summation 3i 2 − 3i + 1 = n^3 for all n ≥ 1.

2. Let X be the set of all natural numbers x with the property that x = 4a + 13b for some natural numbers a and b. For example, 30 ∈ X since 30 = 4(1) + 13(2), but 5 ∈/ X since there’s no way to add 4’s and 13’s together to reach 5. (It’s not a multiple of 4, and adding 13 goes over.) Use strong induction to prove that n ∈ X for all integers n ≥ 36. Hint: it should be easy to show that k + 1 ∈ X if k − 3 ∈ X. You may need multiple base cases for this problem

2.6 Consider all the possible sets of two square roots s, t of 1 (mod 35) where s ≢ t (mod 35) (there are six of them, since addition is commutative (mod 35). For all possible combinations, compute gcd(s + t, 35). Which combinations give you a single prime factor of 35?

2.7 Using CRT notation, show what is going on for all the combinations you considered in #2.6. Explain why gcd(s + t, 35) sometimes gave you a factor, and it sometimes did not.

2.8 Explain how you can make a digital signature that is mathematically equivalent to factoring using the results you considered in this assignment.

A company owns two factories A and B that produce 3 different types of appliances dishwashers, refrigerators and ovens. Each day factory A operates it produces 80 dishwashers, 10 refrigerators and 50 ovens at a cost of $10,000 per day. Each day factory B operates it produces 20 dishwashers, 10 refrigerators and 20 ovens at a cost of $20,000 per day. The company has to fill an order of at least 1600 dishwashers, 500 refrigerators and 1900 ovens. How many days should they operate each factory to fill the orders and minimize cost? What is the minimum cost?

5. A town offers a lottery. To win the grand prize of $1,000,000, your ticket needs to consist of the winning seven numbers chosen from the set {1, 2, …, 55}. To win the lesser prize of $10,000, your ticket needs to have exactly five of the seven.

a) What is the probability of winning the grand prize?

b) What is the probability of winning the lesser prize?

c) If a lottery ticket costs $1, what is the expected value of playing this lottery?

1.Logical equivalence of two English statements.

Define the following propositions:

• j: Sally got the job.
• l: Sally was late for her interview
• r: Sally updated her resume.

Express each pair of sentences using a logical expression. Then prove whether the two expressions are logically equivalent.

(a)

If Sally did not get the job, then she was late for interview or did not update her resume.
If Sally updated her resume and was not late for her interview, then she got the job.

(b)

If Sally got the job then she was not late for her interview.
If Sally did not get the job, then she was late for her interview.

(c)

If Sally updated her resume or she was not late for her interview, then she got the job.
If Sally got the job, then she updated her resume and was not late for her interview.

can you please provide a truth table for each?

1. Define Limit , Continuity, and Differentiation

2. Provide some applications in the business area using the above 3 concepts. How will these concepts be useful in your Major subject?

You can refer to any books or other sources to answer this, except your My Mathlab or its Textbook.

Design a linear-time algorithm which, given an undirected graph G and a particular edge e in it, determines whether G has a cycle containing e. Your algorithm should also return the length (number of edges) of the shortest cycle containing e, if one exists. Just give the algorithm, no proofs are necessary. Hint: you can use BFS to solve this.

Explain the difference between IPv4 and IPv6 addressing schemes and discuss that how many different IPv4 and IPv6 addresses are available for computers on the Internet?

Let n be a positive integer. Let S(n) = n sigma j=1 ((1/3j − 2) − (1/3j + 1)). a) Compute the value of S(1), S(2), S(3), and S(4). b) Make a conjecture that gives a closed form (i.e., not a summation) formula for the value of S(n). c) Use induction to prove your conjecture is correct.

Estimate the dose from CT for a prostate patient (total dose 78 Gy) who undergoes the following:

- 2 CT simulations

- 39 KVCT localization images

- 1 Midcourse Evaluation CT

- 1 Post Treatment CT

Please show all work.

Owen is jumping on a trampoline. When his feet hit the deck of the trampoline, the material depresses to a minimum height of 2cm. On average, Owen is reaching a maximum height of 200cm every 10 seconds. Determine the equation of a sinusoidal function that would model this situation, assuming Owen reaches his first maximum at 6 seconds.