Question:

1. Prove the following:

Claim: Consider a triangle ▵ABC and a point D on the interior of segment BC. If σ(▵ABC) = 180, then σ(▵ABD) = σ(▵ACD) = 180.

Hint: Use the Split Triangle Theorem and/or the Split Quadrilateral Theorem

Some hints: use the definition: f is a function iff a = b implies f(a) = f(b) and recall that in informal proofs we show an implication by assuming the if part of the implication, and then deducing the then part of the implication.

The base case will show that a = b implies f(a) = f(b) when f(x) = c₀ (a constant function). The inductive case will assume a = b implies f(a) = f(b) for degree k, and will deduce it is also true for degree k+1.

1. Show that every polynomial of degree n:
   y = f(x) = c_nxⁿ + c_n-1x^n-1 + . . . + c₂x² + c₁x + c₀
   is a function by mathematical induction on degree n.
   Assume n is a nonnegative integer, all c_is are real, c_n ≠ 0, and x and y are also real.

1. Given the following relation, { ( A, A ), ( A, B ), ( A, D ), ( B, B ), ( B, C ), ( B, E ), ( C, B ), ( C, C ), ( C, D ), ( D, A ), ( D, B ), ( D, C ), ( D, E ), ( E, D ), ( E, E ) }
   i) Draw the digraph of the relation, ii) construct the matrix diagram for the relation, and iii) why or why not is the relation reflexive, symmetric, antisymmetric, transitive?

discrete math

most important is c) and e) and f) statements with nested quantifiers: variables ...

please with a clear and concise explanation on how to do each steps. So not just the answer but the explanation as well because I'm totally lost on how to do this at all.

Question: Discrete Math Most important is c) and e) and f) Statements with nested quantifiers: variables wi...

Discrete Math

Most important is c) and e) and f)

Statements with nested quantifiers: variables with different domains.

The domain for the first input variable to predicate T is a set of students at a university. The domain for the second input variable to predicate T is the set of Math classes offered at that university. The predicate T(x, y) indicates that student x has taken class y. Sam is a student at the university and Math 101 is one of the courses offered at the university. Give a logical expression for each sentence.

(b)

Every student has taken at least one math class.

(c)

Every student has taken at least one class other than Math 101.

(d)

There is a student who has taken every math class other than Math 101.

(e)

Everyone other than Sam has taken at least two different math classes.

(f)

Sam has taken exactly two math classes.

6. The following data represents a company’s revenue in millions of dollars. Year: 2010 2012 2014 2015 2016 2018 2019 Revenue: 30 32 34 35 39 39 45 Let the year 2010 be the base year with x=0.

a) Model the data with a linear function using the points in years 2010 and 2019. Round computed values to 2 decimal places. Also, using your model predict the revenue in the year 2021 accurate to 3 decimal places.

b) Set up a table of (x, y) values to be used for a least squares model and find a linear least squares model y ax b = + for the data. Express the coefficients a and b accurate to 3 decimal places. Also, what is the 2 R value accurate to 3 decimal places?

c) Use your least squares model to predict the revenue in 2020 accurate to 3 decimal places and graph the actual and predicted revenue data superimposed on the same graph.

What straight line y=ax+b best fits the following data in the least-squares sense?

i. Formulate the problem in the form Ax=b for appropriate A and b (matrix form).

We want to fit in the function g(x) = a sinx + b cosx for a data set

x 1 1.5 2 2.5

y 1.902 0.5447 0.9453 2.204

i. Formulate the problem in the form Ax=b for appropriate A and b (matrix form).

use Lagrange multipliers to find the maximum and minimum values of f subject to the given constraint, if such values exist.  f(x, y, z) = xyz, x² + y² + 4z² = 12

Based on dynamics, describe the behavior of a phase portrait of a n-body problem.

Explain how you can use QR factorization to solve the following problem

Find x1 and x2 that minimize ||Ax1 − b1||^2 + ||Ax2 − b2||^2. The problem data are the m× n matrix A,
and the m-vectors b1 and b2. The matrix A has linearly independent columns.

f(x)=x^3-3x-1=0

x=[0,2]

epsilon=5*10^-2

1. perform the bisection method for the root in [0,2] until your root is closer to the real root within epsilon.

Let x_0=1.0, x_1=1.2

2. perform the secant method until your root is closer to the real root within epsilon.

3. do as in 2. with the Newton's method, with x_0=1.1

1. The values of Alabama building contracts (in millions of dollars) for a 12-month period follow.

   250  340  230  250  280  310  210  320  250  300  240  230

   1. Choose the correct time series plot.
      

      (i)	Value (Millions of dollars) Month (t)
      (ii)	Value (Millions of dollars) Month (t)
      (iii)	Value (Millions of dollars) Month (t)
      (iv)	Value (Millions of dollars) Month (t)

      • Plot (i)
      • Plot (ii)
      • Plot (iii)
      • Plot (iv)
      
      
      What type of pattern exists in the data?
      
      
      • Horizontal Pattern
      • Trend Pattern
      
      
   2. Compare a three-month moving average forecast with an exponential smoothing forecast. Use = 0.2. Which provides the better forecasts based on MSE? Do not round your intermediate calculations and round your final answers to two decimal places.
      

      	Moving average	Exponential smoothing
      MSE

      • Moving average
      • Exponential smoothing
   3. What is the forecast for the next month? Round your answer to the nearest whole number.
      
      _____________ millions of dollars?

Solve the following differential equation. "Preferably using the chain rule/ order reduction".

(d) y′y′′−3(y′)^{2}=y^{2} y′′.

(e) (y′)2y′′=y.

Postal regulations specify that a parcel sent by parcel post may have a combined length and girth of no more than 130 in. Find the dimensions of a cylindrical package of greatest volume that may be sent through the mail, using the method of Lagrange multipliers

4. Let n ≥ 8 be an even integer and let k be an integer with 2 ≤ k ≤ n/2. Consider k-element subsets of the set S = {1, 2, . . . , n}. How many such subsets contain at least two even numbers?