Consider the movie ticket and popcorn example discussed in Section 17.7. The theater sells two products, tickets and popcorn. Suppose the weekly demand for movie tickets is

          Qdtix=500−25Ptix−20Ppopcorn,

where P_tix and P_popcorn are the prices of a ticket and a bag of popcorn, respectively. Suppose that each time a moviegoer buys a ticket, his demand for popcorn is

          Qdpopcorn=3−0.4Ppopcorn,

where Q^d_popcorn is the number of bags of popcorn the moviegoer buys. Suppose further that the theater's marginal cost of a ticket is $2, while the marginal cost of popcorn is $3.

Instructions: Round your answer to 2 decimal places.

a. What is the profit-maximizing price of a movie ticket if a bag of popcorn sells for $5 a bag?

    $.

Use the Laplace transform to solve the given system of differential equations.

d2x/dt2 + x − y = 0

d2y/dt2 + y − x = 0

x(0) = 0, x'(0) = −4

y(0) = 0, y'(0) = 1

Determine by hand calculation (show all steps) the clamped cubic spline that interpolates the data f(-3) = 2 ; f(-1) = -3 ; f(0) = 1 ; f(3) = 6 ; f(5) = 3 and satisfies s'(-3) = s'(5) = 0

 

1. Write the negation of each statement:

(a) ∀? ∈ ℝ, ∃? ∈ ℝ such that ? < ? 2.

(b) ∀? ∈ ℚ, ∃?, ? ∈ ℕ such that ? = ??.

(c) ∀ even integers ?, ∃ an integer ? such that ? = 2?.

(d) ∃? ∈ ℝ such that for all real numbers ?, ? + ? = 0.

(e) ∃?, ? ∈ ℝ such that if ? < ? then ? 2 < ? 2.

(f) ∀? > 0, ∃? > 0 such that ∀ ? ∈ (? − ?, ? + ?), |?(?) − ?| < ?. Remember this definition from Calculus?!

2. Which of the following statements are true and which are false. Justify your answers. Hint: This is a great problem. You need to be particularly careful about the order of the quantifiers (∀ and ∃).

(a) ∃? ∈ ℝ such that ∀? ∈ ℝ ? 2 + ? 2 = 9.

(b) ∀? ∈ ℝ, ∃? ∈ ℝ such that ? 2 < ? +1.

(c) ∀? ∈ ℤ +, ∃? ∈ ℤ + such that ? = ? + 1.

(d) ∀? ∈ ℤ, ∃? ∈ ℤ such that ? = ? + 1.

(e) ∃? ∈ ℝ such that ∀? ∈ ℝ, ? = ? + 1.

(f) ∀? ∈ ℝ +, ∃? ∈ ℝ + such that ?? = 1.

(g) ∀? ∈ ℝ, ∃? ∈ ℝ such that ?? = 1.

(h) ∀? ∈ ℤ + and ∀? ∈ ℤ +, ∃? ∈ ℤ + such that ? = ? − ?.

(i) ∀? ∈ ℤ and ∀? ∈ ℤ, ∃? ∈ ℤ such that ? = ? − ?.

(j) ∃? ∈ ℝ + such that ∀? ∈ ℝ +, ?? < ?.

(k) ∀? ∈ ℝ +, ∃? ∈ ℝ + such that ?? < ?.

3. Determine whether the following arguments are valid or invalid (specify if by converse or inverse error).

(a) The product of two rational numbers is rational. ?? is rational. Therefore, ? and ? are rational.

(b) If ? and ? are odd, then their sum is even. ? and ? are odd. Therefore their sum is even.

(c) If a ? × ? matrix, ?, has ? distinct eigenvalues, then it has ? linearly independent eigenvectors. ? does not have ? linearly independent eigenvalues. Therefore, ? does not have ? distinct eigenvalues.

4. Use Venn-Diagram to determine if the following arguments are valid:

(a) Everyone taking discrete mathematics can think logically. Everyone who likes ice cream can think logically. Everyone taking discrete mathematics likes ice cream.

(b) All students who took Linear Algebra took Calculus. All students who took Calculus to know how to integrate. All students who took Linear Algebra know how to integrate.

In Section 1.8 of the text there are standard translation matrices defined. Choose two and determine their product.

Use the product matrix and determine the transformation of one of the geometric shapes created by the following points:

a) (0, 0), (-3, 3), and (3, 3)

b) (0, 0), (-3, -3), and (3, -3)

c) (0, 0), (3, 3), and (3, -3)

d) (0, 0), (-1, 4), and (1, 4)

e) (0, 0), (-4, 2), and (4, 2)

Do not post a duplicate example.

Then, create your own geometric shape and identify its vertices. Explain how your shape would be transformed by the product matrix.

State and prove the Generalised Mean Value Theorem.

Suppose that a “word” is any string of six letters. Repeated letters are allowed. For our purposes, vowels are the letters a, e, i, o, and u. a) How many words are there? b) How many words begin with a vowel? c) How many words begin with a vowel and end with a vowel? d) How many words have no vowels? e) How many words have exactly one vowel?

A professor teaching a Discrete Math course gives a multiple choice quiz that has six
questions, each with four possible responses: a, b, c, d. What is the minimum number of students that
must be in the professor’s class in order to guarantee that at least three answer sheets must be identical?
(Assume that no answers are left blank.)

use variation of parameters to solve y''+y'-2y=ln(x)

Classify (if possible) each critical point of the given plane autonomous system as a stable node, a stable spiral point, an unstable spiral point, an unstable node, or a saddle point. (Order your answers from smallest to largest x, then from smallest to largest y.)

x' = x(1 − x2 − 5y2)

y' = y(5 − x2 − 5y2)

Use the power series method to solve the given initial-value problem. (Format your final answer as an elementary function.) (x − 1)y'' − xy' + y = 0, y(0) = −7, y'(0) = 2

The table shows the specifications of an adjustable rate mortgage​ (ARM). Assume no caps apply. Find​ a) the initial monthly​ payment; b) the monthly payment for the second​ adjustment; and​ c) the change in monthly payment at the first adjustment. ​*The principal balance at the time of the first rate adjustment.

Beginning Balance ​$75000

Term 20 years

Initial index rate 5.4​%

Margin 2.6 ​%

Adjustment period 1 year

Adjusted index rate 6.9​%

​*Adjusted balance $73,414.75

What is the initial monthly​ payment?

​(Round to the nearest​ cent.)

What is the monthly payment for the second adjustment​ period?

​(Round to the nearest​ cent.)

How much is the increase in monthly​ payments?

Let K be a field. Observe that the polynomials in K[x] that are not zero and not units are precisely the polynomials of positive degree.

Suppose f is a degree n polynomial where f(a1) = b1, f(a2) = b2, · · · f(an+1) = bn+1 where a1, a2, · · · an+1 are n + 1 distinct values. We will define a new polynomial g(x) = b1(x − a2)(x − a3)· · ·(x − an+1) / (a1 − a2)(a1 − a3)· · ·(a1 − an+1) + b2(x − a1)(x − a3)(x − a4)· · ·(x − an+1) / (a2 − a1)(a2 − a3)(a2 − a4)· · ·(a2 − an+1) + · · · + bn+1(x − a1)(x − a2)· · ·(x − an) / (an+1 − a1)(an+1 − a2)· · ·(an+1 − an).

Find g(a1), g(a2), · · · g(an+1).

In parts 1–2 below, determine whether or not ? is a subgroup of ?. (Assume that the operation of ? is the same as that of ?.). (2)

1. ? = 〈R, +〉, ? = {log?: ?∈Q, ?>0}.

2. ? = 〈R, +〉, H = {log?: ?∈Z, ?>0}.

Was the Elements an exposition of the most advanced mathematics of its time? (Elements by Euclid)