Ordinary Differential Equations, Tenenbaum & Pollard, Exercise 15A, Problem 12:

The CO2 content of the air in a 7200-cu-ft room is 0.2 percent. What volume of fresh air containing 0.05 percent CO2 must be pumped into the room each minute in order to reduce the CO2 content to 0.1 percent in 15 min?

A spherical storage tank contains oil. The tank has a diameter of 6 feet. You are asked to calculate the height at which an 8-foot-long dipstick would be wet with oil when immersed in the tank when it contains 6 ft³ of oil. The equation giving the height h of the liquid in the spherical tank for the given volume and radius is given by:

f(h) = h³−9h² + 3.8197

Use the secant method to find roots of equations to find the height at which the dipstick is wet with oil.

Perform three iterations to estimate the root of the equation above.

Find the absolute relative approximate error at the end of each iteration and the least correct number of significant digits at the end of each iteration. (h_-1= 0.5 and h₀ = 1)

20. The concept of best-, worst-, and average-case analyses extends beyond algorithms to other counting problems in mathematics. Recall that the height of a binary tree is the number of edges in the longest path from the root to a leaf.
(a) Find the best-case height of a binary tree with five nodes.
(b) Find the worst-case height of a binary tree with five nodes.
(c) Find the average-case height of a binary tree with five nodes. For this problem, you will have to list all possible binary trees with five nodes. Assume that each of these is equally likely to occur.
(d) Find the worst-case height of a binary tree with n nodes. (e) Approximate the best-case height of a binary tree with n nodes.

the following differential equation determine using the method of undetermined coefficients y''-4y'=8x^4+x^2e^-4x+sin3x

According to modern​ science, Earth is about 4.5 billion years old and written human history extends back about​ 10,000 years. Suppose the entire history of Earth is represented with a 10​-meter-long ​timeline, with the birth of Earth on one end and today at the other end. a. What distance represents 1 billion​ years? b. How far from the end of the timeline does written human history​ begin?

A certain antihistamine is often prescribed for allergies. A typical dose for a 100​-pound person is 19 mg every six hours. Complete parts​ (a) and​ (b) below. a. Following this​ dosage, how many 12.4 mg chewable tablets would be taken in a​ week? b. This antihistamine also comes in a liquid form with a concentration of 12.4 ​mg/8 mL. Following the prescribed​ dosage, how much liquid antihistamine should a 100​-pound person take in a​ week?

Let x(t) ∈ [0, 1] be the fraction of maximum capacity of a live-music venue at time t (in hours) after the door opens. The rate at which people go into the venue is modeled by dx dt = h(x)(1 − x), (1) where h(x) is a function of x only. 1. Consider the case in which people with a ticket but outside the venue go into it at a constant rate h = 1/2 and thus dx dt = 1 2 (1 − x). (a) Find the general solution x(t). (b) The initial crowd waiting at the door for the venue to open is k ∈ [0, 1] of the maximum capacity (i.e. x(0) = k). How full is the venue at t?

2. Suppose people also decides whether to go into the venue depending on if the place looks popular. This corresponds to h(x) = 3 2 x and thus dx dt = 3 2 x(1 − x). (a) Find the general solution x(t). (b) What should be the initial crowd x(0) if the band wants to start playing at t = 2 hours with 80% capacity?

3. Consider the two models, A and B, both starting at 10% full capacity. Model A is governed by the process of question (1) and model B is governed by the process described in question (2). Start this question by writing down the respective particular solutions xA(t) and xB(t). (a) Which of the two models will first reach 50% of full capacity? (b) Which of the two models will first reach 99% of full capacity? (c) Plot the curves xA(t) and xB(t). Both curves should be consistent with: (i) your answers to the two previous items; (ii) the rate of change at t = 0 (i.e., dx dt at t = 0); (iii) the values of x in the limit t → ∞.

How do you do question 3?

Discuss the basic questions that must be addressed in a sampling study. [6 Marks]

Describe a personal scenario in which matrices can be used effectively, then explain the benefits as well as the drawbacks to this type of application.

Given that the square matrix, A is nilpotent (A^k = 0 for some positive integer k). If A is n by n, show that Aⁿ = 0.

Your niece is visiting you, and you are serving her ice cream in a conical glass (conical shape whose bottom is the tip of the cone). The diameter of the opening is 3 inches, and the cone is 6 inches tall.
You plan to fill the cone part way to a height h. Your niece requires that you smooth the ice cream so that it lies perfectly flat. Find a formula for the height of ice cream as a function of the volume V.

Prove the following theorem. If n is a positive integer such that n ≡ 2 (mod 4) or n ≡ 3 (mod 4), then n is not a perfect square.

Let A be a 2 x 2 matric with Schur decomposition UTU^H and suppose that t₁₂≠ 0. Show that

(a) the eigenvalues of A are λ₁ = t₁₁ and λ₂ = t_22.

(b) u₁ is an eigenvector of A belonging to  λ₁ = t_11.

(c) u₂ is not an eigenvector of A belonging to λ₂ = t_22.

Conception of the Integral and convergence of the function

1. We know that if fn—->f is (point-wise or uniformly)and every fn in the interval is Riemann integral, then will f be Riemann integrable

on [a,b]?

please answer this question separately in pointwise and uniformly.

Define d to be the set of all pairs (x,y) of natural numbers such that x divides y. Show that N is partially ordered by d. Define d analogously on Z. Is then d also a partial order on Z?