Numerical MethodsPlease show all steps with clear hand writing.

Use numerical methods to Prove that the following equations have at leat one solution in the given intervals.

(a) x − (ln x)³ = 0, [5, 7]

(b) 5x cos(πx) − 2x 2 + 3 = 0, [0, 2]

5. Ternary strings. A ternary string is a word from the alphabet {0,1,2}{0,1,2}. For example, 022101022101 is a ternary string of length 6.

(a) Enumerate all ternary strings of length 2.
(b) Generalize: how many ternary strings have length r?
(c) A ternary string of length 6 is to be made. How many ways can we choose how many 0's, 1's, and 2's to include in the string?
(d) A ternary string of length 6 is to have two 0's, a 1, and three 2's. How many strings have this property?
(e) What is the probability that a ternary string of length 6 has two 0's, a 1, and three 2's?

The cycloid has parametric equations x = a(t + sin t), y = a(1 - cos t). Find
the length of the arc from t = 0 to t = pi. [ Hint: 1 + cosA = 2 cos² A/2 ]. and the arc length of a
parametric

. True or false, 2 pts each. If the statement is ever false, circle false as your answer. No work is required, and no partial credit will be given. In each case, assume f is a smooth function (its derivatives of all orders exist and are continuous).

If f has a constraint g = c, assume that g is smooth and that ∇g is never 0.

(a) If f has a maximum at the point (a, b) subject to the constraint g = c, then we must have f(a, b) ≤ c. TRUE FALSE

(b) If A and B are square matrices and AB is defined, then BA must also be defined. TRUE FALSE

(c) The function f(x, y) = x 2 − 2y has a maximum subject to the constraint x + y = 1. TRUE FALSE

(d) If (x0, y0) is the point where f attains its minimum subject to the constraint g(x, y) = c, then ∇f and ∇g must point in opposite directions at (x0, y0). TRUE FALSE

The formula for a particular solution given in (3.42) applies to the more general problem of solving y" + p(t)y' + q(t)y = f(t). In this case, y₁ and y₂ are independent solutions of the associated homogeneous equation y" + p(t)y' + q(t)y = 0. In the following, show that y₁ and y₂ satisfy the associated homogeneous equation, and then determine a particular solution of the inhomogeneous equation:

b.) ty" - (t + 1)y' + y = t²e^2t; y₁(t) = 1 + t, y₂(t) = e^t (answer should be: 1/2 (t -1) e^2t + 1/2 + t/2 )

c.) t²y" - 3ty' + 4y = t^5/2; y₁(t) = t², y₂(t) = t²ln(t) (answer should be: 4t^5/2 )

Consider the second-order differential equation x2y′′+(x2+ax)y′−axy=0 where a=−2 Is x0=0 a singular or ordinary point of the equation? If it is singular, is it regular or irregular? Find two linearly independent power series solutions of the differential equation. For each solution, you can restrict it to the first four terms of the expansion

Solve ODE (3x - 2y + 1)dx + (-2x + y + 2)dy = 0 with the method of x =u + h and y = v + k

A teacher puts 10,000 lolly pops into a chest weighing 10 pounds and drops it down a well. A group of students are trying to get the lolly pops out the well. A student decided to attach to the chest a 100 ft long chain weighing 10 pounds. Several other students, at the top of the well, start lifting the chest out of the well by pulling on the chain in a hand over hand fashion. As the chest is being lifted, lolly pops fall from an opening in the chest at a constant rate such that the chest will be empty when it reaches the top of the well. Given that an individual lolly Pop weight 0.6 ounces, find the work done in raising the chest to the top of the well. Show integration steps.

ASSUME THE HYPERBOLIC PARALLEL POSTULATE. SHOW AS COMPLETE ARGUMENTS AS POSSIBLE AND INDICATE ALL STEPS OF REASONING. DIAGRAMS SHOULD ILLUSTRATE ALL ARGUMENTS, BUT REMEMBER THAT DIAGRAMS BY THEMSELVES DO NOT CONSTITUTE PROOFS.

Let l be a line and let P be a point not on l.

Prove that there exists a line m through P such that m is parallel to l but l and m do not admit a common perpendicular

(6) Define a binary operation ∗ on the set G = R^2 by (x, y) ∗ (x', y') = (x + x', y + y'e^x)

(a) Show that (G, ∗) is a group. Specifically, prove that the associative law holds, find the identity e, and find the inverse of (x, y) ∈ G.

(b) Show that the group G is not abelian.

(c). Show that the set H= (x*x=e) is a subgroup of G.

A professor gives two types of quizzes, objective and recall. He plans to give at least 15 quizzes this quarter. The student preparation time for an objective quiz is 15 minutes and for a recall quiz 30 minutes. The professor would like a student to spend at least 5 hours (300 minutes) preparing for these quizzes above and beyond the normal study time. The average score on an objective quiz is 7,and on a recall type 5, and the professor would like the student to score at least 85 points on all quizzes. It takes the professor one minute to grade an objective quiz, and 1.5 minutes to grade a recall type quiz. How many of each type should he give in order to minimize his grading time?

Find the inverse of the matrix

A=
2 -1 3

0 1 1

-1 -1 0

A mass of 1 slug, when attached to a spring, stretches it 2 feet and then comes to rest in the equilibrium position. Starting at t = 0, an external force equal to f(t) = 10 sin(4t) is applied to the system. Find the equation of motion if the surrounding medium offers a damping force that is numerically equal to 8 times the instantaneous velocity. (Use g = 32 ft/s² for the acceleration due to gravity.)

x(t) = ____________ft

​Explain why most folks say that alcohol concentration decays linearly until the alcohol concentration decays to the level at which no one cares.​