Consider the group Z/24Z.

(a) Find the subgroups 〈21〉 and 〈10〉.
(b) Find all generators for the subgroup 〈21〉 ∩ 〈10〉.

(c) In general, what is a generator for 〈a〉 ∩ 〈b〉 in Z/nZ? Prove your assertion.

Calculate the first three terms in the power series solutions of the following differential equations taken about x=0.

x^2y''+xy'+(x^2-1/9)y=0

Let U be a subset of a vector space V. Show that spanU is the intersection of all the subspaces of V that contain U. What does this say if U=∅? Need proof

The figure shows a circuit containing an electromotive force, a capacitor with a capacitance of C farads (F), and a resistor with a resistance of R ohms (Ω). The voltage drop across the capacitor is Q/C, where Q is the charge (in coulombs, C), so in this case Kirchhoff's Law gives

RI + Q/C =E(t)

But I = dQ/dt, so we have

R(dQ/dt) + 1/C (Q) = E(t)

Suppose the resistance is 10 Ω, the capacitance is 0.05 F, and a battery gives a constant voltage of 60 V.

If the initial charge is Q(0) = 0 C, use Euler's method with step size 0.1 to estimate the charge, Q after half a second. (Round your answer to two decimal places.)
Q(0.5) = ???

If the salary is paid at the end of the year, how much will John get?

1. He can keep his current job at the management firm D&L. His annual salary at the firm is $65,000 per year and is salary is expected to increase at 3% per year until retirement. He is currently 28 years old and he expects to work for 40 more years. His current job includes a full paid health insurance plan and is current average tax rate is 26%. John has a savings account with enough money to cover the entire cost of the MBA program.

2. The Carlton College offers a one-year MBA program. The tuition cost is $85,000 to be paid upon matriculation. Books and other supplies for the program are expected to cost $4,500. The Carlton program is a full-time one that does not allow students to work in the meantime. John thinks that after the Carlton degree he will be able to receive an offer of $92,000 per year with a $18,000 signing bonus. The salary at this job will increase at 3.5% per year. His average tax rate at this level of income will be 29%.

factor all integers between 2 and 200 using the canonical order

Suppose that your air conditioner fails on Sunday at midnight (t0 = 0), and you cannot afford to have it repaired until payday at the end of the month. Assume that the outside temperature varies according to the function.

A(t)= 80 − 5 cos(π/12t)-5√3sin(π/12t)

and that your inside temperature, u(t) obeys Newton’s law of cooling and is governed by the differential equation

du/dt=−0.2(u − A(t))

(a) If your indoor temperature when the air conditioner failed was 70◦F, determine the dynamics of temperature inside your apart- ment over time. i.e. find a particular solution to the initial value problem.

(b) What will the temperature inside the apartment be, 24 hours after the break down?

(c) In the long run (t → ∞), what is the maximum and minimum temperature you can anticipate inside your apartment?

(d) Plot the graph of the outdoor and indoor temperature on the same axis and comment on how long it takes for the indoor temperature to reach a maximum after the outdoor temperature peaks.

1. (1) Suppose that V is a vector space and that S = {u,v} is a set of two vectors in V. Let w=u+v, let x=u+2v, and letT ={w,x} (so thatT is another set of two vectors in V ). (a) Show that if S is linearly independent in V then T is also independent. (Hint: suppose that there is a linear combination of elements of T that is equal to 0. Then ....). (b) Show that if S generates V then T also generates V . (Hint: try solving for u and v in terms of w and x.). (c) Summarize the results of parts (a) and (b), correctly employing the word “basis”.

#MATLAB

The following table of values gives experimentally-determined values of the angles of incidence and refraction of light through water. The known index of refraction of water is sinθ_i/ sinθ_r = 1.33 in which θ_i is the angle of incidence and θ_r is the angle of refraction. Make a plot that shows the experimental values as well as the theoretical values for 0 ≤ θ_i ≤ π/2. Experimental values should be filled symbols and the theoretical values should be a smooth line. Give the plot appropriate x and y labels, a title, and a legend.

Let p be an odd prime.

(a) (*) Prove that there is a primitive root modulo p² . (Hint: Use that if a, b have orders n, m, with gcd(n, m) = 1, then ab has order nm.)

(b) Prove that for any n, there is a primitive root modulo pⁿ.

(c) Explicitly find a primitive root modulo 125.

Please do all parts.

Thank you in advance

Let S: R²---> R² be a reflection in the line x₂= Ax_1. Find the standard matrix for S in terms of A.

Show that the following two problems are equivalent: P1 : Minimize cx subject to b1 < Ax < b2 where x > =0. and P2 : Minimize cx subject to Ax + s = b2 where x >= 0, 0 <= s < = b2 - bj. Use the simplex method for bounded variables to solve the following problem after reformulating it as above: Minimize 3x1 - 4x2 subject to 3 < xj + x2 < 4 -15 < 3xj - 5x2 < 2 Xl, x2 > 0.

A regular hexagonal foundation 3 in. deep is being poured for a hexagonal building. If the distance across the flats of the foundation is 27 ​ft, how many cubic yards of concrete are​ needed?

Determine the convergence or divergence if each integral by using a comparison function. Show work using the steps below:

A. Indicate the comparison function you are using.

B. Indicate if your comparison function is larger or smaller than the original function.

C. Indicate if your comparison integral converges or diverges. Explain why.

D. State if the original integral converges or diverges. If it converges, you don’t need to give the value it converges to.

16. integral from 0 to 1((e^(-x))/(√x)) dx