The instructions for the given integral have two​ parts, one for the trapezoidal rule and one for​ Simpson's rule. Complete the following parts.

Integral from 0 to pi ∫4sint dt

T=?

​(Simplify your answer. Round to four decimal places as​ needed.)

An upper bound for AbsoluteValueET is ?

​(Round to four decimal places as​ needed.)

b. Evaluate the integral directly and find ET.

Integral from 0 to pi ∫4sint dt =?

​(Type an integer or a​ decimal.)

AbsoluteValueET =?

​(Simplify your answer. Round to four decimal places as​ needed.)

c. Use the formula AbsoluteValueET​/(true value)) times ×100 to express AbsoluteValueET as a percentage of the​ integral's true value.

?​%

​(Round to one decimal place as​ needed.)

S=?

​(Simplify your answer. Round to four decimal places as​ needed.)

An upper bound for

AbsoluteValueES is ?

​(Round to four decimal places as​ needed.)

b. Evaluate the integral directly and find AbsoluteValueES.

Integral from 0 to pi 4 ∫4sint dt=?

​(Type an integer or a​ decimal.)

ES=?

​(Round to four decimal places as​ needed.)

c. Use the formula AbsoluteValueES​/(true value)) times ×100

to express AbsoluteValueES as a percentage of the​ integral's true value.

? ​%

​(Round to one decimal place as​ needed.)

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Using the formulas for geometric progression to prove the formulas of growing annuity and growing perpetuity

1. Suppose that New York fire department receives an average of 12 requests for fire engines each hour, and that these requests occur according to a Poisson process. Each request causes a fire engine to be unavailable for an average of 12 minutes. To have at least a 90% chance of being able to respond to a request, how many fire engines should the fire department have?

2. Two one-barber shop sit side by side in New York. Each shop can hold a maximum of 5 people, and any potential customer who finds a shop full will not wait for a haircut. Barber 1 charges $18 per haircut and takes an average of 20 minutes to complete a haircut. Barber 2 charges $13 per haircut and takes 15 minutes to complete a haircut. On average 12 potential customers arrive per hour at each barber shop. Of course, a potential customer become an actual customer only if he or she finds that the shop is not full. Assuming that inter-arrival times and haircut times are exponential, which barber will earn more money?

Determine a lower bound for the radius of convergence of series solutions about each given point x0 for the given differential equation.

(1 + x^3)y'' + 4xy' + 6xy = 0   

x0 = 0. x0 = 4

1. Prove that if the primal minimisation problem is unbounded then the dual maximisation

   problem is infeasible.

Find a particular solution for

?^2?′′ + ??′ − 4? = ?^3

Given the fact that the general homogeneous solution is ??(?) = ?1(?^2) + ?2t(?^−2)

Study the roots of the nonlinear equation f(x) = cos(x) + (1 /(1 + e^2x)) both theoretically and numerically. (a) Plot f(x) on the interval x ∈ [−15, 15] and describe the overall behaviour of the function as well as the number and location of its roots. Use the “zoom” feature of Matlab’s plotting window (or change the axis limits) in order to ensure that you are identifying all roots – you may have to increase your plotting point density in order to see sufficient detail!

Please provide MATLAB code as well.

13.7 Table EX 13.7 shows the precedence relationships among the activities to complete a project.

Table EX 13.7

1. Construct an activity on node network for the project.
2. Identify the paths and path project durations.
3. Determine the critical path and the expected project completion time.
4. Find the ES, LS, EF, LF, and slack time for each activity.

6. Passwords are composed from lower- and uppercase letters of the English alphabet,digits and 34 special characters. What is the exponential generating function of the sequence an=number of passwords with at least one capital letter, one number and one special character.

Use the Table button in the Rich-Text Editor to provide an adjacency matrix for a simple graph that meets the following requirements:

1) has 5 vertices
2) is maximal planar

NOTE- If it is true, you need to prove it and If it is false, give a counterexample

f : [a, b] → R is continuous and in the open interval (a,b) differentiable.

a) If f(a) ≥ f(b), then exists a ξ ∈ (a,b) with f′(ξ) ≤ 0. (TRUE or FALSE?)

b) If f is reversable, then f −1 differentiable. (TRUE or FALSE?)

c) f is constant ⇐⇒ ∀x∈(a,b): f′(x)=0 (TRUE or FALSE?)

Suppose we modified the Pollard rho method, so the iteration would be f(x) = x^2 mod n instead of f(x) = x^2 + 2 mod n. How well would it perform, and why?

The price of a European call that expires in six months and has a strike price of $30 is $2. The underlying stock price is $29, and a dividend of $0.50 is expected in two months and again in five months. Risk-free interest rates (all maturities) are 10%. What is the price of a European put option that expires in six months and has a strike price of $30?

The price of an American call on a non-dividend-paying stock is $4. The stock price is $31, the strike price is $30, and the expiration date is in three months. The risk-free interest rate is 8%. Derive upper and lower bounds for the price of an American put on the same stock with the same strike price and expiration date. What are the percentage changes in the values of the two portfolios for a 5% per annum increase in yields?

Calculus 2 question. Please explain clearly.

A bank account earns 5% interest compounded monthly. Suppose that $1,000 is initially deposited into the account, but that $10 is withdrawn each month.

a) Show that the amount in the account after n months is A_n=(1+0.05/12)A_n-1-10; A₀=1000

b) How much money will be in the account after 1year?

c) Suppose that instead of $10, a fixed amount of d dollars is withdrawn each month from the account. Find a value of d such that the amount in the account after each month remains at $1,000

Sn = (1+(1/n))^n

(a) Prove Sn is strictly increasing (b) bounded below by 2 and above by 3

(c) Sn converges to e

(d) Obtain an expression for e

(e) Prove e is irrational