Use Laplace transform method to solve the following initial value problems

(a) d2y/dt2 + y = e^ −t ; y(0) = 0, y′ (0) = 0.

(b) d²y/dt²+ y = t subject to the initial conditions y(0) = 0, y′ (0) = 2

(c) dy/dt + 2y = 4e ^3t subject to the initial condition y(0) = 1.

In a population of 200,000 people, 40,000 are infected with a virus. After a person becomes infected and then recovers, the person is immune (cannot become infected again). Of the people who are infected, 5% will die each year and the others will recover. Of the people who have never been infected, 35% will become infected each year. How many people will be infected in 4 years? (Round your answer to the nearest whole number.)

________ people

In parts a-d evaluate the following determinants. show all steps.

a. 2x2 matrix the first row being 1 and 2 the second row being -3 and 4.

b. 3x3 matrix, the first row being 2,1, 5, the second row being 0, 3, 2, the third row being 0, 0, 4.

c. 3x3 matrix, the first row being 3, -1, 4, the second row being 2, -2, 3, the third row being 1, -1, 2

d. 4x4 matrix, the first row being 1, 1, 0, 3, the second row being 0, 2, 0, 0, the third row being 0, 3, -2, 1, the fourth row being 0, 4, 3, 2.

e. Which matrices in parts a-d are invertible? how do you know? show all steps.

for square matrices A and B show that [A,B]=0 then [A^2,B]=0

a) Use Fermat’s little theorem to compute 52003 mod 7,
52003 mod 11, and 52003 mod 13.

b) Use your results from part (a) and the Chinese remainder
theorem to find 52003 mod 1001. (Note that
1001 = 7 ⋅ 11 ⋅ 13.)

Deduce from the Completeness Axiom that there exists a square root of a real number a if and only if a ≥ 0

Write a code to approximate the derivative of a function  f(x) 
using  forward  finite difference quotient

                      f( x + h ) - f( x )
           f'(x)  ≈  -------------------     (for small h).
                               h 

For the function  f(x) = sin(x), at x=1 ,  compute the FD quotients for
          h = 1/2k, k=5,...,N,  with N=30
and compare with the exact derivative  cos(x).
Output  k , h , error.  Where SHOULD the error tend as h → 0 ?

1. Look at the numbers. Does the error behave as expected ? 
   Output to a file "out" (or to arrays in matlab), and plot it 
        [ gnuplot>  plot "out" u 2:3 with lines ]
   Which direction is the curve traversed, left to right or right 
   to left ?  Look at the numbers.  h is decreasing 
   exponentially, so the points pile up on the vertical axis. The 
   plot is poorely scaled.  To see what's happening, use logarithmic
   scale, i.e. output  k , log(h) , log(error) and replot.

2. What is the minimum error ?  at what k ?
   Why does the error get worse for smaller h ?

3. Repeat, using  centered finite differences
   [copy your code to a another file and modify it]

                    f( x + h ) - f( x - h )
           f'(x) ≈ -----------------------     (for small h).
                            2 h 

4. Which formula performs better ?  in what sense ?

Suppose you have a set of real-valued waveforms {s₁(t), s₂(t),..., s_N(t)}, and you want to find a basis for the span of their complex envelopes. The obvious approach would be to first downconvert each of the waveforms, and then apply the Gram-Schmidt procedure to the set of complex envelopes. Will we get the same answer if we first apply Gram-Schmidt, and then downconvert? Justify your answer.

Please drive Spherical cap equation use diameter of sphere D and height of cap h

Use the data from problem 12 BELOW. If the lead time for placing an order is 5 days, and Technology Corporation holds a safety stock equal to a 30-day supply of chips, then at what inventory level should an order be placed? Enter your answer rounded to two decimal places. For example, if your answer is 12.345 then enter as 12.35 in the answer box.

Q12. Technology Corporation expects to order 126,000 memory chips for inventory during the coming year, and it will use this inventory at a constant rate. Fixed ordering costs are $200 per order; the purchase price per chip is $25; and the firm’s inventory carrying costs is equal to 20 percent of the purchase price. (Assume a 360-day year.) What is the economic ordering quantity for chips? Enter your answer rounded to two decimal places. For example, if your answer is 12.345 then enter as 12.35 in the answer box.

A manufacturer of an external hard disc provided the information that when the price per external hard disk is RM89, then 1 million hard disc will sell. When the price of each hard disc is RM79, then 3 million hard disc will sell. The cost of producing and selling ? million hard disc is ?(?) = 19? + 156.25. a) Find a linear demand function where ? is the number of millions of hard discs sold and ? is the price of each hard disc in RM. b) What is the company’s revenue function for this hard disc? c) What is the company’s profit function? d) How many hard discs must be produced and sold so that the company will have a maximum profit? e) What is the price of hard disc when the company is maximizing their profit? (1 mark)

1.

Set-up the appropriate differential equation(s) and solve to derive the general equation of motion for a human sized “dummy” moving vertically (up/down) under the following assumptions:

(a)The initial elevation is h0 ft.

(b)The initial velocity is V0 ft./sec.

(c)All motion vertical (ignore any sideways motion).

(d)The force due to wind is proportional to velocity and in the opposite

direction of velocity.

(e)The “terminal velocity” is 120mph (e.g.   lim t→∞ (V)= 120 mph).

(f)Force = Mass * Acceleration.

(g)Acceleration due to gravity = 32 ft/sec^2

2.

Assume the dummy is ejected from a test balloon and has initial elevation of 10 miles and initial velocity of 100mph (straight up)

(a) Determine and simplify the equation of motion for this situation

(b) Determine the maximum height, hmax, of the dummy

(c)Determine the dummy’s height at time t = 60 sec

(d)Determine when the dummy will hit the ground in seconds

(e)Determine the speed of the dummy at impact

3.

Repeat (2) using Newton’s equations and compare the answers to those obtained from your model

Problem 1: For the following linear programming problem: ???????? ? = 40?1 + 50?2

Subject to constraints:

3?1 − 6?2 ≥ 30

?1 – 15 ≤ 3?2 2

?1 + 3 ?2 = 24

?1, ?2 ≥ 0

1- Find the optimal solution using graphical solution corner points method or iso profit line method. Please, show the values for state variable, decisions variables, and slack and surplus variables

2- Determine the value for basic solution and non-basic solution, binding constraints and nonbinding constrains, and if there are any redundant constraints

3- Identify if there is any special case solution and state it.

solve using linear programming graphical solution

A mass of 50 g stretches a spring 3.828125 cm. If the mass is set in motion from its equilibrium position with a downward velocity of 50 cms, and if there is no damping, determine the position u of the mass at any time t.

Enclose arguments of functions in parentheses. For example, sin(2x).

Assume g=9.8 ms2. Enter an exact answer.

u(t)=     m

When does the mass first return to its equilibrium position?

Enter an exact answer.

t=     s

James Bond, Q, and M have agreed to meet at a pub after work for drinks. Bond cannot remember if they agreed to meet at the “Fanny on the Hill” or at “My Father’s Moustache” - so he tosses a coin to decide which pub to go to. Q is also in the same predicament; he tosses a coin to decide between “My Father’s Moustache” and “The Quiet Woman”. M faced with same quandary flips a coin first to decide whether or not he needs to head to the “Fanny on the Hill”. If the answer is “no”, then he flips again to decide between “My Father’s Moustache” and “The Quiet Woman”. What is the probability that

(a) Bond and Q meet? (b) Q and M meet? (c) all three meet? (d) all three go to different places? (e) at least two meet?