#1. Represent the function: e(t)=5sin31.4t+2sin44t+6sin15.74t+2sin37.68t as a discrete set of N=128 numbers seperated by a time increment of (1/N). Using MatLab construct an amplitude spectrum from the data set.

#2. Repeat problem #1 using a data set of 256 numbers at sigma(t)=(1/N) and sigma(t)=(1/2N) seconds. Please show the full code used in matlab to solve the problems.

The Department of Foreign Languages of a liberal arts college conducted a survey of its recent graduates to determine the foreign language courses they had taken while undergraduates at the college. Of the 550 graduates 207 had at least one year of Spanish. 174 had at least one year of French. 143 had at least one year of German. 34 had at least one year of Spanish and French. 26 had at least one year of Spanish and German. 22 had at least one year of French and German. 3 had at least one year of all three languages.

(a) How many of the graduates had at least 1 yr of at least one of the three languages?
  

(b) How many of the graduates had at least 1 yr of exactly one of the three languages?


(c) How many of the graduates had less than 1 yr of any of the three languages?
  

How many colors are needed to color a planar graphs? Give a proof that each planar graph can be colored with at most 6 colors. (Hint:induction. Use the fact that each planar graph has a vertex of degree no more than 5.) Please include all explanantion and steps. Also draw the diagram too so i understand it better. Thanks!

2. Two images, f(x, y) and g(x, y), have histograms hf and hg. Give the conditions under which you can determine the histograms of,

   • f(x,y) + g(x, y)

   • f(x,y) / g(x,y)
     in terms of hf and hg. Explain how the histogram can be obtained (in terms of hf and hg)

Review the article, The Rise of Big Data by Kenneth Cutier. Discuss the opportunities and challenges, especially social and ethical challenges, related to big data as described in the article.

Problem 2. Give an example of an open set O such that the boundary of the closure of O has positive Lebesque measure. (Hint: try to play with the union of open intervals taken out of [0, 1] during the construction of Cantor set on... ODD steps.)

Problem 6. Show that a closed set is a G_δ and open set is F_δ.

Consider the equation x^2+(y-2)^2 and the relation “(x, y) R (0, 2)”, where R is read as “has distance 1 of”. For example, “(0, 3) R (0, 2)”, that is, “(0, 3) has distance 1 of (0, 2)”. This relation can also be read as “(x, y) belongs to the circle of radius 1 with center (0, 2)”. In other words: “(x, y) satisfies this equation if, and only if, (x, y) R (0, 2)”. Does this equation determine a relation between x and y? Can the variable x can be seen as a function of y, and y be seen as a function of x? What are the domains for these two functions? What are the graphs of these two functions? Are there points of the coordinate axes that relate to (0, 2) by means of R? Your Discussion should be a minimum of 250 words in length and not more than 750 words.

Write a function called ReturnOddEntries.m that accepts as input a column or row array (vector) and returns only the odd index entries. Do this by first setting the even entries to 0, and then removing the 0 entries by using a logical array. The first line of your code should read

function p = ReturnOddEntries(p)

For example, if you run in the command window p = ReturnOddEntries([1.2 7.1 8.4 -42 100.1 7 -2 4 6]), then you should get

p = [1.2 8.4 100.1 -2 6]  

1) Use Euler's method to obtain a four-decimal approximation of the indicated value. Carry out the recursion of (3) in Section 2.6

y_{n + 1} = y_n + hf(x_n, y_n)        (3)

by hand, first using h = 0.1 and then using h = 0.05.

y' = 2x − 3y + 1, y(1) = 4;   y(1.2)

2)Use Euler's method to obtain a four-decimal approximation of the indicated value. Carry out the recursion of the equation

y_{n + 1} = y_n + hf(x_n, y_n)

by hand, first using h = 0.1 and then using h = 0.05.

y' = x + y², y(0) = 0;   y(0.2)

3) Use a numerical solver and Euler's method to obtain a four-decimal approximation of the indicated value. First use

h = 0.1 and then use h = 0.05.

y' = e^−y, y(0) = 0;   y(0.5)

Solve the given initial-value problem.

(x + y)² dx + (2xy + x² − 2) dy = 0,   y(1) = 1

A 4 foot spring measures 8 feet long after a mass weighing 8 pounds is attached to it. The medium through which the mass moves offers a damping force numerically equal to 2^1/2 times the instantaneous velocity. Find the equation of motion if the mass is initially released from the equilibrium position with a downward velocity of 9 ft/a. (Use g =32 ft/s^2 for the acceleration due to gravity.) a) Find the time at which the mass attains its extreme displacement from the equilibrium position. b) What is the position of the mass at this instant? c) The extreme displacement is equal to..

Nothing to add

L: P3 → P1 is a linear mapping and L (g (t)) is the remainder of g (t) divided by t ^ 2 + t - 2.

1. Assume that the matrix A = [L] BB` representing the mapping L when the ordered basis of P3 is B = {t, t ^ 3, t ^ 2} and the oredered basis B of P1 = {1, t} Please.

2. Find the nullity of matrix A from above.

how to derive wave equation?
(not second order, but first order)