Prove that if f, g are integrable, then the function (f(x) + cos(x)g(x))2is integrable.

From a shipment of 75 transistors, 5 of which are defective, a sample of 6 transistors is selected at random.

(a) In how many different ways can the sample be selected?

(b) How many samples contain exactly 3 defective transistors?

(c) How many samples do not contain any defective transistors?

I have no clue how to do the math here and get the values asked? Please help by working out the equation.  

Two species of algae (called Blue and Green) compete. Their dynamics can be described using the Lotka-Volterra competition equations:

dN_B/dt = r_BN_B(K_B-N_B- a_BGN_G)/K_B

dN_G/dt = r_GN_G(K_G-N_G- a_GBN_B)/K_G                                                                     

The following information is known:

dN_B/dt = 0 when N_B = 100 and N_G = 0

dN_B/dt = 0 when N_B = 50 and N_G = 25

dN_G/dt = 0 when N_B = 50 and N_G = 25

dN_G/dt = 0 when N_B = 0 and N_G = 225

a. (6 pts.) Using this information, determine the values of:

K_B =

K_G =

a_BG =

a_GB =

Consider the differential equation:

y'(x)+3xy+y^2=0.     y(1)=0.    h=0.1

Solve the differential equation to determine y(1.3) using:

a. Euler Method
b. Second order Taylor series method
c. Second order Runge Kutta method
d. Fourth order Runge-Kutta method
e. Heun’s predictor corrector method
f. Midpoint method

Calculate the Euler method approximation to the solution of the initial value problem at the given x-values. Compare your results to the exact solution at these x-values.

y' = y+y^2; y(1) = -1, x = 1.2, 1.4, 1.6, 1.8

Determine the reasonable form of the particular solution for each non homogeneous differential equation. Do not solve it.

a) y''-y'-2y= e^-x+xcos2x+e^xsin2x.

b) D^2[y] +4y =1+x^2+xsin2x.

Let a and b be integers and consider (a) and (b) the ideals they generate. Describe the intersection of (a) and (b), the product of (a) and (b), the sum of (a) and (b) and the Ideal quotient (aZ:bZ).

1. Prove the Heine-Borel Theorem (Theorem 3.35).
2. Suppose f: X → Y maps from the metric space X to the metric space Y, and x ∈ X.
Prove that f is continuous at x if and only if, for any sequence {xn} in X that converges to x, f(xn) → f(x).

Solve SVM for a data set with 3 data instances in 2 dimensions: (1,1,+), (-1,1,-),(0,-1,-). Here the first 2 number are the 2-dimension coordinates. ‘ +’ in 3rd place is positive class. And ‘-‘ in 3rd place is negative class . Your task is to compute alpha’s, w, b. Then, Solve SVM when data are non-separable, using k=2 when minimizing the violations of the mis-classification, i.e., on those slack variables.

For p = 5 , find a number x such that x^2 is congurent to -1 mod p. here x is denoted by sqrt(-1). Determine if sqrt(-1) exists mod p for p = 7, 11, 13 , 17, 23, 29.

How many ways are there to choose three different numbers each between one and a
hundred so that their sum is even? Explain

Prove the following:

Let V and W be vector spaces of equal (finite) dimension, and let T: V → W be linear. Then the following are equivalent.

(a) T is one-to-one.

(b) T is onto.

(c) Rank(T) = dim(V).

2.31. Show that for each of the following values of a and b, there exists x, y in Z satisfying ax + by = 11. (i) a = 11, b = 0, (ii) a = 22, b = 11, (iii) a = 33, b = 22, (iv) a = 451, b = 33, (v) a = 484, b = 451.

2.39. Prove that gcd(ad, bd) = |d|gcd(a, b).

2.44. Does the Diophantine equation 12x + 33y = 1 have an integer solution? If so, can you list all integer solutions?

2.47. For nonzero integers a, b, c, gcd(a, b, , c) denotes the largest integer that divides all of them. Show that gcd(a, b, c) = gcd(a, gcd(b, c)).

Coding: Use MATLAB to figure out the following problem, if you do not know how to use MATLAB then please do not answer. Coding is required for the exercise.

For f(x) = arctan(x), find its zeros by implimenting Newtons method and the Secant method in Matlab. (Hint: Use Newtons method to calculate x1 for Secant method)

Comment all code please since I would like to learn how to do this correctly in MATLAB. Thank you.

Let V be the vector space of all functions f : R → R. Consider the subspace W spanned by {sin(x), cos(x), e^x , e^−x}. The function T : W → W given by taking the derivative is a linear transformation

a) B = {sin(x), cos(x), e^x , e^−x} is a basis for W. Find the matrix for T relative to B.

b)Find all the eigenvalues of the matrix you found in the previous part and describe their eigenvectors. (One of the factors of the characteristic polynomial will be λ 2+1. Just ignore this since it has imaginary roots)

d) Use your answer to the previous part to find all the eigenvalues of T and describe their eigenvectors. Check that the functions you found are indeed eigenvectors of T.