4. Show that the set A = {f_m,b : R → R | m does not equal 0 and f_m,b(x) = mx + b, m, b ∈ R} forms a group under composition of functions. (The set A is called the set of affine functions from R to R.)

For each of the following groups, find all the cyclic subgroups: (a) Z₈, (b) Z₁₀, (c) Z ^x₁₀, (d) S₄.

You bought 500 Forever stamps just before the price went up in January of 2014 at $0.46/stamp, a $0.03 savings per stamp. If you could have paid off a credit card charging 12% per year instead of investing in stamps, how fast must you use the stamps to breakeven? Assume the $0.49 price never changes before you run out of stamps. (Hint use 12% /year as your effective interest rate per year)

a. 3 months

b. 6 months

c. 10 months

d. 12 months

e. 36 months

We saw in class that each LP can be transformed into an equivalent LP in any of the following two forms below: (1) maxcTx: Ax=b, x≥0 (2) maxcTx: Ax≤b. Can we always transform any LP in an LP of the form Prove your answer correct. maxcTx: Ax=b?

For the wave equation, utt = c2uxx, with the following boundary and initial conditions,

u(x, 0) = 0

ut(x, 0) = 0.1x(π − x)

u(0,t) = u(π,t) = 0

(a) Solve the problem using the separation of variables.

(b) Solve the problem using D’Alembert’s solution. Hint: I would suggest doing an odd expansion of ut(x,0) first; the final solution should be exactly like the one in (a).

Project proposal "Exploring the role of small and medium enterprises in Zimbabwe".

Two chemicals A and B are combined to form a chemical C. The rate of the reaction is proportional to the product of the instantaneous amounts of A and B not converted to chemical C. Initially there are 23 grams of A and 47 grams of B, and for each gram of B, 1.3 grams of A is used. It has been observed that 17.5 grams of C is formed in 15 minutes. How much is formed in 40 minutes? What is the limiting amount of C after a long time?

Let x, y, z be (non-zero) vectors and suppose w = 12x + 18y + 4z

If z = − 2x − 3y, then w = 4x + 6y

Using the calculation above, mark the statements below that must be true.

A. Span(w, x, y) = Span(w, y)
B. Span(x, y, z) = Span(w, z)
C. Span(w, x, z) = Span(x, y)
D. Span(w, z) = Span(y, z)
E. Span(x, z) = Span(x, y, z)

1. Prove or disprove: if f : R → R is injective and g : R → R is surjective then f ◦ g : R → R is bijective.

2. Suppose n and k are two positive integers. Pick a uniformly random lattice path from (0, 0) to (n, k). What is the probability that the first step is ‘up’?

Determine the edge-connectivity of the petersen graph

Given x = [0, 0.05, 0.1, 0.15, 0.20, ... , 0.95, 1] and f(x) = [1, 1.0053, 1.0212, 1.0475, 1.0841, 1.1308, 1.1873, 1.2532, 1.3282, 1.4117, 1.5033, 1.6023, 1.7083, 1.8205, 1.9382, 2.0607, 2.1873, 2.3172, 2.4495, 2.5835, 2.7183], write a Matlab script that computes the 1st and 2nd derivatives of O(h^2).

Compute the first partial derivatives of h(x,y,z)=(1+9x+4y)^z

Solve the differential equation 2x^2y"-x(x-1)y'-y = 0 using the Frobenius Method

Find all integer solutions to the equation:

a) 105x + 83y = 1

b) 105x + 83y = 8

3. In R⁴ , does the set {(1, 1, 1, 0,(1, 0, 0, 0),(0, 1, 0, 0),(0, 0, 1, 1)}, span R⁴? In other words, can you write down any vector (a, b, c, d) ∈ R⁴ as a linear combination of vectors in the given set ? Is the above set of vectors linearly independent ?

4. In the vector space P₂ of polynomials of degree ≤ 2, find explicitly a polynomial p(x) which is not in the span of the set {x + 2, x² − 1}.

5. Let S be the subspace of P₂ defined by S := {ax² + bx + 2a + 3b : a, b ∈ R}, for different choices of real numbers a and b (you don’t need to show here that S is indeed a subspace, and can assume. But is a good practice problem). Find a basis, and hence dimension for S.