1: Let X be the set of all ordered triples of 0’s and 1’s. Show that X consists of 8 elements and that a metric d on X can be defined by ∀x,y ∈ X: d(x,y) := Number of places where x and y have different entries.

2: Show that the non-negativity of a metric can be deduced from only Axioms (M2), (M3), and (M4).

3: Let (X,d) be a metric space. Show that another metric D on X can be defined by ∀x,y ∈ X: D(x,y) := d(x,y)/(1 + d(x,y)).

4: Let (X,d) be a metric space.

1. Show that every open d-ball is a d-open subset of X.
2. Show that every closed d-ball is a d-closed subset of X.

5: Let (X,d) be a metric space. Show that a subset A of X is d-open if and only if it is the union of a (possibly empty) set of open d-balls.

Let X be a subset of R^n. Prove that the following are equivalent:

1) X is open in R^n with the Euclidean metric d(x,y) = sqrt((x1 - y1)^2+(x2 - y2)^2+...+(xn - yn)^2)

2) X is open in R^n with the taxicab metric d(x,y)= |x1 - y1|+|x2 - y2|+...+|xn - yn|

3) X is open in R^n with the square metric d(x,y)= max{|x1 - y1|,|x2 - y2|,...,|xn -y n|}

(This can be proved by showing the 1 implies 2 implies 3)

(TOPOLOGY)

Let C(R) be the vector space of continuous functions from R to R with the usual addition and scalar multiplication. Determine if W is a subspace of C(R). Show algebraically and explain your answers thoroughly.

a. W = C^n(R) = { f ∈ C(R) | f has a continuous nth derivative}

b. W = {f ∈ C^2(R) | f''(x) + f(x) = 0}

c. W = {f ∈ C(R) | f(-x) = f(x)}.

Prove or disprove each of the followings.

1. If f(n) = ω(g(n)), then log₂(f(n)) = ω(log₂g(n)), where f(n) and g(n) are positive functions.

2. ω(n) + ω(n²) = theta(n).

3. f(n)g(n) = ω(f(n)), where f(n) and g(n) are positive functions.

4. If f(n) = theta(g(n)), then f(n) = theta(20 g(n)), where f(n) and g(n) are positive functions.

5. If there are only finite number of points for which f(n) > g(n), then f(n) = O(g(n)), where f(n) and g(n) are positive functions.

Determine which subsets are subspaces of M 2x2 (R) and prove your answer.

a. W = {A ∈ M 2x2 (R) | a12 = -a21}

b. W = {A ∈ M 2X2 (R) | a12 = 1}

c. Fix B ∈ M 2x2 (R). Let W ={ A ∈ M 2x2 (R) | AB = BA

Let F3={cos⁡(t),sin⁡(t),cos⁡(3t),sin⁡(3t)} and T3={cos3(t),cos2(t)sin(t),cos(t)sin2(t),sin3(t)}. Use the power reduction formulas and the triple angle identities to show the following:

1. Show T3⊆Span(F3).
2. Show F3⊆Span(T3).

Janine is considering buying a water filter and a reusable water bottle rather than buying bottled water. Will doing so save her money?
First, determine what information you need to answer this question, then click here to display that info (along with other info).

• How much water does Janine drink in a day? She normally drinks 6 bottles a day, each 16.9 ounces.
• How much does a bottle of water cost? She buys 24-packs of 16.9 ounce bottles for $3.19.
• How much does a reusable water bottle cost? About $10.
• How long does a reusable water bottle last? Basically forever (or until you lose it).
• How much does a water filter cost? How much water will they filter?
  • A faucet-mounted filter costs about $28 and includes one filter cartridge. Refill filters cost about $33 for a 3-pack. The box says each filter will filter up to 100 gallons (378 liters)
  • A water filter pitcher costs about $22 and includes one filter cartridge. Refill filters cost about $20 for a 4-pack. The box says each filter lasts for 40 gallons or 4 months
  • An under-sink filter costs $130 and includes one filter cartridge. Refill filters cost about $60 each. The filter lasts for 500 gallons.

Which option is cheapest over one year (365 days)?
The cheapest option saves her $______ over a year.
Give your answer to the nearest cent. Pro-rate the costs of additional filters (so if you only use part of a filter, only count the corresponding fraction of the filter cost).

y = c1 cos(5x) + c2 sin(5x) is a two-parameter family of solutions of the second-order DE y'' + 25y = 0. If possible, find a solution of the differential equation that satisfies the given side conditions. The conditions specified at two different points are called boundary conditions. (If not possible, enter IMPOSSIBLE.)

y(0) = 1, y'(π) = 7

y =

Calculate the first three terms in the power series solutions of the following differential equations taken about x=0.

x^2y''+xy'+(x^2-1/9)y=0

Differential Equations

1. Create a direction field for y 0 = y − y 2 .

(a) You should find any equilibrium solutions by hand and at least a few other solutions. Feel free to make a direction field with some piece of technology and share a picture of it.

(b) Find at least a few solution curves and describe the behavior of y as x → ∞, for different ranges of initial values y(x0) = y0.

(c) Use your direction field to approximate the value of y(0.5) if the initial condition is y(0) = 0.5.

(d) Use Euler’s method with h = 0.1 to approximate y(0.5) when the initial condition is y(0) = 0.5.

(e) Bonus: Find the analytical solution to this differential equation with initial condition y(0) = 0.5 and then find the exact value of y, what is your percent error from your Euler method approximation?

1. Show that if λ1 and λ2 are different eigenvalues of A and u1 and u2 are associated eigenvectors, then u1 and u2 are independent. More generally, show that if λ1, ..., λk are distinct eigenvalues of A and ui is an eigenvector associated to λi for i=1, ..., k, then u1, ..., uk are independent.

2. Show that for each eigenvalue λ, the set E(λ) = {u LaTeX: \in∈Rn: u is an eigenvector associated to λ} is a subspace of Rn.