Solve the given initial-value problem.

X' =

X +

, X(0) =

X(t) =

You may have used shortcuts to see if a number is divisible by another. Where do they come from? Why are some numbers left out? Could you prove these shortcuts?

Math201:

Directions: Annual Percentage Yield (APV). Find the annual percentage yield (to the nearest 0.01%) in each case.

1.) A bank offers an APR of 3.2% compounded monthly.

Directions: Continuous Compounding. Use the formula for continuous compounding to compute the balance in each account after 1,5, and 20 years. Also, find the APY for this account.

1.) A $2,000 deposit in an account with an APR of 3.1%

Solve the following IVPs:

A. y"" +2y′′+y = 4et, y(0)=1,  y′(0)=1, y′′(0)=1, y′′′(0)=1.

B. y′′′+25y′ = 325e−t, y(0) = 0, y′(0) = 0, y′′(0) = 0.

ssume you lend $10.000 for a five (5) year period. The current the real rate at the time you lend the money is 2.3%. You charge no
risk premium on the loan.
At the end of the 5-years loan period you receive back your $10,000 and then decide to determine your rate of return. You collect the following information for your calculation.

1)Using the Fisher Equation, what is your expected required rate of return on the loan? Show clearly all work, carrying all calculations out to four (4) decimal places. Highlight in bold your answer.

2) Using the Fisher Equation, what is your realized required rate of return on the loan? Show clearly all work, carrying all calculations out to four (4) decimal places. Highlight in bold your answer.

Thank you.

Please justify below questions (its a topology question) with valid arguments:

1. Show that an infinite set is not finite.

2. Is there an infinite set which is not countably in finite?

Thanks in advance.

List examples of equations similar to the Pell's equation, ie. any  Diophantine equation other then Pell's.

Find a general solution of the following equations:

A. y′′′ +6y′′ +11y′ +6y = −e^-t

B. y′′′ + y′ = t.

V and W are finite dimensional inner product spaces,T:V→W is a linear map, and∗represents the adjoint.

1A: Let n be a positive integer, and suppose that T is defined on C^n (with the usual inner product) by T(z1,z2,...,zn) = (0,z1,z2,...,zn−1). Give a formula for T*.

1B: Show that λ is an eigenvalue of T if and only if λ is an eigenvalue of T*.

Use stars and bars to solve each counting problem. You may leave your answers as binomial coefficients.

(a) How many collections of 6 (not necessarily distinct) coins can be made from an infinite supply of pennies, nickels, dimes, and quarters?

(b) A social security number is a sequence of 9 digits. How many social security numbers are there n1n2n3 . . . n9 such that ni ≤ ni+1 for i = 1 to 8? For example, 024455888 would count but 254180419 would not count

Find the intersection of the line passing through P=(21,21) and Q(−63,168) and the line passing through points R(21,0) and S(0,21) using vectors.

For each of the following simplicial complexes ,X = {[a], [b], [c], [d], [a, b], [c, d]}, and X = {[a], [b], [c], [d], [e], [a, b], [b, c], [c, d], [a, d], [a, c], [a, e], [b, e], [a, b, c]},give a basis for each non-zero Hj(X).

Find the distance from

(2, −7, 7)

to each of the following.

(a) the xy-plane

(b) the yz-plane

(c) the xz-plane

(d) the x-axis

(e) the y-axis

(f) the z-axis

195 dots are placed in a square that is 4 x 4 units.

This configuration creates an issue where at least two points need to be very close to one another, no farther than a distance of D. What is the minimum distance D? (Hint: Consider using pigeonhole principle to solve min distance)