Consider the curve traced out by the parametric equations: { x = 1 + cos(t) y = t + sin(t) for 0 ≤ t ≤ 4π.

1. Show that that dy dx = − 1 + cos(t)/sin(t) = − csc(t) − cot(t).

2. Make a Sign Diagram for dy dx to find the intervals of t over which C is increasing or decreasing.

• C is increasing on: • C is decreasing on:

3. Show that d2y/dx2 = − csc2 (t)(csc(t) + cot(t))

4. Make a Sign Diagram for d 2 y dx2 to find the intervals of t over which C is concave up or concave down.

• C is concave up on: • C is concave down on:

5. Using all of your work from numbers 1 through 4, sketch a detailed graph of C. Label the points (x, y), if any, where C has horizontal tangents, vertical tangents, or where C is not smooth.

Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule to approximate the given integral with the specific value of n. (Round your answer to six decimal places).  
∫₁³ sin (?) / ? ?? , ? = 4
Please show all work.

You want to be able to withdraw the specified amount periodically from a payout annuity with the given terms. Find out how much the account needs to hold to make this possible. Round your answer to the nearest dollar.

regular withdraw: $1900
interest rate: 2%
frequency: quarterly
time: 28 years

Given y 1 ( t ) = t2 and y2 ( t ) = t ^− 1 satisfy the corresponding homogeneous equation of

t^2 y ' ' − 2 y = − 3 − t , t > 0

Then the general solution to the non-homogeneous equation can be written as y ( t ) = c1y1(t)+c2y2(t)+yp(t)

Use variation of parameters to find y p ( t ) .

Discrete Structures

Use a proof by contraposition to show if x³ + 3x is an irrational number then so is x, for any real number x.

Show that the indicial roots of the singularity x=0 differ by an integer. Use the method of Frobenius to obtain the first four terms of at least one series solutions of the DE:

xy''+2y'-xy=0

A tank originally contains 120 gal of fresh water. Then water containing 1/2 lb of salt per gallon is poured into the tank at a rate of 2 gal/min, and the mixture is allowed to leave at the same rate. After 11 min the process is stopped, and fresh water is poured into the tank at a rate of 2 gal/min, with the mixture again leaving at the same rate. Find the amount of salt Q(11) in the tank at the end of an additional 11 min.

Round your answer and intermediate answer to two decimal places.

Amount of salt in the tank at the end of an additional 11 min is Q(11)= _____ lbs.

Expand the function, f(x) = x, defined over the interval 0 <x <2, in terms of:
A Fourier sine series, using an odd extension of f(x)

and A Fourier cosine series, using an even extension of f(x)

4) Laplace Transform and Solving second order Linear Differential Equations with Applications The Laplace transform of a function, transform of a derivative, transform of the second derivative, transform of an integral, table of Laplace transform for simple functions, the inverse Laplace transform, solving first order linear differential equations by the Laplace transform

Applications: a) Series RLC circuit with dc source b) Damped motion of an object in a fluid [mechanical, electromechanical] c) Forced Oscillations [mechanical, electromechanical]

You should build the theory portion of your report on what you have learned in the math courses including Mathematics 1, Applied Mathematics, Differential calculus and Integral Calculus. Any additional material you need in order to begin or complete your project must be included and discussed within the report. Special attention is paid to the consistency of the derivation of formulae and concepts in your report. Once the mathematical foundations are laid in a proper way, you need to introduce the topic you have been assigned from your program where the mathematical concepts you have explored are applied. It is important you support your derivations and conclusions by real world engineering applications

Dean takes his boat out fishing every weekend. His current boat is still in okay condition, but he decides he’d like to buy a new one. He finds the boat of his dreams for $20,950. He does some research and finds that his credit union will give him a 5-year loan with an APR of 4.25% if he makes a down payment of 18%.

1. What is the amount of Dean’s down-payment?   

2. What is the loan amount that Dean will be financing?

3. What will his monthly payments be on the loan?

4. What is the total amount Dean will pay for his boat if he finances the purchase?

5. How much interest will Dean pay the credit union over the life of the loan?   

Sam, Dean’s younger brother, tries to convince Dean to save up for a new boat instead of getting a loan. He tells Dean that the credit union has a savings account option that offers an APR of 1.2% compounded monthly, so long as the account maintains a minimum balance of $2000.

6. Since Dean has the down payment amount already, if he uses that to start a savings account with the credit union, then adds $200 to the account each month, how much will he have at the end of 5 years?

7. If Dean starts the savings account with his down payment money, how much would he have to deposit each month for 5 years to end up with the $20,950 he needs for the boat of his dreams?

8. If Dean follows the savings plan from problem 7, how much total will he have deposited into the savings account at the end of the 5 years?

9. If Dean follows the savings plan from problem 7, how much interest will he have earned in the savings account over the course of the 5 years?

10. Compare the total Dean would pay on the loan to the total Dean would deposit into the savings account. How much would Dean save by going with the savings option?

11. If Dean decides he wants to save up for the boat as fast as possible, how long would it take the account to reach the full value of $20,950 if he uses the down payment money to start the account then adds $400 each month? Give the answer as years with 2 decimal places!

Suppose V and V0 are finitely-generated vector spaces and T : V → V0 is a linear transformation with ker(T) = {~ 0}. Is it possible that dim(V ) > dim(V0)? If so, provide a specific example showing this can occur. Otherwise, provide a general proof showing that we must have dim(V ) ≤ dim(V0).

A Professor decides to hide their hoard money, 100 identical gold coins, in 20 unique locations hidden across the city. It is assumed that multiple coins or even none can be stored at each location.

How many ways can he distribute these coins if neither the Mining Lab location nor the Office location can have more than 20 coins (e.g., 25 coins to the Mining Lab and 25 coins to the Office with 50 coins to other locations would not be legal)? It is assumed they cannot trust any of his colleagues with their precious and hard-earned research funds.

a) How many integers in between 1 and 10⁶ have an even number of divisors?

Show work proving your answer.

Express your answer in prime factorized form.

b) With proof, determine all integer solutions to the following equation:

1935x + 2322y = 177