I have decided to perform squat jumps as a good at home exercise during the stay at home order. In the morning I ate two donuts (190 kilocalories each) and would like to know how many squat jumps I need to perform in order to burn off the donuts? For each repetition, I squat down so my center-of-mass is 0.5 m above the ground and at the top of my jump my center-of-mass is 1.6 m above the ground. One kilocalorie equals 4,184 Joules. My body mass is 84 kg. My gross efficiency is 23%. One repetition is jumping up to 1.6 m and coming back down to 0.5 m (so I am performing concentric and eccentric work).

Over the next few weeks I become very good at performing squat jumps and my efficiency increases to 28%. Will I need to perform more or less repetitions now to burn off those donuts? Describe one strategy to reduce the number of required squat jumps. Be sure your description appropriately incorporates biomechanical concepts.

2. Find the first four nonzero terms and the general term for the two fundamental power series solutions about x0 = 0.

Write out the series for each of the two fundamental solutions.

2y’’ + xy’ + y = 0

Please show all steps

1.a.)Use the assumed Babylonian square root algorithm (also known as Archimedes’ method) √ a 2 ± b ≈ a ± b/2a to show that √ 3 ≈ 1; 45 by beginning with the value a = 2. Find a three-sexagesimal-place approximation to the reciprocal of 1; 45 and use it to calculate a three-sexagesimal-place approximation to √ 3.

1.b)An iterative procedure for closer approximations to the square root of a number that is not a square was obtained by Heron of Alexandria (ca. 75 CE). In his work Metrica he merely states a rule that amounts to the following in modern notation: If A is a non square number, and a^2 is the nearest perfect square to it, so that A = a^2 ± b, then approximations to √ A can be obtained using the recursive formula:

x0 = a

xn = 1/2 ( Xn−1 + A/( Xn−1)), n ≥ 1

(i) Use Heron’s method to find approximations through n = 3 to √ 720 and √ 63.

(ii) Show that Heron’s approximation x1 is equivalent to the Babylonian’s square root algorithm.

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 differential equation

y''+y'-2y=3, y(0)=2, y'(0) = -1

Suppose that we put 4 rooks on a standard 8 × 8 chess board so that none of the rooks can capture the others. This means that no two rooks can appear in the same row or column. Furthermore, suppose that we do not put a rook in the upper left corner. How many ways can we do this?

Prove that all rotations and translations form a subgroup of the group of all reflections and products of reflections in Euclidean Geometry. What theorems do we use to show that this is a subgroup?

I know that I need to show that the subset is

closed

identity is in the subset

every element in the subset has an inverse in the subset.

I don't have to prove associative property since that is already proven with Isometries. What theorems for rotations and translations so that they are closed, identity is in the subset and every element is the subset has an inverse in the subset.

The following four methods are commonly used for solving systems of equations:
1.    Graphing
2.    Substitution
3.    Addition
4.    Determinants and Cramer’s Rule

Pick one method and discuss the pros and cons of that method. Provide an example of a problem that can be easily solved using your chosen method and an example of a problem that would be more difficult to solve using your method. Review your classmates’ responses to find a classmate who chose a different method. Discuss that alternative method with your classmate.

Prove that if ∑a_n converges absolutely, then both ∑P_n and ∑N_n converge

1a. Find the number of ways to rearrange each of the following words a. GUIDE b. SCHOOL c. SALESPERSONS

1b. A handful of 6 jellybeans is drawn from a jar that contains 5 different flavors: blueberry, popcorn, pineapple, apple, lemon. a. What outcome does × × ×| × | | × ×| represent? b. How many ways are there to select a handful of 6 jellybeans from the jar?

1c. How many integer solutions are there to the equation x + y + z = 8 where x, y, and z are all greater than or equal to zero?

1d. How many ways are there to choose a dozen donuts from the 21 varieties at a donut shop?

7.              Determine the first 4 nonzero terms of the Taylor series for the solution y = φ(x) of the given initial value problem, y^’’ + cos(x)y^’ + x²y = 0; y(0) = 1, y^’(0) = 1.

What do you expect the radius of convergence to be? Why?

please show all steps

A given field mouse population satisfies the differential equation

d⁢p/d⁢t=0.2p-390.

where p is the number of mice and t is the time in months.

(a) Find the time at which the population becomes extinct if

   p(0)=1940⁢   .

Round your answer to two decimal places.

tf=

(c) Find the initial population p0 if the population is to become extinct in 1 year.

Round your answer to the nearest integer.

 p0⁢= 

This is a linear algebra question.

Determine whether the given system has a unique solution, no solution, or infinitely many solutions. Put the associated augmented matrix in reduced row echelon form and find solutions, if any, in vector form. (If the system has infinitely many solutions, enter a general solution in terms of s. If the system has no solution, enter NO SOLUTION in any cell of the vector.)

Given a natural number q ≥ 1, define a relation ∼ on the set Z by x ∼ y
if x - y is divisible by q.
(i) Show that ∼ is an equivalence relation.
We will denote the set of equivalence classes defined by ∼ with Z=qZ. Also
let x mod q denote the equivalence class to which an integer x belongs.
(ii) Check that the operations

are well-defined on Z=qZ.
(iii) With the operations as defined above, show that Z=qZ is not a field
if q is not prime.

3. How many strings can be made using 9 or more letters of MISSISSIPPI?