Please answer all questions:

1)Write the formula for the updating function mt+1 = f(mt) in the following scenario, and then find the solution function mt = f(t). During a particularly dry season, the volume of water in a lake increases by 3% each day from precipitation, and then 8% of the volume of water is removed through a river. On day t0, the lake has 20,000 acre feet of water.

2)Use the solution function from the above example to determine the time it would take under these conditions for the lake’s volume to be reduced by half.

3) Identify the average, amplitude, period, and phase of the following oscillating functions.

(a) g(t) = cos(5(t + π)) − 3.

(b) h(t) = 1 ?8+6cos(2π(2t−1))?

4)The function f(x) has the following properties: f(3) = 5, f(4) = 2, and f′(3) = −2. Write the equations for the secant line of f between x = 3 and x = 4, and the tangent line at x = 3.

5) Identify the critical points and state where the function is increasing and decreasing for the function f(x)=x^3-3x. The find the derivative of f(x) and sketch it

6)Suppose a function f(x) = g(x)/h(x) . Use the following table to calculate f′(3), and write the equation of the tangent line to f at x = 3.

1. Solve the equations 256? ≡ 442(??? 60), 3? + 4 ≡ 6(??? 13).

5. Prove that ?2 + ? + 1 is an irreducible polynomial of degree 2.

3. (a) Let n ∈ N with n ≥ 2 and consider (Zn, ⊕). If a ∈ Zn, show that a = a −1 if and only if n | 2a. What conditions on n would guarantee that no element is equal to its own inverse?

(b) Let p be prime and consider (Up, ). Under what conditions does a ∈ Up satisfy a = a −1 ? Can you specifically identify the elements that satisfy this condition?

Let T : V → V be a linear map. A vector v ∈ V is called a fixed point of T if Tv = v. For example, 0 is a fixed point for every linear map T. Show that 1 is an eigenvalue of T if and only if T has nonzero fixed points, and that these nonzero fixed points are the eigenvectors of T corresponding to eigenvalue 1

Use Strong induction to show that any counting number can be written as the sum of distinct powers of 2.

1. Consider the differential equation dy/dt = 2?square root(absolute value of y) with initial condition y(t0)=y0

• For what values of y0 does the Existence Theorem apply?
• For what values of y0 does the Uniqueness theorem apply?
• Verify that y1(t) = 0 solves the initial value problem with y0 = 0
• Verify that y2(t) = t2 solves the initial value problem with y0 = 0
• Does this violate the theorems from this section 1.5? Why or why not?

Let Σ a_n and Σ b_n be series with positive terms such that lim(a_n/b_n) = λ ∈ (0, ∞). Prove that the two series have the same behavior, that is they both converge or they both diverge to +∞.

G ilbert Moss and Angela Pasaic spent several summers during their college years working at

archaeological sites in the Southwest. While at those digs, they learned how to make ceramic tiles from

local artisans. After college they made use of their college experiences to start a tile manufacturing firm

called Mossaic Tiles, Ltd. They opened their plant in New Mexico, where they would have convenient

access to special clay they intend to use to make a clay derivative for their tiles. Their manufacturing

operation consists of a few relatively simple but precarious steps, including molding the tiles, baking, and

glazing.

Gilbert and Angela plan to produce two basic types of tile for use in home bathrooms, kitchens,

sunrooms, and laundry rooms. The two types of tile are a larger, single- colored tile and a smaller,

patterned tile. In the manufacturing process, the color or pattern is added before a tile is glazed. Either a

single color is sprayed over the top of a baked set of tiles or a stenciled pattern is sprayed on the top of a

baked set of tiles.

The tiles are produced in batches of 100. The first step is to pour the clay derivative into specially

constructed molds. It takes 18 minutes to mold a batch of 100 larger tiles and 15 minutes to prepare a

mold for a batch of 100 smaller tiles. The company has 60 hours available each week for molding. After

the tiles are molded, they are baked in a kiln: 0.27 hour for a batch of 100 larger tiles and 0.58 hour for a

batch of 100 smaller tiles. The company has 105 hours available each week for baking. After baking,

the tiles are either colored or patterned and glazed.

This process takes 0.16 hour for a batch of 100 larger tiles and 0.20 hour for a batch of 100 smaller

tiles. Forty hours are available each week for the glazing process. Each batch of 100large tiles requires

32.8 pounds of the clay derivative to produce, whereas each batch of smaller tiles requires 20 pounds.

The company has 6,000 pounds of the clay derivative available each week.

Mossaic Tiles earns a profit of $190 for each batch of 100 of the larger tiles and $240 for each batch

of 100 smaller patterned tiles. Angela and Gilbert want to know how many batches of each type of tile

to produce each week to maximize profit. In addition, they have some questions about resource usage

they would like answered.

A. Formulate a linear programming model for Mossaic Tiles, Ltd., and determine the mix of tiles it

should manufacture each week.

B. Transform the model into standard form.

C. Solve the linear programming model graphically.

D. Determine the resources left over and not used at the optimal solution point.

E. Determine the sensitivity ranges for the objective function coefficients and constraint quantity values

by using the graphical solution of the model.

F. For artistic reasons, Gilbert and Angela prefer to produce the smaller, patterned tiles. They also

believe that in the long run, the smaller tiles will be a more successful product. What must the profit

be for the smaller tiles in order for the company to produce only the smaller tiles?

G. Solve the linear programming model by using the computer and verify the sensitivity ranges

computed in (E).

H. Mossaic believes it may be able to reduce the time required for molding to 16 minutes for a batch of

larger tiles and 12 minutes for a batch of smaller tiles. How will this affect the solution?

I. The company that provides Mossaic with clay has indicated that it can deliver an additional 100

pounds each week. Should Mossaic agree to this offer?

J. Mossaic is considering adding capacity to one of its kilns to provide 20 additional glazing hours per

week, at a cost of $90,000. Should it make the investment?

K. The kiln for glazing had to be shut down for 3 hours, reducing the available kiln hours from 40 to 37.

What effect will this have on the solution?

When the velocity v of an object is very​ large, the magnitude of the force due to air resistance is proportional to v squared with the force acting in opposition to the motion of the object. A shell of mass 5 kg is shot upward from the ground with an initial velocity of 500 ​m/sec. If the magnitude of the force due to air resistance is ​(0.1​)v squared​, when will the shell reach its maximum height above the​ ground? What is the maximum​ height? Assume the acceleration due to gravity to be 9.81 m divided by s squared.

(1 point) Use Euler's method with step size 0.1 to estimate y(2), where y(x) is the solution of the initial-value problem y′=−5x+sin(y), y(0)=−1.

Pam retires after 28 years of service with her employer. She is 66 years old and has contributed $84,000 to her employer's qualified pension fund. She elects to receive her retirement benefits as an annuity of $8,400 per month for the remainder of her life.

Click here to access Exhibit 4.1 and Exhibit 4.2.

a. Assume that Pam retired in June 2018 and collected six annuity payments that year. What is her income from the annuity payments in the first year?
$

b. Assume that Pam lives 25 years after retiring. What is her income from the annuity payments in the twenty-fourth year?
$.

c. Assume that Pam dies after collecting 160 payments. She collected eight payments in the year of her death. What are Pam's income and deductions from the annuity contract in the year of her death?
Income from the annuity payments: $
Loss deduction: $

Using lambda calculus, perform the aplha transformation and beta reduction of the following expression: λx . x y λz . x z

Then, explain why the transformation was needed and how the reduction was performed.

Derive the unit vectors ρ, θ, and φ as functions of the spherical coordinates <ρ, θ, φ> and the Cartesian unit vectors i, j, and k.

1. Suppose that ρ, θ, and φ depend on time t. Compute dρ/dt , dθ/dt , and dφ/dt , leaving results in terms of the spherical coordinates and the Cartesian unit vectors.

2. Express the derivatives dρ/dt , dθ/dt , and dφ/dt in terms of the spherical coordinates and spherical unit vectors.

3. Compute the second derivatives d^2ρ/dt^2 , d^2θ/dt^2 , and d^2φ/dt^2 in terms of the spherical coordinates and spherical unit vectors.

6. Consider the initial value problem dy/ dt = t ^2 y , y(1) = 1 .

(a) Use Euler’s method (by hand) to approximate the solution y(t) at t = 2 using ∆t = 1, ∆t = 1/2 and ∆t = 1/4 (use a calculator to approximate the answer for the smallest ∆t). Report your results in a table listing ∆t in one column, and the corresponding approximation of y(2) in the other.

(b) The direction field of dy/dt = t ^2y is shown at right. In this figure, add a sketch of ◦ the exact solution y(t) to the given initial value problem, for t ∈ [1, 2] ◦ the numerical approximation using Euler’s method that you found above (indicate the solution obtained at each step, and a line showing how it is obtained)