Show that if y is a reparametrization of a curve x, then x is a reparametrization of y.

Give an example of a math problem that you have seen in a previous math class that would not be considered a discrete math problem. In order to receive full credit for this question you must: state the problem, and give a brief explanation as to why the problem is not a discrete math problem. You DO NOT have to give the solution to the problem

Prove that the rank of the Jacobian matrix does not depend on the choice of the generators of the ideal.

South Shore Construction builds permanent docks and seawalls along the southern shore of Long Island. Although the firm has been in business only five years, revenue has increased from $308,000 in the first year of operation to $1,084,000 in the most recent year. The following data show the quarterly sales revenue in thousands of dollars: (Answer 1-3

1.Below is a simple linear regression analysis for this forecasting problem. Is this a valid model to use to forecast quarterly revenue for South Shore? Why or why not? Explain completely.

2.Based on the model in question 1, what is the forecast for Quarter 1 of Year 6?

3.Is there a seasonal trend in this data? Find the appropriate seasonal factor for Quarter 1 of Year 6 and apply it to get a new forecast for revenue in that quarter. Does this provide a better model than the one used in questions 1 and 2? Why or why not? Provide support for your answer (not necessarily numerical data).

For each of the following, subsets of R decide whether or not the set has an infimum and/or a supremum. If they exist, write down the infimum and supremum, and say whether it is an element of the set. (a) [3,7)

(b) {x ∈ Q : 1 ≤ x < 5}

(c) {1, 2, 3, . . . }

(d) {. . . , 1/4, 1/2, 1, 2, 4, . . . }

Find the mass and the center of mass of the solid E with the given density function ρ(x,y,z).

E lies under the plane z = 3 + x + y and above the region in the xy-plane bounded by the curves
y=√x, y=0, and x=1; ρ(x,y,z) = 10.

m =

x =

y =

z =

Use false position method to find the root of ?(?) = −sin(? − 5) + ? with initial guesses of 0.2 and 1. Show up to three iterations and calculate the relative percent error ?? for each iteration possible? Show full details for at least one iteration to get full points. Also, if three significant figure accuracy is required, show if the value after third iteration is acceptable or not.

0. Lon Lee has a collection of twenty photos of friends and family that he keeps to cheer himself

   up. Each of the following questions refer to this collection.

   1. Lon wants to hang 8 of his 20 photos in a line on the wall above his desk. How many ways can he do this?

   2. How many ways can Lon mount 4 of his 20 photos in a picture frame (2 rows with 2 pictures each).

   3. Lon wants to select 9 photos of his 20 to take on a trip. How many ways can he do this?

   4. Lon has decided to give his photo collection to his three children. In how many ways can he

      partition his photo collection among this three children?

   5. Lon also has 500 dollars he wants to give his three children. In how many ways can he devide it among his three children. Let’s assume that it is OK that a child does not get any money.

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.