Consider the function, ? (?) = 20?² + 28? - 17

Determine both real roots of the quadratic equation as follows:

i. Graphically and determine the intervals for both roots to be used in iii & iv.

ii. Using the quadratic formula

iii. Using the bisection method until ?? is less than ?? = 0.5%

iv. Using the false-position method until ?? is less than ?? = 0.5%

Let A be such that its only right ideals are {¯0} (neutral element) and A. Show that A or is a ring with division or
A is a ring with a prime number of elements in which a · b = 0 for any, b ∈ A.

Approximately how many flops are needed to find the LU factorization of an n x n matrix using Doolittle’s method? If a computer requires 1 second to find an LU factorization of a 500 x 500 matrix, what would you estimate is the largest matrix that could be factored in less than 1 hour?

For each of the following statements, determine whether it is true or false and justify your answer.
a. If the function f + g: IR --> IR is continuous, then the functions f :IR --> IR and g :IR --> IR also are continuous.
b. If the function f^2 : IR --> R is continuous, then so is the function f :R --> IR.
c. If the functions f + g: IR and g: IR --> IR are continuous, then so is the function f :IR --> IR.
d. Every function f: N --> IR is continuous, where N denotes the set of natural numbers.

For each of the following statements, determine whether it is true or false and justify your answer. a. Every function f : [0, 1] ~ lR has a maximum. b. Every continuous function f :[a, b] ~ lR has a minimum. c. Every continuous function f : (0, 1) ~ lR has a maximum. d. Every continuous function f : (0, 1) ~ lR has a bounded image. e. If the image of the continuous function f: (0, 1) ~ lR is bounded below, then the function has a minimum.

Humid air at 155 kPa, 40°C, and 70 percent relative humidity is cooled at constant pressure in a pipe to its dew-point temperature. Calculate the heat transfer, in kJ/kg dry air, required for this process. Use data from the tables.

The heat transfer is  kJ/kg dry air.

Prove the following:

(a) Let A be a ring and B be a field. Let f : A → B be a surjective homomorphism from A to B. Then ker(f) is a maximal ideal.

(b) If A/J is a field, then J is a maximal ideal.

2. Let f(x) ≥ 0 on [1, 2] and suppose that f is integrable on [1, 2] with R 2 1 f(x)dx = 2 3 . Prove that f(x 2 ) is integrable on [1, √ 2] and √ 2 6 ≤ Z √ 2 1 f(x 2 )dx ≤ 1 3 .

A Ferris wheel has a radius of 10 feet and is boarded in the 6 o’clock position from a platform that is 3 feet above the ground. The wheel turns counterclockwise and completes a revolution every 3 minutes. At t = 0 the person is at the 6 o’clock position.

1. Draw a diagram and impose coordinates

2. Find a function, F(t), for the height of the person above the ground after t minutes

3. Find two times when a passenger is at a height of 18.0 feet.

4. How far does a rider travel along the circle if they travel from the boarding point on the bottom of the Ferris wheel to the top of the Ferris wheel?

Assigned Exercise IX.1. IX.1. (a) Suppose that f : [a, b] → R is continuous. Define A := 1/b−a integral of f from a to b, and B := 1/b-a integral of f² from a to b . Show that 1/b − a integral from a to b of (f(x) − A)² dx = B − A ² . Conclude that A² ≤ B. (b) Assume the Cauchy–Schwarz Inequality for Integrals of Exercise 6.3 #2, which we state here for continuous functions f : [a, b] → R and g : [a, b] → R: integral from a to b of (fg)² ≤ integral from a to b  (f ² ) integral from a to b  (g ² ) . How does this Cauchy–Schwarz inequality imply the inequality A² ≤ B of part (a)?

2a. Find the orthogonal projection of [9,40,-29,4] onto the subspace of R4 spanned by [1,6,5,6] and [5,1,5,5].

Answer choices: [2,14,-15,7] [-32,13,-10,7] [0,9,12,6] [-5,-2,3,2] [-12,0,-9,-9] [-16,20,0,4] [27,29,29,21] [-3,1,2,7] [-23,7,-3,-9] [-15,5,-15,30]

2b. Find the orthogonal projection of [17,18,-10,24] onto the subspace of R4 spanned by [2,7,1,6] and [3,7,3,4].

Answer choices: [-34,-22,-29,-34] [-6,4,-2,0] [-12,36,21,33] [3,21,-3,24] [7,-14,-12,1] [5,3,32,45] [14,32,12,11] [9,13,18,11] [20,2,-3,19] [-2,-6,1,-7]

Give examples–a formula and an illustration–of two-dimensional vector fields F⃗(x,y) with each of the following properties. You could do the illustrations by hand.

a) The direction of F⃗ is constant but the magnitude is not constant.

b) The magnitude |F⃗| is constant but the direction is not constant.

c) All the vectors F⃗ along a horizontal line are equal, but F⃗ is not constant overall.

d) F⃗ (x, y) is perpendicular to xˆi + yˆj at every point (x, y).

e) F⃗ is a force field which repels from the origin. It is strongest near the origin, and weaker farther

away.

After studying the material in this module, please solve and submit the following problems from the module reading assignment on Sets from Math in Society, by D. Lippman (v. 2.4): #6, 14, 16, 20, 22, 26, 30, 34, 36, 44

q) a)Parametrize the following paths from (−2, 0) to (2, 0) in the xy-plane:
• A straight line path.
• A path consisting of three lines: (−2, 0) to (−2, 2), from (−2, 2) to (2, 2), and from (2, 2) to (2, 0). (Give three parametrizations.)
• A path counterclockwise along a circle centered at the origin.

b) Compute the integral F⃗ · d⃗r along each of these paths for F⃗ (x, y) = (−y, x).

c) Compute the integral G⃗ · d⃗r along each of these paths for G⃗ (x, y) = (y2, 2xy).

d)What do you notice about your answers to parts (b) and (c)? Are either F⃗ or G⃗ a gradient field?

The Weemow Lawn Service wants to start doing snow removal in winter when there are no lawns to maintain. Jeff and Julie Weems, who own the service, are trying to determine how much equipment they need to purchase, based on various job types they have. They plan to work themselves and hire some local college students on a per-job basis. Based on historical weather data, they estimate that there will be 6 major snowfalls next winter. Virtually all customers want their snow removed no more than 2 days after the snow stops falling. Working 10 hours per Day (into the night), Jeff and Julie can remove the snow from a normal driveway in about 1 hour, and it take about 4 hours to remove snow from a business parking lot and sidewalk. The variable cost (mainly for labor and gas) per job is $12 for a driveway and $47 for a parking lot. Using their lawn service customer base as a guideline, they believe they will have demand of no more than 40 homeowners and 25 businesses. They plan to charge $35 for home driveway and $120 for a business parking lot, which is slightly less than the going rate. They want to know how many jobs of each type will maximize their profit.

Problem 2

A: Alternatively, hiring additional people on a per job basis will increase Jeff and Julie’s variable cost to $16 for a driveway and $53 for a parking lot, but it will lower the time it takes to clear a driveway to 40 mins and a parking lot to 3 hours. Will this affect their profit?

B: If Jeff and Julie combine the two alternatives (b) and (c), will this affect their profit?

Only problem 2...Specifically, why does the time constraint change to be less than 100? The correct answers for 2a is 2435 and for 2b is 2073