Suppose a > b are natural numbers such that gcd(a, b) = 1.

Compute each quantity below, or explain why it cannot be determined (i.e. more than one value is possible).

(a) gcd(a³, b²)

(b) gcd(a + b, 2a + 3b)

(c) gcd(2a,4b)

Prove a cubic is reducible over R (using conjugate pairs) if necessary

This is a question regarding Statistics.

Calculate and interpret a confidence interval for a population mean. given a normal distribution with 1) a known variance2) an unknown population variance

or 3) an unknown variance and a large sample size

when sampling from a normal distribution, why test statistic

no matter small(n<30) or large(n>=30) we choose z-statistic?(please give an example)

Thanks  

Use the simplex method to solve the linear programming problem. The maximum is ___ when x1= ___ and x2=___

a.)

Maximize : z= 24x1+2x2

Subject to: 6x1+3x2<=10, x1+4x2<=3

With: x1>=0, x2>=0

b.)

Maximize: z=2x1+7x2

Subject to: 5x1+x2<=70, 7x1+2x2<=90, x1+x2<=80

With: x1,x2>=0

c.)

Maximize: z=x1+2x2+x3+5x4

Subject to: x1+3x2+x3+x4<=55, 4x+x2+3x3+x4<=109

With: x1>=0, x2>- 0, x3>=0, x4>=0

d.)

Maximize: z=4x1+7x2

Subject to: x1-4x2<=35 , 4x1-3x2<=21

With: x1>=0, x2>=0

Write the form of yp that is needed for method of undetermined coefficients. Use the annihilator method with the lowest order differential operator. DO NOT SOLVE for the coefficients. You must show the corresponding homogeneous solution and the differential operator you are applying to each side of the equation as part of your answer.

a. y′′ + 6y′ + 9y = 4cos x

b. y′′ + 7y′ +10y = 2e^6x + 3cos(5x)

c. y′′ + 9y′ + 20y = 5xe^4x

Matlab

Consider f(x) = x^3 - x. Plot it. (a) Use Newton's method with p_0 =1.5 to find an approximation to a root with tolerance TOL = 0.01. (b) Use secant method with p_0 = 0.5 , p_1 =1.5, to answer same question as (a). (c) Use Method of False position with initial approximate roots as in (b).

Need both answered in a copy and paste manner and must be 175 words

1. What is the relationship between exponentials and logarithms? How can you use these to solve equations? Provide an example in your explanation.

2. What is a vertical asymptote? How can you determine if a function has a vertical asymptote?

Explain the behavior of the function as the variable x increases without bound. Provide an example of your vertical asymptote in your explanation.

Find the laplace transform of the following functions, using the definition of Laplace transforms:

f(t)=-2cos4t

f(t)=2 sin^2(t)

g(t)=3e^tcos(t)

Gamma Corp. manufactures and sells widgets for $6.00 each. Therefore, its total revenue, TR, from the sale of X widgets in a year is

TR=$6X

It costs Gamma $2 for materials and labour to produce each widget. In addition, Gamma expects to incur $80,000 of other costs during a year. Therefore, Gamma’s total costs for the year, TC, are expected to be

TC =$2X+ $80,000

Gamma’s expected net income, NI, for the year will be

NI =TR -TC

=$6X - ($2X+$80,000)

=$4X-$80,000

PLEASE ANSWER ALL QUESTIONS WILL FULL WORKINGS

a. What is the slope and TR-intercept of a TR vs. X plot? (NEED ANSWERS WILL FULL AND CLEAR WORKINGS)

b. What is the slope and TC-intercept of a TC vs. X plot? (NEED ANSWERS WILL FULL AND CLEAR WORKINGS)

c. What is the slope and NI-intercept of an NI vs. X plot?(NEED ANSWERS WILL FULL AND CLEAR WORKINGS)

d. Which of the three plotted lines is steepest? (NEED ANSWERS WILL FULL AND CLEAR WORKINGS)

e. How much does NI increase for each widget sold?(NEED ANSWERS WILL FULL AND CLEAR WORKINGS)

f. If Gamma were able to reduce its materials and labour cost to $1.75 per widget, state for each plotted line whether its slope would increase, decrease, or remain unchanged. (NEED ANSWERS WILL FULL AND CLEAR WORKINGS)

PLEASE ANSWER ALL QUESTIONS WILL FULL WORKINGS. PLEASE SHOW HOW THE ANSWER WAS DERIVED

Use ALL the data found in the Excel file for your analysis. Calculate the following:

• Run the descriptive statistics for BMI (Body Mass Index) and SysBP (Systolic Blood Pressure), including graphing a scatter plot of BMI by SysBP.
• Calculate the correlation between BMI and SysBP.
• Calculate the regression with SysBP as the response variable (y) and BMI as the explanatory (sometimes called “Independent”) variable (x). In other words, we want to see how much BMI predicts SysBP. Is there a relationship between weight and blood pressure? Yes, we know there is, so we strongly suspect BMI will significantly predict SysBP. We can better understand this relationship using linear regression

BMI 28 27 33 28 29 24 34 30 36 30 31 30 38 32 27 23 26 38 32 29 26 37 24 27 33 32 33 42 30 27 42 34 24 29 28 35 41 26 35 20 28 27 40 31 45 37 35 45 29 24 28 32 34 31 30 34 30 40 36 41 30 39 36 34 38 36 39 32 28 30 25 39 25 40 21 26 24 24 31 27 29 31 24 39 31 36 21 41 23 33 26 27 26 30 23 35 52 26 35 30 47 27 30 47 23 25 32 39 28 28 35 28 32 32 30 29 32 30 27 35 36 35 44 31 31 34 25 28 29 35 37 27 28 30 24 31 38 26 32 33 31 29 38 30 38 38 36 28 31 43 24 24 31 31 36 26 30 29 35 32 24 34 39 38 35 28 38 37 33 28 25 35 33 38 36 30 28 35 31 26 64 40 32 28 45 29 25 31 34 30 35 29 33 31 22 30 38 29 35 26 41 26 28 32 28 29 27 27 32 38 31 30 27 33 28 38 37 29 25 23 38 27 39 20 29 29 25 28 39 31 35 31 39 40 30 29 37 31 33 34 25 24 42 23 42 26 34 38 38 44 26 32 30 24 31 30 24 30 31 26 42 27 24 27 28 30 24 22 28 32 44 25 31 25 24 36 27 32 25 33 28 30 33 34 36 30 30 40 29 39 27 27 31 27 36 35 23 27 24 32 24 30 29 32 29 40 31 43 36 40 33 35 38 26 28 34 47 33 25 31 34 34 30 33 33 24 28 34 29 29 35 25 40 35 29 28 31 42 38 41 27 37 38 33 40 33 32 30 41 27 25 27 34 32 39 24 32 35 24 32 31 29 33 30 32 30 26 28 31 30 27 34 23 41 30 45 25 30 30 24 38 36 44 31 30 36 47

SysBP 128 108 108 138 118 114 132 110 162 144 142 158 126 128 114 116 134 122 150 138 134 110 138 118 124 158 114 120 110 100 142 109 108 124 118 132 126 118 136 148 122 128 135 140 128 110 108 136 138 126 114 108 152 126 118 120 146 109 136 120 129 120 130 136 130 136 126 112 128 110 122 104 112 138 120 126 122 112 132 122 142 136 122 102 112 110 100 152 125 138 130 118 114 120 142 128 138 124 132 130 152 126 118 154 118 122 115 122 146 146 124 114 134 140 118 100 124 110 124 134 136 138 117 112 112 118 130 128 120 122 142 132 112 128 132 120 144 108 122 128 160 124 150 128 130 150 102 110 126 120 118 130 144 128 168 134 132 136 130 126 128 134 120 100 110 132 155 140 128 112 128 154 114 156 118 142 112 100 143 124 114 132 124 114 120 130 118 98 120 158 118 124 140 124 126 140 122 126 132 116 122 112 108 136 122 124 122 150 128 142 98 126 116 112 114 136 138 142 138 130 122 120 124 108 118 128 122 118 126 136 134 120 136 120 142 118 124 136 134 134 118 114 120 126 120 118 112 110 130 124 108 114 122 113 112 108 108 124 132 108 134 104 122 114 118 114 118 164 116 126 130 120 120 122 132 106 124 142 110 144 136 118 112 128 130 128 120 146 136 124 148 116 132 124 116 136 128 127 120 112 112 130 107 134 118 112 110 164 128 136 108 130 112 122 116 130 126 120 124 120 112 132 118 120 148 118 124 148 140 110 115 114 136 158 122 120 108 134 138 122 130 118 138 118 118 136 118 140 120 108 106 120 120 134 122 112 130 120 112 120 126 120 134 118 130 104 100 120 122 120 96 124 104 112 128 138 100 136 120 118 128 118 128 115 120 132 118

A hash table works as follows. We allocate a table of m slots. All the items we intend to store in the table belong to a large set U of items. We adopt a function f: U ->{0,1...m-1) which maps an item to a slot.

Suppose we seek to store n items from U in a table of m slots.

(a) Suppose f maps every item in U with equal probability to one of the m slots. What is the probability, given two distinct items i1,i2∈ U, that f(i1),f(i2)?

(b) Prove that if |U| > nm, then no matter what f is, there exist n items in U all of which are mapped by f to the same slot.

An automorphism of a group G is an isomorphism from G to G. The set of all automorphisms of G forms a group Aut(G), where the group multiplication is the composition of automorphisms. The group Aut(G) is called the automorphism group of group G.

(a) Show that Aut(Z) ≃ Z2. (Hint: consider generators of Z.)

(b) Show that Aut(Z2 × Z2) ≃ S3.

(c) Prove that if Aut(G) is cyclic then G is abelian.

A 180-lb person jumps out of an airplane with initial upward velocity of 5 ft/s at a height of 3000 ft (assuming the air resistance is proportional to its falling speed). The air resists the body's motion with a force of 2 lb for each ft/s of speed. Assuming the constant gravity is 32 ft/s^2 (1) find its velocity v = v(t) (2) find its terminal velocity (3) find the distance of falling x = x(t) (4) find the duration in seconds until the body hits the ground

FOR EAICH PAIR OF PROPOSITIONS P AND Q STATE WHETHER ON NOT p=q

p=(s→(p ∧¬r)) ∧ ((p→(r ∨ q)) ∧ s), Q=p ∨ t

The construction of a dual D(G) can be applied in any plane graph G: draw a vertex of D(G) in the middle of each region of G and draw an edge e* of D(G) perpendicular to each edge e of G; e* connects the vertices of D(G) representing the regions on either side of e.

a) A dual need not be a graph. It might have two edges between the same pair of vertices or a self-loop edge (from a vertex to itself). find two planar graphs with duals that are not graphs because they contain these two forbidden situations.

C) Show that the degree of a vertex in dual graph D(G) equals the number of boundary edges of the corresponding region in the planar graph G.

E) show for any plane depiction of a graph G that the vertices of G correspond to regions in D(G)