Construct the Dual and solve the Dual by the graphical and simplex method.

Minimize Z = 1x1 + 1x2 + 2x3

Subject to:

2x1 + 2x2 + 1x3 >= J

5x1 + 6x2 + 7x3 >= K

x1, x2, and x3 >= 0

constant resources:

J = 15

K = 17

Please do on paper..

Prof. X has two brilliant students Y and Z in his class. He introduces the
concept of vector spaces, bases and dimension in his 5th session. As an exercise,
he gives a vector space V of dimension n and asks Y and Z to find a basis. Y
produces a set S with n elements. But Z being lazy, takes the set S, removes
a vector and adds a new vector to it creating a new set T. Prof. X looks at set
T and confirms to class that it is a basis. He then asks the class if the set S
produced by Y could be a basis without telling them what it is. While student
U says yes, student W says need not and Prof. X says that both U and W
could be correct. Justify the statement of Prof. X with suitable examples of V
over F, n, S and T.

Q2a) Consider a transformation T : R
2×2 → R
2×2
such that T(M) = MT
.
This is infact a linear transformation. Based on this, justify if the following
statements are true or not. (2)
a) T ◦ T is the identity transformation.
b) The kernel of T is the zero matrix.
c) Range T = R
2×2
d) T(M) =-M is impossible.
b) Assume that you are given a matrix A = [aij ] ∈ R
n×n with (1 ≤ i, j ≤ n)
and having the following interesting property:
ai1 + ai2 + ..... + ain = 0 for each i = 1, 2, ...., n
Based on this information, prove that rank(A) < n. (2)
c) Let A ∈ R
m×n be a matrix of rank r. Suppose there are right hand sides
b for which Ax = b has no solution, which of following expression(s) is/are
correct: r < m, r = m, r > m.
Now, consider the linear system AT y = 0. Do you think this linear system can
have non-zero solutions, that is y 6= 0 such that AT y = 0. Give justification for
all your answers.

-When first observed, an oil spill covers 8 square miles. Measurements show that the area is tripling every 6 hrs. Find an exponential model for the area A (in mi²) of the oil spill as a function of time t (in hr) from the beginning of the spill. (Enter a mathematical expression.)

A(t)=

-An internet analytics company measured the number of people watching a video posted on a social media platform. The company found 129 people had watched the video and that the number of people who had watched it was increasing by 30% every 3 hours.

A=

-A restaurant owner deposits $6,000 into an account that earns an annual interest rate of 6% compounded monthly. Find an exponential growth model for A, the value of the account (in dollars) after t years. (Enter a mathematical expression)

A=

**please show work

Let T = (V,E) be a tree, and letr, r′ ∈ V be any two nodes. Prove that the height of the rooted tree (T, r) is at most twice the height of the rooted tree (T, r′).

Linear algebra Thank you.

Extra Credit (No partial credit - 1%): It is often useful to perform a change of basis to examine an object from a different perspective. In the process of a larger work the conic  work the conic 3 x square − 10 x y + 3 y square + 16 x root 2 − 32 = 0 needs to be identified; this can be accomplished by a rotation. Identify the given conic and give its equation in standard form.

Ralph's Electronics purchased 75 sound bars from the manufacturer for $250 each less discounts of 25% and 12%. The regular markup on the sound bar is 60% of the regular selling price, and Ralph's Electronics overhead is 10% of the regular selling price. On the Black Friday sale, the sales price was reduced to $199.

The regular selling price =?

Profit or loss (please indicate which one) on each sound bar at the sale price = ?

A hotel rental service needs to have clean towels for each day of a three-day period. Some of the clean towels may be purchased new and some may be dirty towels from previous days that have been washed by a laundry service. The cost of new towels is $1 per towel, the cost of a fast one-day laundry serice is 40¢ per towel, and the cost of a slow two-day laundry service is 25¢ per towel. If the rental service needs 300,200, and 400 clean towels for each of the next three days (respectively), how many towels should the rental service buy new and how many should the rental service have washed by the different laundry services so as to minimize total costs?

The solution is minC=570 at (x,y,z,w)=(400,100,200,200), please use hand-writing for the process.

An eccentric Computing Science Professor decides to hide their hoard of NSERC funded money, 100 identical gold coins, in 20 unique locations hidden across campus. It is assumed that multiple coins or even none can be stored at each location.
a) How many ways can they distribute these coins without any restrictions?
b) How many ways can they distribute these coins if each location gets at least 5 coins?
c) How many ways can they distribute these coins if each location only gets an even number of coins?
d) How many ways can they distribute these coins if the Sessional Instructor Lab location can't have more than 20 coins distributed to it?
BONUS: e) How many ways can he distribute these coins if neither the Data-Mining Lab location nor the Sessional Instructor office location can have more than 20 coins (e.g., 25 coins to the Data-Mining Lab and 25 coins to the Sessional Instructor Office with 50 coins to other locations would not be legal)? It is assumed they cannot trust any of his colleagues with their precious and hard-earned research funds.

ONLY ANSWER d) and BONUS.

Please make sure everything is clear and understandable, if it's good I will give thumbs up 100%! Thanks.

1a) Theasymptotes for the graphs of tangent, cotangent, secant, and cosecant are difficult to remember, especially if there is a change in the period. Discuss how to find the asymptotes from the equation of the function. Give an example to illustrate.

1b) Discuss the restricted quadrants for the inverse trigonometric functions.

2) Write down all the identities.

2a) sum and difference Identities

2b) Double-Angle Identities

2c.) Half-Angle Identities

2d) Power-Reduction Formulas

Find the best quadratic approximation to f(x,y) = e^5ycos(-4x) about the origin (0,0) and estimate the value when this approximation is used to calculate f(0.1,0.1).Do not do the error approximation. Also, find the extreme values of the function f(x,y,z) = 8y²-4xz +16z - 35 on the surface 2x²+4y²+4z² = 32

Thank you for your help

3. Parametric Curves (7 marks total) (a) Explain in your own words the similarities and differences between the following two parametric curves. i. x = t 2 , y = t 4 ii. x = cost, y = cos2t (b) . Consider the curve defined parametrically by x = at2 + bt and y = ct3 + dt2 + et, where a, b, c, d, e are all constants. Give one example of what each of the constants a, b, c, d, e could equal so that your curve has exactly one vertical tangent line and no horizontal tangent lines. Show all of your work! Hint: start by finding a formula for dy dx !

1.Identify the open intervals on which the function is increasing or decreasing. (Enter your answers using interval notation.) f(x) = sin(x) + 5 0 < x < 2π increasing decreasing

2. Identify the open intervals on which the function is increasing or decreasing. (Enter your answers using interval notation.) f(x) = x + 2 cos(x), 0 < x < 2π increasing decreasing

3. Consider the following function. f(x) = x + 1 x2 (a) Find the critical numbers of f. (Enter your answers as a comma-separated list.) x = (b) Find the open intervals on which the function is increasing or decreasing. (Enter your answers using interval notation. If an answer does not exist, enter DNE.) increasing decreasing (c) Apply the First Derivative Test to identify the relative extremum. (If an answer does not exist, enter DNE.) relative maximum (x, y) = relative minimum (x, y) =

4. s(t) = t3 − 5t2 + 3t − 290 (a) Find the velocity function v(t) of the particle at any time t ≥ 0. v(t) = (b) Identify the time interval(s) on which the particle is moving in a positive direction. (Enter your answer using interval notation.) (c) Identify the time interval(s) on which the particle is moving in a negative direction. (Enter your answer using interval notation.) (d) Identify the time(s) at which the particle changes direction. (Enter your answers as a comma-separated list.) t =

5. An air traffic controller spots two airplanes at the same altitude converging to a point as they fly at right angles to each other. One airplane is 75 miles from the point and has a speed of 450 miles per hour. The other is 100 miles from the point and has a speed of 600 miles per hour. (a) At what rate is the distance between the planes changing?-------- mph (b) How much time does the controller have to get one of the airplanes on a different flight path? -------h

6. An airplane flies at an altitude of y = 5 miles toward a point directly over an observer (see figure). The speed of the plane is 600 miles per hour. Find the rates (in radians per hour) at which the angle of elevation θ is changing when the angle is θ = 30°, θ = 60°,and θ = 80°. (a) θ = 30° rad/hr (b) θ = 60° rad/hr (c) θ = 80° (Round your answer to two decimal places.) rad/hr

7. The formula for the volume of a cone is given below. Find the rate of change of the volume for each of the radii given below if dr/dt is 8 inches per minute and h = 18r. V = (1/3)πr2h (a) r = 9 in V' = π in3/min (b) r = 30 in V' = π in3/min

What are the connected components and the path components of the product space R x R_l where R has the standard topology and R_l has the lower limit topology?

A Ferris wheel with a diameter of 20 m and makes one complete revolution every 90 seconds. Determine an equation that models your height, in metres, above the ground as you travel on the Ferris Wheel over time, t in  seconds. Assume that at time t=0 the Ferris Wheel is at the lowest position of 3 m.