Suppose you are climbing a hill whose shape is given by the equation

z = 1400 − 0.005x² − 0.01y²,

where x, y, and z are measured in meters, and you are standing at a point with coordinates (60, 40, 1366). The positive x-axis points east and the positive y-axis points north.

(a) If you walk due south, will you start to ascend or descend?

At what rate?

vertical meters per horizontal meter

(b) If you walk northwest, will you start to ascend or descend?

At what rate? (Round your answer to two decimal places.)

vertical meters per horizontal meter

(c) In which direction is the slope largest?

What is the rate of ascent in that direction?

vertical meters per horizontal meter

At what angle above the horizontal does the path in that direction begin? (Round your answer to two decimal places.)

°

Obtain a spectral decomposition for the symmetric matrix A = [0 2 2, 2 0 2, 2 2 0] (that means the first row is 022, then below that 202, etc.) , whose characteristic polynomial is −(λ + 2)^2 (λ − 4)

If you could provide a step-by-step way to solve this I'd greatly appreciate it.

1. Perform two iterations of the gradient search method on f(x,y)= x^2+4xy+2y^2+2x+2y. Use (0,0) as a starting point. Please find the optimal λ* by taking the derivative and setting it equal to 0.

4(a)Verify that the equation y=lnx-lny is an implicit solution of the IVP,

                  ydx=x(y+1)dy    

                            y(e)=1                                          

   (b) Consider the IVP, test whether it is exact and solve it.    

             x(x-y)dy=-(3xy-y^2)dx                   

                      y(-1)=1                                        

(c) Determine the degree of the following homogeneous function                                                 

       (i) f(x,y)=4x^2+2y^4 ,                                     

       (ii) f(x,y)=√5x^6-3y^6+4x^2y^4 ,           

Using a real world example, state the domain and range of the function or relationship. What are the practical things to consider when limiting the domain and range of a real world function?

If u(t) = < sin(5t), cos(5t), t > and v(t) = < t, cos(5t), sin(5t) >, use the formula below to find the given derivative.

d/dt[ u(t) * v(t)] = u'(t) * v(t) + u(t)* v'(t)

d/dt [ u(t) x v(t)] = ?

olve the linear programming problem by sketching the region and labeling the vertices, deciding whether a solution exists, and then finding it if it does exist. (If an answer does not exist, enter DNE.) Maximize P = 20x + 30y Subject to 2x + y ≤ 16 x + y ≤ 10 x ≥ 0, y ≥ 0

A biologist must make a medium to grow a type of bacteria. The percentage of salt in the medium is given by S=0.01x2y2zS=0.01x2y2z , where SS is the percentage expressed as a decimal. And where xx, yy, and zz are the amounts in liters of 3 different nutrients mixed together to create the medium. The ideal salt percentage for this type of bacteria is 34.7%. The costs of the xx, yy, and zz nutrient solutions are respectively 10 , 4, and 10 dollars per liter. Determine the minimum cost that can be achieved.

(Round your answers to the nearest 4 decimal places.)

Using the power series method solve the given IVP. (The answer will include the first four nonzero terms.)

(x + 1)y'' − (2 − x)y' + y = 0, y(0) = 4, y'(0) = −1

Prove or disprove the following statements.
(a) There is a simple graph with 6 vertices with degree sequence (3, 3, 5, 5, 5, 5)?
(b) There is a simple graph with 6 vertices with degree sequence (2, 3, 3, 4, 5, 5)?

I need a decision tree

Consider a scenario in which a college must recruit students. They have three options. They can do nothing. Alternatively, they could make their own ad campaign, spending $200,000. Their campaign has an 70% chance of being successful, which would mean bringing in 100 new students at $40,000 per student in tuition. Or the college could decide to spend $800,000 to hire a social media company to do the ads. This social media campaign has a 60% chance of being successful. If it is successful, there is an 75% chance in will bring in 100 new students. If it is not successful, there is a 10% chance it will bring in 100 new students. Otherwise it will bring in no new students. What should the college do to maximize expected value? Show your work including expected values.

1) Solve the given initial-value problem.

(x + y)² dx + (2xy + x² − 3) dy = 0,   y(1) = 1

2) Find the general solution of the given differential equation.

x dy/dx + (4x + 1)y = e^−4x

y(x) =

Give the largest interval over which the general solution is defined. (Think about the implications of any singular points. Enter your answer using interval notation.)

Determine whether there are any transient terms in the general solution. (Enter the transient terms as a comma-separated list; if there are none, enter NONE.)

please show steps

Give a recursive algorithm to compute a list of all permutations of a given set S. (That is, compute a list of all possible orderings of the elements of S. For example, permutations({1, 2, 3}) should return {〈1, 2, 3〉, 〈1, 3, 2〉, 〈2, 1, 3〉, 〈2, 3, 1〉, 〈3, 1, 2〉, 〈3, 2, 1〉}, in some order.)

Prove your algorithm correct by induction.