a)

Find the value of the Wronskian of the functions  f = x^7

and g = x^8 at the piont  x = 1.

b)

Let  y be the solution of the equation y ″ − 5 y ′ + 6 y = 0

satisfying the conditions y ( 0 ) = 1 and y ′ ( 0 ) = 2.

Find ln ⁡ ( y ( 1 ) ).

c)

Let y be the solution of the equation  y ″ + 2 y ′ + 2 y = 0

satisfying the conditions  y ( 0 ) = 0 and y ′ ( 0 ) = 1.

Find the value of  y at x = π.

d)

Let y be the solution of the equation  y ″ + 6 y ′ + 9 y = 0

satisfying the conditions  y ( 0 ) = 0 and y ′ ( 0 ) = 1.

Find the value of the function   f ( x ) = ln [ (y(x))/(x)] at  x = 1.

e)

One of solutions of the equation y ″ − y ′ + y = x^2 + 3x + 5

is a function of the form y = A*x^2 + B*x + C.

Find the value of the coefficient C.

f)

One of solutions of the equation y ″ − 2 y ′ + 2 y = ( x + 1 ) e^x

is a function of the form y = ( A*x + B ) e^x.

Find the value of the coefficient B.

Shane (famous Western movie name of the star character) settles down in Wyoming Territory. He lives the same distance from General Store (-78,202), the Saloon is (111, 193) and the Courthouse (202, -106). What are coordinates of Shane's home?

How far is Shane's house from the place mentioned?

What is the equation passing through the General Store, Saloon and Courthouse?

A manufacturer estimates that production​ (in hundreds of​ units) is a function of the amounts x and y of labor and capital​ used, as f(x,y)=[ 1/3x^-1/3+1/3y^-1/3]^-3. Find the number of units produced when

8 units of labor and 27 units of capital are utilized. Find and interpret

f_x (8,27) and f_y (8,27).

What would be the approximate effect on production of increasing labor by 1​ unit?

A) The number of units produced when 64 units of labor and 125 units of capital are utilized is ___

B) f_x( , ) =

c) f_y=( , )

d) The approximate effect on production of increasing labor by 1 unit would be ___ units

perform a rotation of axes with a suitable angle of rotation (no xy term) and identify the related conic.

1. x²-xy+y²=2

2. x²-3y²-8x+30y=60

A lamina occupies the region inside the circle x^2 + y^2 = 6x, but outside the circle x^2 + y^2 =9.

Find the center of mass if the density at any point is inversely proportional to its distance from the origin.

A man gets a job with a salary of $38,900 a year.
He is promised a $2,430 raise each subsequent year

During a 9-year period his total earnings are________

A lamina occupies the region inside the circle x^2 + y^2 = 6x, but outside the circle x^2 + y^2 =9.

Find the center of mass if the density at any point is inversely proportional to its distance from the origin.

In this assignment, we will explore four subspaces that are connected to a linear transformation. For the questions below, assume A is an m×n matrix with rank r, so that T(~x) = A~x is a linear transformation from R n to R m. Consider the following definitions:

• The transpose of A, denoted A^T is the n × m matrix we get from A by switching the roles of rows and columns – that is, the rows of A^T are the columns of A, and vice versa.

• The column space of A, denoted col(A), is the span of the columns of A. col(A) is a subspace of R m and is the same as the image of T.

• The row space of A, denoted row (A), is the span of the rows of A. row (A) is a subspace of Rⁿ .

• The null space of A, denoted null(A), is the subspace of Rⁿ made up of vectors x such that Ax = 0 and is equal to the kernel of T.

• The left null space of A, denoted null( A^T) , is the subspace of R^m made up of vectors y such that A^Ty = 0.

col(A), row (A), null(A), and null (A^T ) are called the four fundamental subspaces for A.

We showed in class that the pivot columns of A form a basis for col(A), and that the vectors in the general solution to Ax = 0 form a basis for null(A). Likewise, the vectors in the general solution to A^Ty = 0 form a basis for null (A^T ).

Q3: Show that row (A) and null(A) are orthogonal complements.

Q4: Show that col(A) and null ( A^T ) are orthogonal complements.

Q5: Assuming A is an m × n matrix with rank r, what are the dimensions of the four subspaces for A?

if two parallel planes, vector n1* vector x = a, vector n2* vector x = b. How to describe their distance?

Approximate the area under the graph of f(x)=0.03x4−1.44x2+58 over the interval [2,10] by dividing the interval into 4 subintervals. Use the left endpoint of each subinterval.

The area under the graph of f(x)=0.03x4−1.44x2+58 over the interval [2,10] is approximately ...

A firm produces two commodities, A and B. The inverse demand functions are:

pA =900−2x−2y, pB =1400−2x−4y

respectively, where the firm produces and sells x units of commodity A and y units of commodity B. Its costs are given by:

CA =7000+100x+x^2 and CB =10000+6y^2

where A, a and b are positive constants.

(a) Show that the firms total profit is given by:

π(x,y)=−3x^2 −10y^2 −4xy+800x+1400y−17000.

(b) Assume π(x, y) has a maximum point. Find, step by step, the production levels that maximize profit by solving the first-order conditions. If you need to solve any system of linear equations, use Cramer’s rule and provide all calculation details.

(c) Due to technology constraints, the total production must be restricted to be exactly 60 units. Find, step by step, the production levels that now maximize profits – using the Lagrange Method. If you need to solve any system of linear equations, use Cramer’s rule and provide all calculation details. You may assume that the optimal point exists in this case.

(d) Report the Lagrange multiplier value at the maximum point and the maximal profit value from part (c). No explanation is needed.

(e) Using new technology, the total production can now be up to 200 units (i.e. less or equal to 200 units). Use the values from part (c) and part (d) to approximate the new maximal profit.

(f) Calculate the true new maximal profit for part (e) and compare with its approximate value you obtained. By what percentage is the true maximal profit different from the approximate value?

I only need answers for part(e) and (f)!!!!!! Thanks for your help!!!!!

g(x, y) = 2x ³ + 9xy² + 15x ² + 27y²

Find all the critical points of the following functions. For each critical point of g(x, y), determine whether g has a local maximum, local minimum, or saddle point at that point.

Determine the slope of the tangent line, then find the equation of the tangent line at t=−1 .

x= 9t,

y=t^2

A water tank is spherical in shape with radius of 90 feet. Suppose the tank is filled to a depth of 140 feet with water. Some of the water will be pumped out and, at the end, the depth of the remaining water must be 40 feet.

i) Set up a Riemann sum that approximates the volume of the water that is pumped out of the tank. (use horizontal slicing). You have to draw a diagram and choose a coordinate system to associate with the diagram.

ii) Set up, but do NOT evaluate, the integral for the volume of the water that is pumped out of the tank.

iii) Suppose the tank is filled at the depth of 140 feet with water. (Use the fact that water weighs approximately 62.4 lbs) . Some of the water is to be pumped out through a spout that is 6 feet over the top of the tank. The depth of the water remaining at the end is 40 feet. Set up (do not evaluate) the definite integral for the work required to pump the water out of the spout.