Find the local maximum and minimum values and saddle point(s) of the function.

f(x,y)=5-10x+12y-5x^2-4y^3

Find the Taylor series or polynomial generated by the following functions

a. )f(x) √ x centred at x=4 , of order 3

b.) f(x) cosh x= e^x+e^-x/(2), centred at x=0

c.) f(x) = x tan^-1x^2 , centred at x=0

d.) f(x) = 1/(√1+x^3) , centred at x=0 , of order 4

e.) f(x) = cos(2x+pie/2) centred at x= pie/4

We consider a rectangular parallelepiped-shaped box based on a rectangle and open from above. The height of the box is 4 dm. The base of the box has a fixed perimeter 20 dm and one side of it is x with 0 <x <10.
a. Prove that the total area of the box as a function of x is E(x)=-x²+10x=80, x belongs to (0,10)
b. Find for which value of x the box has a maximum area.
c. Show that E^'[E(X) / 40] <6, for each x belongs to (0,10).
d. Find the value of x for which it holds E(x-2)=absolute value of x-7 + 105.

Find the Taylor series for f (x) = 2/(1+3x) at x = a where a = 0

9) R is the region bounded by the curves ? = x^3 , y=2x+4 , And the y-axis.

a) Find the area of the region.

b) Set up the integral you would use to find the volume of a solid that has R as the base and square cross sections perpendicular to the x-axis.

The National Bank, like most other large banks, found that using automatic teller machines (ATMs) reduces the cost of routine bank transactions. National installed an ATM in the corporate offices of the Fun Toy Company. The ATM is for the exclusive use of Fun's 595 employees. After several months of operation, a sample of 100 employees revealed the following use of the ATM machine by Fun employees in a month:

a. What is the estimate of the proportion of employees who do not use the ATM in a month?

Proportion %:

b-1. For estimate of the proportion of employees who do not use the ATM in a month, develop a 80% confidence interval. (Round the final answers to 3 decimal places.)

80% confidence interval is _____and ______ .

b-2. Can National be sure that at least 40% of the employees of Fun Toy Company will use the ATM?

Yes or No?

c. How many transactions does the average Fun employee make per month? (Round the final answer to 2 decimal places.)

Total transactions ___________

d. Develop a 80% confidence interval for the mean number of transactions per month. (Round the final answers to 3 decimal places.)

80% confidence interval for the mean number of transaction per month is ____ and ______ .

e. Is it possible that the population mean is 0?

Yes or No?

A cereal box, in the shape of a rectangular prism and with a closed top, is to be
constructed so that the base is twice as long as it is wide. Its volume is to be 8000cm3。

Find the dimensions that will minimize the amount of cardboard required to make the box.

A roofer requires 10 h to shingle a roof. After the roofer and an apprentice work on a roof for 5 h, the roofer moves on to another job. The apprentice requires 13 more hours to finish the job. How long would it take the apprentice, working alone, to do the job?

Calculate the arc length of the indicated portion of the curve r(t).

r(t) = i + (9t sin t)j + (9t cos t)k ; -3 ≤ t ≤ 7

The motion of a particle in space is described by the vector equation

⃗r(t) = 〈sin t, cos t, t〉

Identify the velocity and acceleration of the particle at (0,1,0) How far does the particle travel between t = 0 & t= pi

What's the curvature of the particle at (0,1,0) & Find the tangential and normal components of the acceleration particle at (0,1,0)

Solve each inequality and graph the solution. Thank you!

1. -3/8(x) - 20 +2x > 6

2. 2/3(x) + 14 - 3x > -7

3. 0.5x - 4 - 2x ≤ 2

4. 4x + 1 + 2x ≥ 5

Consider the linear transformation T : P2 ? P2 given by T(p(x)) = p(0) + p(1) + p 0 (x) + 3x 2p 00(x). Let B be the basis {1, x, x2} for P2.

(a) Find the matrix A for T with respect to the basis B.

(b) Find the eigenvalues of A, and a basis for R 3 consisting of eigenvectors of A.

(c) Find a basis for P2 consisting of eigenvectors for T.

Determine the amplitude, the period and the phase shift of the function

1. y = -3 cos ( 4 x - PIE ) + 2

2. y = 2 + 3 cos ( PIEx - 3 )

3. y = 5 - 2 cos ( PIE/2 x + PIE/2 )

4. y = - 1/2 cos (2 PIE x ) + 2

5.  y = - 2 sin ( - 2 x + PIE ) - 2

6. y = - sin ( ( ½ x - PIE/2 ) + 1/2

Thank you.

12a Find an equation of the tangent plane to the surface ? = 2? 2 + ? 2 − 5?, ?? (1, 2, −4).

12b If ? = ? 2 − ?? + 3? 2 and (?, ?) changes from (3, −1) to (2.96, −0.95), compare ∆? and ??.

Calculus 3 question. Please help.