Find the Maclaurin series using the definition of a Maclaurin series as well as the interval of convergence.

a) f(x) = 1/((1-x/3)^2)

b) f(x) = sin(2x)

A primal maximization problem is given.

Maximize

f = 60x₁ + 30x₂

subject to

(a) Form the dual minimization problem. (Use y₁ and y₂ as the variables and g as the function.)
Minimize g =  

  subject to

(b) Solve both the primal and dual problems with the simplex method.

Check that each of the following functions solves the corresponding differential equation, by computing both the left-hand side and right-hand side of the differential equation

A) y=tan(x) solves dy/dx =y^2 / sin^2 (x)

B) y=x^2 +5x solves (x)dy/dx= y+x^2

C)y= 1/ C+x^2 solves dy/dx= -2xy^2

D) y= xln(x) solves (x) dy/dx = x+y

Find the extreme values of the function and where they occur.

y= (x+1)/(x^2+3x+3)

I have a logistic growth equation for a population where K = 820 carrying capacity, and P(0) =550 initial population with a growth rate r of 0.25. Using the equation P= K/(1+A.e^-rt) i can find out the Value of A from initial conditions as A=(K-P(0))/P(0) and apply the equation to calculate values of P at different time intervals between 0 ans 12 years. If I have to use the same parameters as a logistic decay starting with an initial population of 820 and a final population of 550, what would be the value of A. If I apply initial conditions then the value of A comes as negative and if we use -r as negative growth rate in the exponent of the denominator, there result does not follow a logistic decay. However if we keep A positive and r negative then we get a logistic decay. Can you please clarify this confusion.

integrate e^3tsint

Find the second-degree Taylor polynomial of f(x)=ln(1+sinx) centered at x=0.

3. State thoroughly and precisely Euclids fifth postulate.
According to Playfair, "Given a line l and a point P not on l, there is exactly one line through P that is parallel to l"
Are Euclids fifth postulate and Playfair's claim logically equivalent? If yes, justify your stance. If no, produce a counter example.

Find all second order derivatives for r(x,y)=(xy)/2x+5y.

(I need most detail at the rx/ry to rxx/ryy part, I can't get the top correct)

Determine whether the lines and are parallel, skew, or intersecting.

L1:X=1+2t, Y=2+3t , z=3+4t

L2: X=-1+6s, Y=3-s ,z=-5+2s

An object is thrown vertically upward with an initial velocity of 10 m/sec from a height of 3 meters. In meters, find the highest point it reaches. (Round your answer to three decimal places, in m)

Find when it hits the ground. (Enter your answer in seconds. Round your answer to three decimal places in seconds)

A company's total sales (in millions of $) t months from now is given by the function S(t)=2√t+10

1. Find the average rate of change in sales in four to six months from now.

2. Find the S'(t) with appropriate units. Use the limit definition of the derivative (with rationalizing) or technology

3. Find the sales and the instantaneous rate of change of sales in three months. Write a brief explanation of these results