4. (Oblique Trajectory Problem) Let F(x, y) = x 2 + xy + y 2 . Find a formula for G(x, y) such that every curve in the one-parameter family defined by F(x, y) = c intersects every curve in the one-parameter family defined by G(x, y) = c at a sixty degree angle

Using the epsilon-delta definition prove

Lim of x^2 =4 when x approaches 2

Fill out table 4 for all pH measurements made in table 2.

Table 2: pH of acids and conjugate bases

Table 4: pH for acids and bases (include in this table all solutions from table 2)

a.maximize and minimize −2xy on the ellipse x^2+4y^2=4

b.Determine whether or not the vector function is the gradient ∇f (x, y) of a function everywhere defined. If so, find all the functions with that gradient. (x^2+3y^2)i+(2xy+e^x)j

(a). Let S be the surface defined by x^4 − 3x^2y^2 + 4z^2 = 17. Find the equation of the tangent plane to the surface S at (1, 0, 2).

(b)

Let f(x,y) = e^2xy

. Let P = (2,0) and Q = (3,2). Find the directional derivative of f at P in the

direction of PQ.

Balance the following redox reaction, assume in basic conditions. SO 3 2- + CrO 4 2-  SO 4 2- + CrO 2 1-

F(x) = 0 + 2x + (4* x^2)/2! + (3*x^3)/3! + .....

This is a taylors series for a function and I'm assuming there is an inverse function with an inverse taylors series, I am trying to find as much of the taylors series of the inverse function (f^-1) as I can

Consider the following polynomial

?(?)=(?+2)(?−10)(?+4)(?−2)

a)Use Matlab to multiply these terms in order to form a 4thorder polynomial

b)Apply the ‘roots’ function to your answer in part a to verify your calculation was correct.

c)Use Matlab to divide your polynomial answer in part aby the term (s+2

)d)Apply ‘roots’ to your answer in part c to verify your calculation was correct.

4) If the US has 250,000 hours available to produce 2 goods: Wheat which takes 2 hours per ton of wheat to produce and Airplanes which takes 1,000 hours per airplane to produce. If China has 500,000 hours available to produce 2 goods: Wheat which take 5 hours per ton of wheat to produce and airplanes which takes 3,000 hours per airplane to produce. a. Draw the PPF diagram for the US assume they consume half of each product without trade. Demonstrate all relevant points. b. Draw the PPF diagram for China assume they consume half of each product without trade. Demonstrate all relevant points c. Is there a basis for trade between US and China? If so what is the economic theory for the basis of trade? d. If they trade and the terms of trade are 50,000 Wheat for 100 planes, demonstrate and prove the gains from trade.