For each of the following vector fields F, decide whether it is conservative or not by computing the appropriate first order partial derivatives. Type in a potential function f (that is, Vf F) with f(0, 0) = 0. If it is not conservative, type N. A. F(x, y) = (6x+2y)i + (2x+12y)j f(x,y) = B. F(x, y) =3yi+ 4xj f(x,y)= C. F(x, y) = (3 sin y)i + (4y+ 3x cos y)j f(x, y) = Note: Your answers should be either expressions of x and y (e.g. "3xy + 2y"), or the letter "N"

Let f1(x)=3x and f2(x)=|x|. Graph both functions on the interval 2x2. Show that these functions are linearly dependent in the vector space C[0,1], but linearly independent in C[1,1].

Repeat the instruction of Exercise 11 for the function. f(x)=x3+x For part d, use i. a1=0.1 ii a1=0.1 11. Consider the function f(x)=4x2(1x) a. Find any equilibrium points where f(x)=x. b. Determine the derivative at each of the equilibrium points found in part a. c. What does the theorem on the Stability of Equilibrium points tell us about each of the equilibrium points found in part a? d. Find the next four iterations of the function for the following starting values. i. a1=0.4. ii. a2=0.7 e. Describe the behavior of successive iteration found in part d. f. Discuss how the behavior found in part d relates to the results from part c.