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= Which of the following is a diffeomorphism from R2 to R2? O g(x,y) + (x,...
4. Consider the functions f : R2 R2 and g R2 R2 given by f(x, y) (x, xy) and g(x, y)-(x2 + y, x + y) (a) Prove that f and g are differentiable everywhere. You may use the theorem you stated in (b) Call F-fog. Properly use the Chain Rule to prove that F is differentiable at the point question (1c). (1,1), and write F'(1, 1) as a Jacobian matrix.
4. Consider the functions f : R2 R2 and...
Consider the following subsets of R2:
C1 ={(x,y)∈R2 :x+y≤3,x≥0,y≥0}
C2 ={(x,y)∈R2 :4x+y≤4,x≥0,y≥0}
Algebra Consider the following subsets of R2. Draw a sketch of the intersection CinC2 and the union C1UC2. State whether each set is convex or not. If the set is not convex, give an example of a line segment for which the definition of convexity breaks down
Algebra Consider the following subsets of R2. Draw a sketch of the intersection CinC2 and the union C1UC2. State whether each...
Suppose (X,Y)∼Unif(A), whereA⊆R2 is the triangular region connecting vertices (0,2),(2,0), and (2,2). FindCor(X,Y).
Which of the following are linear transformations? f: R3 R2 [x, y, z] [7x - 2y, 0 h R R x > sin x g R2R [x, y] [y- x, 2 the map T R > R< described by reflection in a line L: 2x + 7y = 0 through the origin.
EXERCISE 57 (2-27). Define g, h: {x E R2· 1} → R3 by g(x, y) = (x, y, V1-x2ー),2) , h(x, y) = (x, y,-V1-x2-y2) Show that the maximum off on {x E R3 : 11x11 = 1} is either the maximum of fo g or the maximum offo h on {x e R2· 1} .
4. A random point (X, Y ) is chosen uniformly from within the unit disk in R2, {(x, y)|x2+y2< 1} (a) Let (R, O) denote the polar coordinates of the point (X,Y). Find the joint p.d.f. of R and . Compute the covariance between R and 0. Are R and e are independent? (b) Find E(XI{Y > 0}) and E(Y|{Y > 0}) (c) Compute the covariance between X and Y, Cov(X,Y). Are X and Y are independent?
4. A random...
Which of the following transformations T R2 → R is linear? A. T(x,y)=5 OB. T(x,y) = x2 OC. None of the given options OD. TIX.Y)=3x+4y OE. T(x,y) = x2 + y2
5) Let P(1,2,2) be a point, and f(x,y,z) and g(x,y,z) be two differentiable functions satisfying the following conditions. 1) f(P)=1 and g(P)=4 og IT) = -2 Oz IP III) The direction in which f increases most rapidly at the point Pis ū=4i - +8k , and the derivative in this direction is 3. IV) Equation of the plane tangent to the surface f(x,y,z)+3g(x,y,z)=13 at the int P is x+4y + 5z =19 According to this, calculate og Ox . (20P)
Which one of the following description is wrong?* 1 point x 8,8 ca Y X >0,0 2,2 B Y 6,6 5, 5 O There are 2 pure-strategy Nash Equilibria. There exists some strictly dominated strategy. O There are infinite mixed-strategy Nash Equilibria. 0 (0,0,1),(1/4, 3/4)) is a mixed-strategy Nash Equilibrium.
Which one of the following is a linear transformation? O F(x, y) = xy O F(x,y) = (22,32) O F(x, y, z) = (x,y – 2,0) O F(x, y, z) = (2,4 – 2,1)