Given the set A = {(x, y) ∈ R2 | x2 + y2 < 1 and...

Given the set A = {(x, y) ∈ R² | x² + y² < 1 and y ≥ 0}. Draw sketches of cl A, int A, ∂A, (cl(A^c))^c, the limit points of A, and the isolated points of A. Try to be clear about what the sketch is describing. (The answer does not depend on whether one uses the Euclidean distance or the taxi distance on R².)

Expert Solution

Colby Messinger answered 3 months ago

1. Let ρ: R2 ×R2 →R be given by ρ((x1,y1),(x2,y2)) = |x1 −x2|+|y1 −y2|. (a) Prove...

1. Let ρ: R2 ×R2 →R be given by ρ((x1,y1),(x2,y2)) = |x1 −x2|+|y1 −y2|. (a) Prove that (R2,ρ) is a metric space. (b) In (R2,ρ), sketch the open ball with center (0,0) and radius 1. 2. Let {xn} be a sequence in a metric space (X,ρ). Prove that if xn → a and xn → b for some a,b ∈ X, then a = b. 3. (Optional) Let (C[a,b],ρ) be the metric space discussed in example 10.6 on page 344...

The optimal solution to the following problem is: Max X+Y2 S.T. X2+Y2 <= 200 X-Y <=...

The optimal solution to the following problem is: Max X+Y2 S.T. X2+Y2 <= 200 X-Y <= 20 3Y2-X <= 50 X = 5.34, Y=4.29 X = 13.37, Y=4.59 X = 11.31, Y=3.14 X = 13.31, Y=3.14

Let x,y ∈ R3 such that x = (x1,x2,x3) and y = (y1,y2,y3) determine if <x,y>=...

Let x,y ∈ R3 such that x = (x1,x2,x3) and y = (y1,y2,y3) determine if <x,y>= x1y1+2x2y2+3x3y3 is an inner product

Find and classify all the extrema of the function f(x; y) = Exp(-x2 -y2 )*(x2 +...

Find and classify all the extrema of the function f(x; y) = Exp(-x2 -y2 )*(x2 + 2y2).

Curvilinear integral of the function f (x, y) = x2 + y2 on a (3,0) centered...

Curvilinear integral of the function f (x, y) = x2 + y2 on a (3,0) centered and 3 radius circle. a)Calculate the curvilinear integral by expressing the curve in parametrically. b)Calculate the curvilinear integral by expressing the curve in polar coordinates. c)Calculate the curvilinear integral by expressing the curve in cartesian coordinates.

Sketch the region enclosed by the given curves and find its area. a) y=4/x,y=4x,y=(1/4)x,x>0 b) x=y2−4y,x=2y−y2

a. Consider d on R, the real line, to be d(x,y) = |x2 – y2|. Show...

a. Consider d on R, the real line, to be d(x,y) = |x2 – y2|. Show that d is NOT a metric on R. b.Consider d on R, the real line, to be d(x,y) = |x3 – y3|. Show that d is a metric on R. 2. Let d on R be d(x,y) = |x-y|. The “usual” distance. Show the interval (-2,7) is an open set. Note: you must show that any point z in the interval has...

The value of f depends on two independent variables x and y as defined below: f(x,y)=x2+y2−x+3cos(x)sin(y)...

The value of f depends on two independent variables x and y as defined below: f(x,y)=x2+y2−x+3cos(x)sin(y) Function f has a minimum in the neighborhood of the origin (i.e. [0 0]). Find x and y which minimize f. Note: You are not allowed to use MATLAB built-in functions for optimization. Follow the logic of the derivative test: The minimum of f occurs where: ∂f∂x=0∂f∂y=0

Evaluate or solve the following A) dy/dx= -(2x2+y2)/(2xy+3y2) B)dy/dx=(1+y2)/(1+x2)xy C) (x2+1)dy/dx+2xy=4x2 given that when x=3,y=4 Already...

Evaluate or solve the following A) dy/dx= -(2x2+y2)/(2xy+3y2) B)dy/dx=(1+y2)/(1+x2)xy C) (x2+1)dy/dx+2xy=4x2 given that when x=3,y=4 Already rated.Best chegg expert

(1 point) In general for a non-homogeneous problem y′′+p(x)y′+q(x)y=f(x) assume that y1,y2 is a fundamental set...

(1 point) In general for a non-homogeneous problem y′′+p(x)y′+q(x)y=f(x) assume that y1,y2 is a fundamental set of solutions for the homogeneous problem y′′+p(x)y′+q(x)y=0. Then the formula for the particular solution using the method of variation of parameters is yp=y1u1+y2u2 where u′1=−y2(x)f(x)W(x) and u′2=y1(x)f(x)W(x) where W(x) is the Wronskian given by the determinant W(x)=∣∣∣y1(x)y′1(x)y2(x)y′2(x)∣∣∣ So we have u1=∫−y2(x)f(x)W(x)dx and u2=∫y1(x)f(x)W(x)dx. NOTE When evaluating these indefinite integrals we take the arbitrary constant of integration to be zero. In other words we have...

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