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 a ball centered at z, and that ball is completely contained within the interval (-2,7).

Solutions

Expert Solution

Colby Messinger answered 11 months ago

Next > < Previous

Related Solutions

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

f(r,?) f(x,y) r(cos(?)) = x r(cos(2?)) = ? r(cos(3?)) = x3-3xy2/x2+y2 r(cos(4?)) = ? r(cos(5?)) =...

f(r,?) f(x,y) r(cos(?)) = x r(cos(2?)) = ? r(cos(3?)) = x3-3xy2/x2+y2 r(cos(4?)) = ? r(cos(5?)) = ? Please complete this table. I am having trouble converting functions from polar to cartesian in the three dimensional plane. I understand that x=rcos(?) and y=rsin(?) and r2 = x2 + y2 , but I am having trouble understanding how to apply these functions.

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.

Consider a region R bounded by the y-axis, the line segment y=8-x for x from 0...

Consider a region R bounded by the y-axis, the line segment y=8-x for x from 0 to 8, and part of the circle y=-sqrt(64-x^2) for x from 0 to 8. Find the centroid.

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

Question