key term - X² + y² < 1

Definition

The expression $$x^2 + y^2 < 1$$ represents the interior of a circle with a radius of 1 centered at the origin (0,0) in the Cartesian coordinate system. This inequality indicates all the points (x,y) that lie inside this circle, excluding the boundary. Understanding this concept is essential when analyzing the domains and ranges of multivariable functions, as it helps identify which pairs of values for x and y are valid inputs.

5 Must Know Facts For Your Next Test

1. The area represented by $$x^2 + y^2 < 1$$ is a two-dimensional region inside the circle with radius 1.
2. The boundary of the circle is given by $$x^2 + y^2 = 1$$, which is not included in the solution set for the inequality.
3. Points such as (0,0), (0.5, 0.5), and (-0.6, -0.4) satisfy the inequality, while points like (1,0) and (0,-1) do not.
4. In terms of multivariable functions, any function defined within this circular region will have its domain restricted to the area described by $$x^2 + y^2 < 1$$.
5. This expression can be visualized graphically, where the shaded area inside the circle represents all valid (x,y) pairs for certain functions.

Review Questions

• How does the inequality $$x^2 + y^2 < 1$$ define a specific region in the Cartesian plane?
  • The inequality $$x^2 + y^2 < 1$$ defines a circular region in the Cartesian plane that includes all points whose distance from the origin is less than 1. This means any point (x,y) that satisfies this condition lies within the interior of a circle with radius 1 centered at (0,0). Understanding this helps in visualizing how functions behave within bounded regions and aids in finding valid input values for multivariable functions.
• Discuss how knowing the area defined by $$x^2 + y^2 < 1$$ can impact the determination of function ranges in multivariable calculus.
  • Knowing the area defined by $$x^2 + y^2 < 1$$ allows us to clearly understand the limitations on input values when determining the range of functions that depend on two variables. For instance, if we have a function $$f(x,y)$$ that only takes inputs from this region, we can analyze how it behaves within this bounded area. This knowledge aids in calculating limits, continuity, and integration over defined regions in multivariable calculus.
• Evaluate how changing the inequality to $$x^2 + y^2 eq 1$$ alters its implications for domain restrictions in multivariable functions.
  • Changing the inequality to $$x^2 + y^2 eq 1$$ modifies the implications significantly because it now excludes only points on the boundary of the circle but allows all other points inside and outside. This means functions defined under this new condition can take on values from both within and outside the circular area, except for those exactly on the perimeter. Such changes affect calculations related to convergence or divergence of integrals and dictate different behavior in function limits when analyzing continuity near boundary points.

Related terms

Circle: A set of points in a plane that are equidistant from a fixed point known as the center.

Inequality: A mathematical statement indicating that one expression is less than or greater than another.

Domain:

The set of all possible input values (x,y) for which a function is defined.