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∞calculus iv review

F(x,y) = x^2 + y^2

Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025

Definition

The function f(x,y) = x^2 + y^2 defines a surface in three-dimensional space where the value of f corresponds to the square of the distance from the origin in the xy-plane. This function is important because it showcases how values of x and y contribute to the overall output, illustrating concepts like limits and continuity in multiple dimensions. The surface formed by this function is a paraboloid, which can help understand behavior around points and how functions approach certain values.

5 Must Know Facts For Your Next Test

  1. The function f(x,y) = x^2 + y^2 represents a continuous surface over the entire xy-plane since it does not have any breaks or holes.
  2. As (x,y) approaches (0,0), the value of f(x,y) approaches 0, demonstrating that the limit exists at this point.
  3. The value of f increases without bound as either x or y moves away from 0, indicating that the function is unbounded.
  4. For any line approaching the origin (such as y=mx), the limit of f(x,y) as (x,y) approaches (0,0) is consistently 0, showing uniform behavior regardless of path.
  5. This function is differentiable everywhere in its domain, meaning it has well-defined partial derivatives at all points.

Review Questions

  • How does the function f(x,y) = x^2 + y^2 illustrate the concept of limits in multiple variables?
    • The function f(x,y) = x^2 + y^2 demonstrates limits by showing that as (x,y) approaches (0,0), f(x,y) consistently approaches 0 regardless of the path taken. This uniformity is crucial because it indicates that the limit exists at that point, which is essential when discussing continuity and differentiability in multiple dimensions.
  • In what ways do level curves help in understanding the behavior of the function f(x,y) = x^2 + y^2?
    • Level curves for the function f(x,y) = x^2 + y^2 are concentric circles centered at the origin. Each curve represents points where the function equals a constant value. Analyzing these curves reveals how the output of the function increases with distance from the origin, helping visualize concepts like gradients and contours in relation to limits and continuity.
  • Evaluate how the properties of continuity and differentiability are demonstrated through f(x,y) = x^2 + y^2 when considering its application to real-world problems.
    • The properties of continuity and differentiability in f(x,y) = x^2 + y^2 allow for smooth transitions between points on its surface. In real-world applications, such as optimization problems, these properties ensure that small changes in input will yield predictable changes in output. This reliability is crucial when modeling scenarios like minimizing distance or maximizing area, where understanding how outputs behave as inputs vary is key to finding optimal solutions.