F(x) = x^2 on (1,∞)

Definition

The function f(x) = x^2 defined on the interval (1,∞) represents a parabolic curve that opens upwards, starting from the point (1, 1) and extending infinitely as x increases. This function showcases specific properties such as continuity and differentiability over the specified domain, which are crucial for understanding concepts like uniform continuity. The characteristics of this function allow for analysis of its behavior as x approaches large values and how it interacts with the concept of uniform continuity.

5 Must Know Facts For Your Next Test

1. The function f(x) = x^2 is continuous on the interval (1,∞), meaning there are no breaks or jumps in the graph.
2. As x increases within the interval (1,∞), f(x) grows without bound, demonstrating that f(x) approaches infinity as x approaches infinity.
3. The derivative f'(x) = 2x shows that the function is increasing on its entire domain, which impacts its uniform continuity.
4. Since f(x) is a polynomial function, it is uniformly continuous on any closed interval, but it's essential to evaluate conditions on unbounded intervals like (1,∞).
5. For the function f(x) = x^2 on (1,∞), uniform continuity can be shown by proving that the differences in outputs can be controlled by choosing appropriate distances in inputs.

Review Questions

• How does the behavior of f(x) = x^2 on (1,∞) illustrate the concept of uniform continuity?
  • The behavior of f(x) = x^2 on (1,∞) demonstrates uniform continuity because for any ε > 0, we can find a corresponding δ > 0 such that for all x1, x2 in (1,∞), if |x1 - x2| < δ then |f(x1) - f(x2)| < ε. This holds true because as both inputs move further from 1, the differences in outputs become increasingly manageable due to the function's smooth and predictable growth.
• Discuss how the properties of continuity and differentiability affect the analysis of f(x) = x^2 on (1,∞).
  • The properties of continuity and differentiability of f(x) = x^2 on (1,∞) ensure that it behaves predictably. Continuity implies there are no interruptions in its graph. Differentiability indicates that the slope at any point is well-defined and positive since f'(x) = 2x is always greater than zero for x > 1. This means that as you move right along the interval, both outputs and rates of change consistently increase, contributing to our understanding of how it meets criteria for uniform continuity.
• Evaluate how understanding the limit at infinity for f(x) = x^2 affects interpretations of its behavior and uniform continuity.
  • Understanding the limit at infinity for f(x) = x^2 provides valuable insight into its growth behavior. As x approaches infinity, f(x) also approaches infinity. This unbounded behavior indicates that despite being continuous and differentiable across its domain, we need to be careful when discussing uniform continuity since it relates to how distances in outputs behave compared to inputs. Recognizing this limit helps clarify why uniform continuity can still be established even though we deal with an infinite range.

Related terms

Uniform Continuity: A function is uniformly continuous if, for every ε > 0, there exists a δ > 0 such that for any two points in the domain, if their distance is less than δ, then the distance between their images is less than ε.

Continuous Function: A function is continuous if, intuitively, you can draw its graph without lifting your pencil from the paper; more formally, it means that small changes in input lead to small changes in output.

Limit at Infinity: The value that a function approaches as the input approaches infinity; it helps to describe the end behavior of functions as x becomes very large.