Differentiable Function

Definition

A differentiable function is one that has a derivative at each point in its domain. This means that the function is smooth and continuous at those points, allowing us to determine the rate at which the function changes. The concept of differentiability connects closely with continuity and provides the foundation for understanding how functions behave, especially when applying various rules and theorems related to calculus.

5 Must Know Facts For Your Next Test

1. For a function to be differentiable at a point, it must be continuous at that point; however, continuity alone does not guarantee differentiability.
2. If a function has a sharp corner or cusp at a point, it will not be differentiable there, even if it is continuous.
3. The derivative gives the instantaneous rate of change of the function, which can provide valuable insights into its behavior and trends.
4. Differentiability implies that the function can be locally approximated by a linear function near any point in its domain.
5. Rolle's Theorem states that if a function is continuous on a closed interval and differentiable on an open interval, then there exists at least one point where the derivative is zero.

Review Questions

• How does differentiability relate to continuity, and why is this connection important?
  • Differentiability is inherently tied to continuity because for a function to be differentiable at a certain point, it must first be continuous at that point. This connection is essential because it ensures that we can analyze how functions behave and apply differentiation rules effectively. If a function fails to be continuous at a point, we cannot ascertain its derivative there, which complicates our understanding of its behavior in that region.
• Discuss how the presence of sharp corners or cusps in a function affects its differentiability and provide an example.
  • Sharp corners or cusps indicate points where a function may change direction abruptly, making it non-differentiable at those points. For example, the absolute value function, $$f(x) = |x|$$, has a cusp at $$x = 0$$. Although it is continuous everywhere, it is not differentiable at this specific point because there is no unique tangent line that can be drawn; instead, there are two different slopes approaching from either side.
• Evaluate how differentiability plays a crucial role in applying the Mean Value Theorem and its implications for understanding function behavior.
  • Differentiability is critical when applying the Mean Value Theorem because this theorem requires functions to be continuous on a closed interval and differentiable on an open interval. It assures us that there exists at least one point where the instantaneous rate of change (the derivative) equals the average rate of change over that interval. This relationship not only helps in analyzing slopes but also offers deeper insights into the behavior of functions between given endpoints, allowing for predictions about maxima and minima based on their derivatives.

Related terms

Continuous Function: A continuous function is one where small changes in the input lead to small changes in the output, meaning there are no breaks or jumps in the graph of the function.

Derivative: The derivative of a function is a measure of how the function's output changes as its input changes, representing the slope of the tangent line to the graph at a given point.

Cauchy Mean Value Theorem: The Cauchy Mean Value Theorem generalizes the Mean Value Theorem by relating the derivatives of two functions to their respective differences over an interval.