Differentiability

Definition

Differentiability is a fundamental concept in calculus that describes the smoothness and continuity of a function. It is closely related to the idea of the derivative, which measures the rate of change of a function at a particular point.

5 Must Know Facts For Your Next Test

1. A function is differentiable at a point if it is continuous at that point and the limit of the difference quotient (the slope of the secant line) exists as the input approaches that point.
2. Differentiability implies continuity, but continuity does not necessarily imply differentiability. There are functions that are continuous but not differentiable at certain points.
3. The derivative of a differentiable function represents the slope of the tangent line to the function at a given point, and it can be used to analyze the behavior of the function, such as its rates of change and critical points.
4. The differentiability of a function is important in understanding its behavior, as differentiable functions have many desirable properties, such as the ability to be optimized and the existence of antiderivatives.
5. The concept of differentiability is crucial in the study of rates of change and the behavior of graphs, as it allows for the analysis of how a function is changing at a particular point and the prediction of its future behavior.

Review Questions

• Explain the relationship between differentiability and continuity, and provide an example of a function that is continuous but not differentiable.
  • Differentiability is a stronger condition than continuity. A function must be continuous at a point in order to be differentiable at that point, but continuity does not necessarily imply differentiability. An example of a function that is continuous but not differentiable is the absolute value function, $|x|$. This function is continuous for all real numbers, but it is not differentiable at $x = 0$ because the left and right limits of the difference quotient do not exist at that point.
• Describe how the derivative of a differentiable function is related to the behavior of the function, including its rates of change and critical points.
  • The derivative of a differentiable function represents the slope of the tangent line to the function at a given point. This information can be used to analyze the behavior of the function, such as its rates of change and critical points. The derivative can be used to determine the points where the function is increasing, decreasing, or has a local maximum or minimum. Additionally, the derivative can be used to find the points where the function's rate of change is changing, which are known as critical points. Understanding the differentiability and derivative of a function is crucial for studying its behavior and characteristics.
• Explain the importance of differentiability in the context of rates of change and the behavior of graphs, and discuss how it relates to the concept of the limit.
  • Differentiability is a crucial concept in the study of rates of change and the behavior of graphs because it allows for the analysis of how a function is changing at a particular point and the prediction of its future behavior. The derivative of a differentiable function represents the instantaneous rate of change of the function at a given point, which is essential for understanding the function's behavior. Additionally, differentiability is closely related to the concept of the limit, as a function must be differentiable at a point if the limit of the difference quotient (the slope of the secant line) exists as the input approaches that point. The differentiability of a function, and its connection to limits and derivatives, is fundamental for the study of rates of change and the analysis of graph behavior.

Related terms

Continuity:

A function is continuous if it has no sudden jumps or breaks in its graph, and the function value at a point can be approached from either side.

Derivative:

The derivative of a function represents the instantaneous rate of change of the function at a given point, and it is the foundation for the study of rates of change.

Limit:

The limit of a function describes the behavior of the function as the input approaches a particular value, and it is a crucial concept in determining differentiability.