Continuity and Rates of Change

Continuity

Continuity is a fundamental concept in calculus, describing whether a function can be drawn without interruption. Understanding continuity is essential for analyzing functions in business applications, such as cost and revenue models.

• Intuitive Explanation: A function is continuous at a point if you can draw its graph at that point without lifting your pencil.

• Formal Definition: A function f(x) is continuous at x = a if the following holds:

• Discontinuity: A function is discontinuous at a point if the above condition fails. Discontinuities are classified as:

  • Removable: The discontinuity can be fixed by redefining the function at that point.

  • Nonremovable: The discontinuity cannot be fixed by redefining the function at that point.

Example: Given a graph, identify points of discontinuity and classify them as removable or nonremovable.

Continuity of Common Functions

Different types of functions have different continuity properties. Recognizing these helps in modeling business scenarios.

• Polynomials: are continuous for all .

• Rational Functions: where and are polynomials. Continuous where .

• Square Root Functions: are continuous where .

• Exponential Functions: where are continuous for all .

• Logarithmic Functions: where and are continuous for all .

Example: Determine where the following functions are continuous:

• (a) (continuous everywhere)

• (b) (continuous where , i.e., )

• (c) (continuous where , i.e., )

• (d) (continuous where and , i.e., )

Piecewise Functions and Continuity

Piecewise functions are defined by different expressions over different intervals. Continuity must be checked at the boundaries where the pieces meet.

• Check continuity within each piece, considering domain restrictions.

• Check continuity at the "gluing" points by evaluating limits from both sides.

Example: Where is the following function continuous?

• Check continuity at and within each interval.

Example: For which value of is the following function continuous for all ?

• Set and solve for .

Rates of Change

Average Rate of Change

The average rate of change measures how a function changes between two points. In business, this can represent average cost, revenue, or profit per unit.

• Definition: The average rate of change of from to is:

• It is the slope of the line connecting and .

Example:

• (a) from to :

  • Average rate:

• (b) from to :

  • Average rate:

Instantaneous Rate of Change

The instantaneous rate of change describes how a function changes at a specific point. This is the foundation of the derivative, which is central to calculus and business applications.

• Definition: The instantaneous rate of change of at is:

  • Alternatively, using as a small increment:

Example: Find the instantaneous rate of change of at .

• Compute

• Use the definition:

  • Expand and simplify numerator:

  • Sum:

  • Subtract 24:

  • So,

Business Applications: Marginal Cost

In business calculus, the instantaneous rate of change is used to calculate marginal cost, marginal revenue, and marginal profit. Marginal cost is the cost of producing one additional unit.

• Marginal Cost: If is the cost function, marginal cost at is .

Example: The cost function to produce units is , for . Find the marginal cost when 8 units are made.

• Compute derivative:

• Marginal cost at :

Additional info: Marginal cost, revenue, and profit are key concepts in business calculus, allowing companies to optimize production and pricing strategies.