Functions and Their Properties

Graphical Conditions and Asymptotes

This section explores the properties of functions, including limits, continuity, differentiability, and asymptotes.

• Function Value: The value of the function at a specific point, e.g., .

• Limit: The behavior of the function as the input approaches a certain value. For example, means as approaches 0, approaches 1.

• Continuity: A function is continuous at a point if its limit at that point equals its value there.

• Differentiability: A function is differentiable at a point if its derivative exists there. If not, the graph may have a sharp corner or cusp.

• Vertical Asymptote: A line is a vertical asymptote if the function grows without bound as approaches .

Example: Sketch a function with the following properties:

• is continuous but not differentiable at

• Vertical asymptote at

Such a function might have a jump or cusp at , and the graph would approach infinity near .

Average Rate of Change

Interpreting the Average Rate of Change

The average rate of change of a function over an interval is given by:

• Application: For on , calculate:

Average rate:

• Interpretation: The average rate of change represents how much decreases per unit increase in over the interval.

Applications of Derivatives

Profit Rate of Change

Given a profit function , the rate at which average profit per machine changes is found by differentiating the average profit function:

• Average profit per machine:

• Find at :

First, Simplify: Then, At :

• Interpretation: When 5 machines have been sold, the average profit per machine is increasing at approximately $19.97 per machine.

Definition of the Derivative

Using the Limit Definition

The derivative of a function at is defined as:

• Application: For , find :

• Interpretation: gives the rate of change of copper consumption at time .

Evaluating at Specific Years

• Find consumption in 2005: (since is 1990)

• Find rate in 2005:

Interpretation: In 2005, copper consumption was 67,200 thousand metric tons, increasing at 8,160 thousand metric tons per year.

Motion Under Gravity

Vertical Motion Equations

The height of a stone thrown vertically is given by .

• Velocity:

• Acceleration: (constant, due to gravity)

Evaluating at Specific Times

• Velocity at : m/s

• Interpretation: Negative velocity means the stone is falling.

• Acceleration: Always m/s

• Velocity at impact: Find when ; solve for , then compute at that .

Example: If is found to be approximately , substitute to find the exact time and velocity.

Summary Table: Key Calculus Concepts