Derivatives and Their Uses

Average Rate of Change and Secant Lines

The average rate of change of a function f(x) from x = a to x = a + h measures how much the function changes over an interval. This is also the slope of the secant line connecting the points (a, f(a)) and (a + h, f(a + h)) on the graph of y = f(x).

• Formula for Average Rate of Change:

• Secant Line: A line passing through two points on the graph of a function.

• Interpretation: The average rate of change is the slope of the secant line between x = a and x = a + h.

• Example: For f(x) = x^2, the average rate of change from x = 1 to x = 1 + h is .

Instantaneous Rate of Change and Tangent Lines

As h approaches zero, the average rate of change approaches the instantaneous rate of change at x = a. This is the slope of the tangent line to the graph at that point.

• Limit Definition of the Derivative:

• Tangent Line: A line that touches the graph at a single point and matches the graph's slope at that point.

• Interpretation: The derivative at x = a gives the instantaneous rate of change of f(x) at that point.

The Derivative Function

The derivative function f'(x) gives the instantaneous rate of change of f(x) at any point x in its domain (where the limit exists).

• Definition:

• Notation: f'(x) is read as "f prime of x".

• Application: The derivative allows for short-term predictions, such as marginal cost, revenue, or profit in business contexts.

Examples of Finding Derivatives Using the Limit Definition

Example 1: Derivative of a Quadratic Function

Find the slope of the tangent line to f(x) = x^2 - 6x at x = 2 using the limit definition. Then, find the equation of the tangent line.

• Step 1: Compute

• Step 2: Compute

• Step 3: Compute

• Step 4: Divide by :

• Step 5: Take the limit as :

• Conclusion: The slope of the tangent line at is .

• Tangent Line Equation: Using point-slope form with point :

Example 2: Derivative of a Rational Function

Find the derivative of using the limit definition.

• Step 1:

• Step 2:

• Step 3: Combine fractions:

• Step 4: Divide by :

• Step 5: Take the limit as :

Example 3: Derivative of a Square Root Function

Find the derivative of using the limit definition.

• Step 1:

• Step 2:

• Step 3: Multiply numerator and denominator by to rationalize:

• Step 4: Divide by :

• Step 5: Take the limit as :

Applications of Derivatives in Business

Cost Function Interpretation

• Total Cost: means the total cost to produce 100 units is $80,000.

• Marginal Cost: means the cost to produce one more unit after 100 units (the 101st unit) is approximately $300$.

Revenue Function and Marginal Revenue

Given , find the marginal revenue function using the limit definition.

• Step 1:

• Step 2: Expand and simplify:

• Step 3:

• Step 4: Divide by :

• Step 5: Take the limit as :

Interpretation of Marginal Revenue

• At :

• Meaning: When 2000 containers are sold, the revenue from selling one more container (the 2001st) is approximately $12.

• Actual Change: dollars, confirming the marginal revenue approximation.

Summary Table: Limit Definition Derivatives

Key Takeaways

• The derivative measures the instantaneous rate of change of a function and is foundational for business applications such as marginal cost and marginal revenue.

• The limit definition of the derivative provides a systematic way to compute derivatives from first principles.

• Understanding how to interpret derivatives in context is essential for making short-term business predictions.