Rates of Change as Limits

Average Rate of Change

The average rate of change of a function f on the interval [a, b] is the slope of the secant line that joins the two points (a, f(a)) and (b, f(b)) on the graph of f. It is given by:

• Formula:

• Interpretation: Measures how much the function changes per unit interval between a and b.

• Application: Used to estimate the overall change in a function over a specified interval.

Instantaneous Rate of Change (Slope of Tangent Line)

The instantaneous rate of change at a point a is the slope of the tangent line to the graph at (a, f(a)). It is defined as the limit of the average rate of change as b approaches a:

• Formula:

• Alternate Form:

• Application: Used to find the exact slope at a single point, which is fundamental in calculus.

Example: Finding the Slope of the Tangent Line

Let f(x) = x^2. To find the slope of the tangent line at x = a:

• Apply the limit definition:

• Simplify:

• The slope at x = a is 2a.

Equation of the Tangent Line

Once the slope is found, the equation of the tangent line at x = a is:

• Point-slope form: , where m is the slope at x = a.

• For f(x) = x^2 at x = 1: Slope is 2, so .

The Derivative Function

Limit Definition of the Derivative

The derivative of a function f at a point a is the limit of the difference quotient as h approaches 0. This measures the instantaneous rate of change of f at a:

• Definition:

• Interpretation: The derivative function f' gives the slope of the tangent line at any point x.

Derivative Notation

Several notations are used for the derivative:

• Leibniz notation: or

• Lagrange notation:

• Newton notation: (rarely used in calculus courses)

• Application: Notation is chosen based on context, such as physics or mathematics.

Examples: Using the Limit Definition

Find the derivative of f(x) = 3x^2 - 2x + 2 using the limit definition:

• Apply:

• Expand and simplify:

• Result:

The Derivative at a Point

Evaluating the Derivative at a Specific Point

The derivative at a point a is found by evaluating the limit definition at x = a:

• Formula:

• Application: Used to find the slope of the tangent line at a specific point.

• Example: For f(x) = 3x^2 - 2x at x = 1:

• Result: Slope at x = 1 is 1.

Equation of Tangent Line at a Point

Once the derivative at a point is found, the tangent line equation is:

• Formula:

• Example: For f(x) = 3x^2 - 2x at x = 1:

Summary Table: Derivative Concepts

Additional info: The notes provide foundational concepts for Chapter 3 (Derivatives) and Chapter 4 (Applications of the Derivative) in a Calculus course, including definitions, formulas, and worked examples for finding derivatives and tangent lines using limits.