Limits, Continuity, and Derivatives

Section 2.1: Average Velocity and Secant Lines

This section introduces the concept of average velocity and its connection to the slope of secant lines for a function. These ideas lay the foundation for understanding instantaneous rates of change.

• Average Velocity: The average velocity of an object over a time interval is given by: where is the position function.

• Secant Line Slope: The slope of the secant line between points and on the graph of is:

• Application: Secant lines approximate the rate of change over an interval; as the interval shrinks, the secant line approaches the tangent line.

Section 2.2: Numerical and Graphical Limits

Limits describe the behavior of functions as inputs approach specific values. This section covers finding limits using tables and graphs, including one-sided limits and infinite limits.

• Numerical Limits: Estimate by evaluating at values close to and observing the trend.

• One-Sided Limits: (from the left), (from the right).

• Graphical Limits: Analyze the graph near to determine the limit.

• Infinite Limits and Vertical Asymptotes: If increases or decreases without bound as approaches , then . Vertical asymptotes occur at such points.

• Limits That Do Not Exist: A limit may not exist if the left and right limits differ, or if the function oscillates or diverges.

• Example: For , as , ; as , .

Section 2.3: Algebraic Limits and Indeterminate Forms

Algebraic techniques are used to evaluate limits, especially when direct substitution leads to indeterminate forms such as .

• Direct Substitution: If is defined and continuous, .

• Indeterminate Forms: When substitution yields , use algebraic methods:

  • Factoring: Factor numerator and denominator to simplify.

  • Conjugate Method: For expressions with square roots, multiply by the conjugate to simplify.

• One-Sided Limits: Apply the same techniques for left/right limits.

• Relationship Between Limits: exists if and only if both one-sided limits exist and are equal.

• Squeeze Theorem: If near , and , then .

• Example: (using the Squeeze Theorem).

Section 2.5: Continuity and the Intermediate Value Theorem

Continuity at a point ensures that a function behaves predictably without jumps or holes. The Intermediate Value Theorem (IVT) is a fundamental result for continuous functions.

• Definition of Continuity: is continuous at if:

  1. is defined

  2. exists

• Showing Continuity/Discontinuity: Check the above three conditions.

• Intermediate Value Theorem: If is continuous on and is between and , then there exists such that .

• Example: If and , then must take the value $0[1, 3]$.

Section 2.6: Limits at Infinity and Horizontal Asymptotes

Limits at infinity describe the behavior of functions as grows without bound. Horizontal asymptotes indicate the value a function approaches as .

• Limits at Infinity: means approaches as increases.

• Horizontal Asymptotes: The line is a horizontal asymptote if or .

• Algebraic Calculation: For rational functions, divide numerator and denominator by the highest power of .

• Key Property: for .

• Example: .

Section 2.7: Definition and Interpretation of the Derivative at a Point

The derivative at a point quantifies the instantaneous rate of change of a function. It is defined as the limit of the average rate of change as the interval shrinks to zero.

• Definition: The derivative of at is:

• Calculation: Use the definition to compute for specific functions.

• Interpretation:

  • Slope of the Tangent Line: is the slope at .

  • Instantaneous Velocity: If is position, is velocity at (units: position/time).

  • Instantaneous Rate of Change: gives the rate at which changes at (units: output/input).

• Example: For , .

Section 2.8: Definition and Properties of the Derivative Function

The derivative function gives the rate of change of at every point. This section covers its definition, notation, and when a function is not differentiable.

• Definition:

• Calculation: Apply the definition to find for various functions.

• Notations:

  • Prime notation:

  • Leibniz notation:

  • Other:

• Non-Differentiability: is not differentiable at points where it is not continuous, has a sharp corner, cusp, or vertical tangent.

• Relationships on Graphs: The graph of shows the slope of ; shows concavity; shows the rate of change of concavity.

• Example: For , does not exist at .

Additional info:

• Sections referenced correspond to standard calculus topics on limits, continuity, and derivatives.

• Examples and definitions have been expanded for clarity and completeness.