Chapter 2: Prelude to Calculus

Section 2.1: Tangent Lines and Slope Predictors

This section introduces the concept of tangent lines to curves and how to determine their slopes using algebraic methods. Tangent lines are fundamental in calculus as they represent the instantaneous rate of change of a function at a point.

• Definition: Tangent Line: A tangent line to the curve y = f(x) at the point x = a is a straight line that touches the curve at that point and has the same slope as the curve at that point.

• Finding the Slope at a Point: The slope of the tangent line at x = a is given by the derivative of f(x) at x = a, denoted f'(a).

• Equation of the Tangent Line: The equation of the tangent line at x = a is:

• Example 1: For f(x) = 4x - 5 at x = 2:

  • Derivative:

  • Slope at x = 2:

  • Point:

  • Tangent line:

• Example 2: For f(x) = 2x^2 - 3x + 4 at x = 2:

  • Derivative:

  • Slope at x = 2:

  • Point:

  • Tangent line:

• Horizontal Tangent Lines: A tangent line is horizontal where the derivative is zero.

  • For y = 10 - x^2, set

  • At , the tangent is horizontal.

• Special Problem: Finding lines through a point on a parabola:

  • Given y = x^2 and point (3, 0), one line is the x-axis. The other can be found by solving for the line through (3, 0) and another point on the parabola.

Section 2.2: The Limit Concept

This section introduces the foundational concept of limits, which is essential for defining derivatives and understanding continuity in calculus.

• Definition: Limit: The limit of a function f(x) as x approaches a is the value that f(x) gets closer to as x gets closer to a. It is denoted as .

• Example 1:

  • Direct substitution:

• Example 2:

  • Evaluate each factor at and multiply.

• Example 3:

  • Substitute :

• Example 4:

  • At , denominator is ; numerator is .

  • Limit is .

• Slope-Predictor Function: For , the slope-predictor (difference quotient) is: For : Expand and simplify to find the derivative.

• Suggested Problems:

  • : Use factoring to simplify and evaluate.

  • For , find the slope-predictor and tangent at .

Section 2.3: More About Limits

This section explores more advanced limit problems, including those involving trigonometric functions and indeterminate forms.

• Trigonometric Limits: Limits involving sine and cosine often require special techniques or known limits.

  • : Use the small angle approximation for small .

  • : Known limit, equals .

  • : Use the substitution ; limit is .

• Key Limit Laws:

  • For

Summary Table: Key Limit Forms

Key Point: Mastery of limits and tangent lines is essential for understanding derivatives and the foundational ideas of calculus.