Lecture Notes on Graphing Derivatives

Key Concepts

• Parent Function: The original function before taking derivatives.
• Derivative (f'): Represents the slope of the parent function at any point.
• Second Derivative (f''): Provides information about the concavity of the original function.

Analyzing Graphs

Graphing the First Derivative (f')

• Determine the Parent Function:
  • Example: If f(x) = x^2, then f'(x) = 2x (a linear function with slope 2).
• Slope Analysis: (f')
  • On the left of the origin, if the function is decreasing, f' is negative.
  • At x = 0 (origin), the slope is zero (horizontal tangent).
  • On the right, if the function is increasing, f' is positive.
  • Where f' crosses x-axis, the slope of f is zero.

Graphing the Second Derivative (f'')

• Example Analysis:
  • If f'(x) = 2x, then f''(x) = 2 (a horizontal line at y=2).
  • The constant slope of f' indicates the y-value of f'' is constant.

Analyzing Specific Examples

• Example 1: f(x) = -x^2
  • f'(x) = -2x (linear, slope -2).
  • f''(x) = -2 (horizontal line at y=-2).
• Example 2: f(x) = x^3
  • f'(x) = 3x^2 (upward parabola).
  • f''(x) = 6x (linear function).

Identifying Key Features

Critical Points

• Zero Slope: Identify where the derivative (f') is zero, which are critical points.
  • Signs of f' change indicate relative maxima/minima.

Concavity and Inflection Points

• Concave Up: When f'' > 0 or f' is increasing.
• Concave Down: When f'' < 0 or f' is decreasing.
• Inflection Points: Where concavity changes; identified by changes in the sign of f'' or slope changes in f'.

Function Behavior

• Increasing/Decreasing Intervals:
  • Increasing: f' > 0.
  • Decreasing: f' < 0.
• Horizontal Tangents: Occur where f' = 0 (critical points).

Example Problem Analysis

• Absolute Value Function (f(x) = |x|)

  • f'(x) is discontinuous at x=0 due to an instantaneous slope change.
  • Graph shows a jump discontinuity.

• Sine Wave Example

  • f'(x) is the cosine function.
  • Identify zero slope points and behavior between them.

Graph Interpretation

• Given f'(x): Determine intervals of function behavior (increasing, decreasing) based on f' above/below x-axis.
  • Critical Points: Where f' = 0.
  • Maxima/Minima: Changes in sign of f'.
  • Concavity and Inflection Points: Analyzed using f'' or the slope of f'.

Practical Applications

• Answering Graph-Based Questions: (f(x) increasing, decreasing, critical points, concavity, inflection points).
• Recognizing Concavity and Discontinuity: Identify intervals of concavity and discontinuous points in derivative graphs.

Conclusion

• Graphing derivatives involves analyzing slope (f'), concavity (f''), critical points, and function behavior.
• Practically, identifying the slope and inflection points aids in rough sketching of derivative graphs.