Lecture Notes: Graphing Derivatives of Functions

Key Concepts

• Derivative: Represents the slope of a function at any given point.
• Parent Function: The original function from which derivatives are calculated.
• Horizontal Tangent Line: Occurs where the slope is zero.
• Inflection Points: Points where the concavity changes.

Graphing First Derivative (f')

• Given f(x): If the graph is similar to a known function (e.g., x^2), find the derivative (e.g., 2x) to sketch f' which is a linear function.
• Analyzing Slopes:
  • Decreasing function = negative slope
  • Increasing function = positive slope
  • At x=0, slope = 0
• Graph Characteristics:
  • Left side: downward slope, f' below x-axis
  • Right side: upward slope, f' above x-axis

Graphing Second Derivative (f'')

• Given f'(x): If f'(x) is 2x, then f'' is a constant 2 (horizontal line).
• Analyzing Concavity:
  • Constant positive slope = positive y-value for f''

Graphing Examples

1. For f(x) = x^2:
   • f'(x) = 2x: linear graph
   • f''(x) = 2: constant line
2. Parabola (downward):
   • f(x) = -x^2
   • f'(x) = -2x
   • f''(x) = -2

Analyzing Specific Graphs

1. Inflection Points & Concavity

   • Identify where the slope is zero (critical points)
   • Inflection points occur where concavity changes.

2. Sine Wave Example

   • f(x) resembles sine function
   • f'(x) resembles cosine function
   • Identify zero slopes and changes in concavity

3. Absolute Value Function

   • f(x) = |x|
   • f'(x) has a jump discontinuity at x = 0

Graphing F' and F'' from F' Graph

• Intervals of Increase/Decrease:
  • F(x) increases where F'(x) > 0
  • F(x) decreases where F'(x) < 0
• Concavity:
  • Concave up: f'(x) is increasing
  • Concave down: f'(x) is decreasing

Analyzing Critical Points and Extrema from F' Graph

• Critical Points: Where f'(x) = 0
• Relative Extrema:
  • Maxima: f'(x) changes from positive to negative
  • Minima: f'(x) changes from negative to positive

Practical Application

• Determine intervals of increase/decrease, maxima/minima, concavity, and inflection points by analyzing the derivative graphs.

Conclusion

• Understanding the relationship between a function and its derivatives is crucial for graph analysis.
• Always identify critical points, analyze slopes, and understand changes in concavity to infer the behavior of original functions.

Summary: This lecture covered how to graph derivatives from a given function, analyze slopes and concavity, and determine critical points and inflection points for better understanding of function behavior.