To determine which graph represents the original function, the first derivative, or the second derivative, let’s analyze the graphs based on their behaviors:

1. General Observations:
   • Original Function: The graph of the original function will typically have turning points (local maxima/minima) where the slope of the tangent (derivative) is zero.
   • First Derivative: The first derivative represents the slope of the original function. Therefore, it crosses the x-axis where the original function has local maxima or minima.
   • Second Derivative: The second derivative represents the slope of the first derivative (or the curvature of the original function). It crosses the x-axis where the first derivative has turning points.

Analyzing Each Graph:

Top Left Graph:

• The curve has turning points (local maxima and minima) at several locations.
• This suggests it could be the original function.

Top Right Graph:

• The curve crosses the x-axis where the top-left graph has local maxima and minima.
• This indicates it represents the first derivative of the top-left graph.

Bottom Graph:

• The curve crosses the x-axis where the top-right graph (first derivative) has local maxima or minima.
• This indicates it represents the second derivative.

Final Identification:

• Top Left Graph: Original Function
• Top Right Graph: First Derivative
• Bottom Graph: Second Derivative

Explanation Summary:

1. The original function (top left) has turning points.
2. The first derivative (top right) crosses the x-axis where the original function has turning points.
3. The second derivative (bottom) crosses the x-axis where the first derivative has turning points.

Let me know if you'd like a more detailed step-by-step explanation or clarification!

Related Questions:

1. What is the relationship between critical points and the first derivative?
2. How can you determine concavity using the second derivative?
3. What does it mean when the second derivative is zero?
4. How do you sketch the first and second derivatives given an original function?
5. What are the practical applications of derivatives in physics?

Tip:

To identify derivatives from graphs, look for x-axis crossings and turning points—these are key indicators of relationships between the function, its first derivative, and its second derivative.