AP Calculus AB 5.9 A Positive or Negative Function, First Derivative, & Second Derivative on a Graph

In AP Calculus AB Topic 5.9, analyzing whether a function, its first derivative \( f'(x) \), and its second derivative \( f''(x) \) are positive or negative at different points on a graph provides essential insights into the behavior of \( f(x) \).

Here's a guide to understanding each part on the graph:

### 1. The Function \( f(x) \):
- **Positive or Negative**: If \( f(x) \) is above the x-axis, the function is positive. If it is below the x-axis, the function is negative.
- **Graph Shape**: When \( f(x) \) is positive, the curve is above the x-axis; when negative, it is below.

### 2. First Derivative \( f'(x) \):
- **Positive or Negative**:
- When \( f'(x) \) is positive at a point, \( f(x) \) is increasing there.
- When \( f'(x) \) is negative, \( f(x) \) is decreasing.
- **Interpretation on the Graph of \( f(x) \)**:
- Positive \( f'(x) \): \( f(x) \) is moving upward.
- Negative \( f'(x) \): \( f(x) \) is moving downward.

### 3. Second Derivative \( f''(x) \):
- **Positive or Negative**:
- When \( f''(x) \) is positive at a point, \( f(x) \) is concave up (curving upwards).
- When \( f''(x) \) is negative, \( f(x) \) is concave down (curving downwards).
- **Inflection Points**: If \( f''(x) \) changes sign, this indicates an inflection point in \( f(x) \), where the concavity changes.

### Summary of Positive/Negative Values on the Graph
- By noting where each derivative is positive or negative, students can interpret whether \( f(x) \) is increasing, decreasing, concave up, or concave down at any given point, helping them understand the overall behavior of the function.

This approach equips students with a straightforward method to connect signs on a graph with the behavior of \( f(x) \), ensuring clarity on how positive or negative derivatives directly influence the function's shape.

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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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