Analyzing Graphs


Each of the figures in Exercises 91 and 92 shows two graphs, the graph of a function 𝔂 = ƒ(x) together with the graph of its derivative ƒ'(x). Which graph is which? How do you know?


<IMAGE>

Verified step by step guidance

1. Examine the behavior of each graph: Identify key features such as increasing or decreasing intervals, local maxima and minima, and points of inflection.
2. Understand the relationship between a function and its derivative: The derivative ƒ'(x) represents the slope of the tangent line to the graph of ƒ(x). Where ƒ(x) is increasing, ƒ'(x) is positive; where ƒ(x) is decreasing, ƒ'(x) is negative.
3. Identify critical points: Look for points where the derivative graph crosses the x-axis (ƒ'(x) = 0). These correspond to potential local maxima or minima on the graph of ƒ(x).
4. Analyze concavity: Determine where the graph of ƒ(x) is concave up or concave down. This can be inferred from the sign of the second derivative, but since we only have ƒ(x) and ƒ'(x), look for changes in the slope of ƒ'(x).
5. Match the graphs: Based on the analysis, match the graph with the characteristics of ƒ(x) and ƒ'(x). The graph with the zero crossings and sign changes corresponding to the behavior of the function is the derivative graph.

Key Concepts

Function and Derivative Relationship

The relationship between a function and its derivative is fundamental in calculus. The derivative of a function, denoted as ƒ'(x), represents the rate of change of the function ƒ(x) at any given point. When analyzing graphs, the behavior of the derivative graph can indicate where the original function is increasing or decreasing, as well as where it has local maxima or minima.

Critical Points and Inflection Points

Critical points occur where the derivative ƒ'(x) is zero or undefined, indicating potential local maxima, minima, or points of inflection in the function ƒ(x). Inflection points are where the concavity of the function changes, which can be identified by changes in the sign of the derivative. Understanding these points helps in distinguishing between the graphs of a function and its derivative.

Graphical Interpretation of Derivatives

Graphically, the derivative of a function can be interpreted as the slope of the tangent line to the function's graph at any point. If the graph of ƒ(x) is increasing, ƒ'(x) will be positive, and if it is decreasing, ƒ'(x) will be negative. Additionally, where the derivative graph crosses the x-axis indicates where the original function has horizontal tangents, aiding in identifying the correct graphs.