5.8 sketching graphs of derivatives

3 min read 01-02-2025

Sketching the graph of a derivative might seem daunting, but with a structured approach and understanding of fundamental calculus concepts, it becomes a manageable and insightful exercise. This guide provides a comprehensive walkthrough, equipping you with the skills to accurately sketch derivative graphs from the original function's graph.

Understanding the Relationship Between a Function and its Derivative

Before diving into sketching techniques, let's solidify the core relationship between a function, f(x), and its derivative, f'(x). The derivative at any point represents the instantaneous rate of change of the function at that point. Geometrically, it's the slope of the tangent line to the curve of f(x) at that point.

This fundamental connection forms the basis of our sketching approach. By analyzing the slopes of the tangent lines across the function's graph, we can infer the behavior of its derivative.

Key Features to Analyze in the Original Function's Graph

To accurately sketch the derivative, meticulously examine the original function's graph for these key features:

1. Increasing and Decreasing Intervals:

Increasing: Where the function's graph rises from left to right, the derivative is positive (above the x-axis).
Decreasing: Where the function's graph falls from left to right, the derivative is negative (below the x-axis).

2. Local Maxima and Minima:

At a local maximum, the function transitions from increasing to decreasing. The derivative will be zero (crossing the x-axis) at this point.
At a local minimum, the function transitions from decreasing to increasing. The derivative will also be zero at this point.

3. Concavity and Inflection Points:

Concave up: The function curves upwards like a "U." The derivative is increasing (rising from left to right).
Concave down: The function curves downwards like an inverted "U." The derivative is decreasing (falling from left to right).
Inflection points: These are points where the concavity changes. The derivative will have a local maximum or minimum at these points.

4. Horizontal Tangents:

Points where the tangent line to the original function is horizontal indicate that the derivative is zero at those points. These are often critical points (maxima, minima, or saddle points).

Step-by-Step Guide to Sketching the Derivative Graph

Identify Increasing/Decreasing Intervals: Mark intervals on the original function's graph where the function is increasing (positive derivative) and decreasing (negative derivative).
Locate Maxima and Minima: Note the x-coordinates of local maxima and minima. These correspond to x-intercepts (where the derivative equals zero) on the derivative's graph.
Determine Concavity: Analyze the concavity of the original function. Concave up indicates an increasing derivative, while concave down indicates a decreasing derivative.
Approximate Slopes: Visually estimate the slope of the tangent line at several points across the original function. Steeper slopes mean larger (positive or negative) values for the derivative. Shallower slopes mean values closer to zero.
Sketch the Derivative: Using the information gathered above, sketch the derivative graph. Ensure its positive/negative values align with the original function's increasing/decreasing intervals, and its increasing/decreasing behavior reflects the original function's concavity.

Example:

Imagine a simple parabola opening upwards. It's decreasing to the left of its vertex and increasing to its right. The vertex is a minimum. The derivative will be a straight line, negative to the left of the vertex (below the x-axis), crossing the x-axis at the vertex (where the derivative is zero), and positive to the right of the vertex (above the x-axis).

Advanced Considerations:

Asymptotes: If the original function has asymptotes, consider how the slopes behave as the function approaches these asymptotes. This might influence the derivative's behavior near the asymptotes.
Sharp Corners: The derivative is undefined at points with sharp corners or cusps in the original function. This will be represented as a discontinuity or break in the derivative graph.
Practice: The best way to master sketching derivatives is through consistent practice. Work through various examples, focusing on interpreting the relationships between the function and its derivative.

By diligently following these steps and understanding the underlying principles, you'll significantly enhance your ability to sketch derivative graphs accurately and efficiently. Remember, it's a skill honed through practice and a keen eye for detail.