Derivative and Anti-derivative Graphs

Deducing the Graph of the Derivative Function

The derivative of a function, \(f'(x)\), represents the instantaneous rate of change of the function \(f(x)\). Graphically, \(f'(x)\) represents the gradient of the tangent to the curve of \(f(x)\) at any point.

Key Principles

• Stationary Points: Where \(f(x)\) has a stationary point (local maximum, local minimum, or stationary inflection point), \(f'(x) = 0\). These points correspond to the x-intercepts of the derivative graph.
• Increasing/Decreasing Intervals:
  • If \(f(x)\) is increasing, \(f'(x) > 0\) (derivative is positive, graph is above the x-axis).
  • If \(f(x)\) is decreasing, \(f'(x) < 0\) (derivative is negative, graph is below the x-axis).
• Concavity:
  • If \(f(x)\) is concave up, \(f'(x)\) is increasing.
  • If \(f(x)\) is concave down, \(f'(x)\) is decreasing.
• Points of Inflection: At points of inflection on \(f(x)\), the derivative \(f'(x)\) has a local maximum or minimum (stationary point).
• Linear Functions: If \(f(x)\) is a linear function, \(f'(x)\) is a constant function (horizontal line).

Steps to Sketch the Derivative Graph

1. Identify Stationary Points: Locate all points where the gradient of \(f(x)\) is zero. These points become the x-intercepts of \(f'(x)\).
2. Determine Intervals of Increase and Decrease: Identify intervals where \(f(x)\) is increasing (positive gradient) and decreasing (negative gradient). This determines where \(f'(x)\) is positive or negative.
3. Analyze Concavity: Determine intervals where \(f(x)\) is concave up or concave down. This indicates whether \(f'(x)\) is increasing or decreasing.
4. Identify Points of Inflection: Points of inflection on \(f(x)\) correspond to local maxima or minima on \(f'(x)\).
5. Sketch the Graph: Use the information gathered to sketch the graph of \(f'(x)\).

Example

Consider a cubic function with a local maximum at \(x = a\) and a local minimum at \(x = b\). The derivative will be a quadratic function with x-intercepts at \(x = a\) and \(x = b\).

Deducing the Graph of an Anti-derivative Function

The anti-derivative of a function, \(F(x)\), is a function whose derivative is \(f(x)\), i.e., \(F'(x) = f(x)\). Unlike differentiation, anti-differentiation results in a family of functions, differing by a constant term (the constant of integration, \(C\)). Graphically, \(F(x)\) represents a function whose gradient function is \(f(x)\).

Key Principles

• X-intercepts of \(f(x)\)**: These correspond to stationary points (local max, min, or stationary points of inflection) on \(F(x)\).
• Positive Intervals of \(f(x)\)**: Where \(f(x) > 0\), \(F(x)\) is increasing.
• Negative Intervals of \(f(x)\)**: Where \(f(x) < 0\), \(F(x)\) is decreasing.
• Area Under the Curve: The area under the curve of \(f(x)\) between two points relates to the change in \(F(x)\) between those points. However, remember this is only a relative anti-derivative graph without knowing initial conditions to determine the constant of integration.
• The Constant of Integration: The graph of the anti-derivative is not unique, as adding a constant to the anti-derivative shifts the graph vertically but does not change its shape.

Steps to Sketch the Anti-derivative Graph

1. Identify X-intercepts of \(f(x)\)**: These indicate where \(F(x)\) will have stationary points.
2. Determine Intervals Where \(f(x)\) is Positive/Negative: This determines where \(F(x)\) is increasing or decreasing.
3. Consider the Area Under the Curve: Estimate the area under the curve of \(f(x)\) to understand the relative change in \(F(x)\). Larger areas correspond to steeper increases or decreases in \(F(x)\).
4. Account for the Constant of Integration: Remember that there are infinitely many possible anti-derivative graphs, differing only by a vertical shift. Unless given an initial condition (e.g., \(F(0) = 2\)), you can only sketch a general shape.
5. Sketch the Graph: Combine all the information to sketch a possible graph of \(F(x)\).

Table Summarizing Relationships

Example

If \(f(x)\) is a linear function with a positive slope and a y-intercept of 2, then \(F(x)\) will be a quadratic function opening upwards. The minimum point of \(F(x)\) will occur at the x-intercept of \(f(x)\), which is where \(f(x) = 0\).

Important Note: When sketching anti-derivative graphs, remember that you are sketching one possible anti-derivative. There are infinitely many others, each shifted vertically by a constant.