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.