Sketching Graphs of Functions and Their Derivatives - AP Calculus AB Study Guide

The Core Idea: Sketching Graphs of Functions and Their Derivatives

The fundamental concept of this topic is understanding the deep, graphical relationship between a function, $f$, and its derivatives, $f^{\prime }$ and $f^{\prime \prime }$. The graph of the derivative, $f^{\prime }$, is not just an arbitrary new function; it is a visual representation of the rate of change—or slope—of the original function, $f$. Similarly, the graph of the second derivative, $f^{\prime \prime }$, describes the rate of change of $f^{\prime }$, which in turn tells us about the concavity of $f$.

Mastering this topic means you can look at the graph of $f^{\prime }$ and deduce the shape and properties of $f$, such as where it is increasing, decreasing, or has local extrema. Conversely, you can look at the graph of $f$ and sketch a qualitatively accurate graph of its derivative, $f^{\prime }$. This skill is a synthesis of many key calculus concepts, requiring you to translate features like the sign (positive/negative) of one graph into features like the direction (increasing/decreasing) of another.

Key Graphical Relationships

The relationship between the graphs of $f$, $f^{\prime }$, and $f^{\prime \prime }$ is governed by a set of direct correspondences. Understanding these connections is essential for sketching and analysis.

Understanding the "Why": Connecting Slopes and Values

The critical nuance to internalize is that the y-value on the graph of $f^{\prime }(x)$ at a specific point $x=c$ is equal to the slope of the tangent line to the graph of $f(x)$ at $x=c$.

Let's break this down:

• If the point $(2,4)$ is on the graph of $f^{\prime }(x)$, it means $f^{\prime }(2)=4$. This tells us that the slope of the function $f(x)$ at $x=2$ is exactly 4. The graph of $f(x)$ is rising steeply at that point.

• If the point $(5,-1)$ is on the graph of $f^{\prime }(x)$, it means $f^{\prime }(5)=-1$. This tells us that the slope of $f(x)$ at $x=5$ is -1. The graph of $f(x)$ is decreasing at that point.

• If the graph of $f^{\prime }(x)$ has an x-intercept at $x=-3$, it means $f^{\prime }(-3)=0$. This tells us that the slope of $f(x)$ at $x=-3$ is zero, indicating a horizontal tangent line and a potential local maximum or minimum for $f(x)$.

A second important nuance is that the graph of $f^{\prime }(x)$ determines the shape of $f(x)$, but not its vertical position. Because the derivative of a constant is zero, $\frac{d}{dx}[f(x)]=\frac{d}{dx}[f(x)+C]$. This means that if you are asked to sketch $f(x)$ from the graph of $f^{\prime }(x)$, there are infinitely many correct answers, all of which are vertical shifts of one another. Without an initial condition (e.g., a known point like $f(0)=5$), you cannot determine the specific vertical placement of the graph of $f$.

Core Concepts & Rules

• Direction from Sign: The sign of the derivative determines the direction of the function.

  • If $f^{\prime }(x)>0$ on an interval, then $f(x)$ is increasing on that interval.

  • If $f^{\prime }(x)<0$ on an interval, then $f(x)$ is decreasing on that interval.

• Concavity from Slope: The slope of the first derivative determines the concavity of the function.

  • If $f^{\prime }(x)$ is increasing on an interval, then $f^{\prime \prime }(x)>0$, and $f(x)$ is concave up on that interval.

  • If $f^{\prime }(x)$ is decreasing on an interval, then $f^{\prime \prime }(x)<0$, and $f(x)$ is concave down on that interval.

• Local Extrema from Roots: Zeros of the first derivative that correspond to a sign change indicate local extrema for the function.

  • A local maximum for $f(x)$ occurs when $f^{\prime }(x)$ changes from positive to negative.

  • A local minimum for $f(x)$ occurs when $f^{\prime }(x)$ changes from negative to positive.

• Inflection Points from Extrema: Local extrema (peaks and valleys) on the graph of the first derivative indicate points of inflection for the function.

  • A point of inflection for $f(x)$ occurs when $f^{\prime }(x)$ changes from increasing to decreasing or from decreasing to increasing. This is because $f^{\prime \prime }(x)$ changes sign at these points.

Step-by-Step Example 1: Sketching $f$ from the Graph of $f^{\prime }$

Problem: The graph of the derivative of a function $f$, denoted $f^{\prime }$, is shown below. Sketch a possible graph of $f$.