Connecting a Function, Its First Derivative, and Its Second Derivative

The Big Picture

In AP Calculus, you should not think of a function and its derivatives as three unrelated graphs. They are three different ways of describing the same behavior.

The original function tells you where the graph is. The first derivative tells you how the graph is changing. The second derivative tells you how that change itself is changing.

Once you understand this connection, you can move back and forth between a graph of , a graph of , and a graph of without necessarily having an equation for any of them.

What the First Derivative Tells You

The first derivative measures the slope of the original function. That means the sign of tells you whether is increasing or decreasing.

This gives you the fundamental connection between the function and its first derivative:

• If , the graph of is increasing.
• If , the graph of is decreasing.
• If , the graph has a horizontal tangent line at that point.

Critical Points and Local Extrema

A critical point occurs where the derivative is zero or does not exist, provided the original function is defined there. Critical points are important because they are the only possible locations for local maxima and local minima.

But a critical point is not automatically a maximum or minimum. You need to examine what happens to the sign of the first derivative on either side of the critical point.

What the Second Derivative Tells You

The second derivative is the derivative of the first derivative:

Because describes how the slope is changing, its sign tells you about the concavity of the original function.

Think of concavity as the direction in which the slopes are moving:

• Concave up: the slopes are increasing as you move from left to right.
• Concave down: the slopes are decreasing as you move from left to right.

Why Concavity Depends on the First Derivative

The second derivative can feel abstract until you connect it directly to the first derivative. Remember that is the slope of .

If those slopes are becoming larger, then is increasing. Therefore is positive and the original function is concave up.

If those slopes are becoming smaller, then is decreasing. Therefore is negative and the original function is concave down.

Points of Inflection

A point of inflection is a point where the concavity of the original function changes.

Because concavity is controlled by the sign of the second derivative, you can often locate possible inflection points by solving:

However, an important AP Calculus detail is that does not automatically mean there is an inflection point. The concavity must actually change from concave up to concave down, or from concave down to concave up.

The Most Important Three-Way Connection

The easiest way to organize all of this information is to remember what each derivative is measuring.

This leads to a very useful chain of reasoning:

1. Look at the sign of the first derivative to determine whether the function is increasing or decreasing.
2. Look at the sign of the second derivative to determine whether the function is concave up or concave down.
3. Use changes in the first derivative to classify critical points as local maxima or local minima.
4. Use changes in the second derivative to identify points of inflection.

Reading the Three Graphs Together

AP Calculus questions frequently give you a graph of one derivative and ask you to make conclusions about the original function. The key is to translate between the graphs systematically.

From to

Suppose you are given a graph of the first derivative. You can use it to determine the behavior of the original function:

• Where is above the horizontal axis, is increasing.
• Where is below the horizontal axis, is decreasing.
• Where changes from positive to negative, has a local maximum.
• Where changes from negative to positive, has a local minimum.

From to

You can also use the graph of the first derivative to determine the concavity of the original function. Remember that is the slope of .

• If is increasing, then is positive and is concave up.
• If is decreasing, then is negative and is concave down.

A Complete Example

The Second Derivative Test for Local Extrema

The second derivative can also help classify a critical point without constructing a complete first-derivative sign chart.

In other words:

• If and , there is a local maximum.
• If and , there is a local minimum.
• If , the second derivative test is inconclusive.

The first derivative test is more general, so it is important to understand both methods. The second derivative test is simply a faster method when its conditions apply.

Common Mistakes

Practice Problems

AP Calculus Takeaway

The most important idea is that the first derivative and second derivative describe different layers of behavior.

If you can move fluently between these three ideas, many AP Calculus graph-analysis questions become much easier. A graph of can tell you about ; a graph of can tell you about ; and the sign changes of the derivatives tell you exactly where important features of the original function occur.