The Big Idea
The first derivative tells you about the instantaneous rate of change of a function. That makes it one of the most useful tools in AP Calculus for understanding the overall behavior of a graph.
The key connection is simple:
So if the derivative is positive, the function is moving upward as you move from left to right. If the derivative is negative, the function is moving downward.
This gives us a systematic way to determine exactly where a function is increasing and decreasing.
What "Increasing" and "Decreasing" Mean
A function is increasing on an interval if its output values get larger as increases. Visually, the graph rises as you move from left to right.
A function is decreasing on an interval if its output values get smaller as increases. Visually, the graph falls as you move from left to right.
The derivative does not need to be large for a function to be increasing. It only needs to be positive. Likewise, a small negative derivative still means the function is decreasing.
The First-Derivative Test for Increasing and Decreasing
The fundamental rule you should remember is:
In practice, this means you can analyze the graph of a function without actually graphing the function itself. You only need to determine where its derivative is positive and where it is negative.
A Step-by-Step Method
- Find the first derivative. Calculate .
- Find the critical numbers. Solve and identify any values where is undefined, provided those values are in the domain of the original function.
- Put the critical numbers on a number line. They divide the domain into intervals.
- Determine the sign of on each interval. Choose one test value from each interval and substitute it into the derivative.
- Translate the signs into behavior. Positive means increasing; negative means decreasing.
- State the intervals carefully. Give your final answer using interval notation.
Why Critical Numbers Matter
A critical number is a value of where the derivative is zero or does not exist, assuming the original function is defined there.
These values are important because the sign of the derivative can change when you pass through them. They therefore divide the number line into regions where the function may have different behavior.
For example, if the only critical number is , then you only need to investigate the intervals and .
You do not need to test every point in an interval. Once you know the derivative is continuous and has no zeros or undefined points inside that interval, its sign cannot change there. One convenient test point is enough.
Worked Examples
Determine the intervals on which is increasing and decreasing.
Determine where is increasing and decreasing.
Suppose the derivative of a function is . Determine where the original function is increasing and decreasing.
Suppose . Determine the intervals on which is increasing and decreasing.
Reading a First-Derivative Sign Chart
A sign chart is often the fastest way to organize this analysis. Consider a derivative with critical numbers at and .
The signs might look like this:
The important part is the sign, not the exact value of the derivative. Once you know the derivative is positive on an interval, you immediately know the function is increasing there.
Increasing and Decreasing vs. Local Extrema
The first derivative also helps identify local maxima and local minima.
If the derivative changes from positive to negative, the function changes from increasing to decreasing. That produces a local maximum.
If the derivative changes from negative to positive, the function changes from decreasing to increasing. That produces a local minimum.
Notice that simply having is not enough to guarantee a local maximum or minimum. You need to examine what happens to the sign of the derivative on either side.
What If the Derivative Is Zero but the Sign Does Not Change?
This is an important AP Calculus distinction. A derivative can equal zero without the function having a local maximum or minimum.
For example, consider .
The derivative is zero at , but it is positive on both sides of zero. The function is therefore increasing on both sides. There is no local maximum or local minimum at .
This is why you should not automatically label every critical number as an extremum. Always check the sign change.
A Fast AP Calculus Strategy
On an AP exam question, you can often solve an increasing/decreasing problem very efficiently using this checklist:
- Find or use the given expression for .
- Find every zero or undefined point of that matters to the domain.
- Put those values in order on a number line.
- Determine the sign of the derivative between consecutive critical numbers.
- Translate into increasing and into decreasing.
- Use interval notation in your final answer.
If a calculator is allowed, a graph of can help you check your sign analysis. But the mathematical reasoning should still come from the derivative.
Common Mistakes
The function being increasing does not mean the function itself is positive. It means its derivative is positive. A function can be negative and still be increasing. For example, if , the function is increasing even though its values are negative.
To determine increasing and decreasing intervals, you are looking for where changes sign. Therefore, the critical numbers come from , not from solving .
Critical numbers can occur where the derivative does not exist. If the original function is defined at that point, it must be considered when constructing the sign chart.
The standard AP Calculus convention is to report intervals of increase and decrease using open intervals. If the critical number is , write intervals such as rather than including as an endpoint of both intervals.
A critical point is a candidate for a local extremum, not a guarantee. Check whether the derivative actually changes sign.
Practice Problems
Determine the intervals on which is increasing and decreasing.
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Determine the intervals on which is increasing and decreasing.
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Suppose . Determine where is increasing and decreasing.
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Determine whether has a local maximum, local minimum, or neither at .
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