What Derivative Information Tells You About a Graph
One of the most important ideas in AP Calculus is that you can learn a remarkable amount about the graph of a function without knowing an explicit formula for the function itself.
The first derivative tells you how the function is moving: whether the graph is increasing or decreasing and where it may have local maxima or minima. The second derivative tells you how that motion is changing: whether the graph is concave up or concave down and where the concavity may change.
When you put this information together, you can sketch the overall shape of from a derivative sign chart or from information about and .
Think of as telling you the direction of the graph and as telling you the curvature. You need both to make a good sketch.
The First Derivative: Increasing and Decreasing
The sign of the first derivative tells you whether the function is increasing or decreasing.
If is positive on an interval, the graph rises as you move from left to right. If is negative, the graph falls.
This gives you the first major piece of your sketch: draw the graph moving upward where the derivative is positive and downward where the derivative is negative.
Critical Points
A critical point occurs where the derivative is zero or where the derivative does not exist, provided the function itself is defined there.
Critical points are important because local maxima and minima can occur there. But a critical point is only a candidate for a local extremum. You must look at the behavior of the derivative on either side.
Using the First Derivative Test
The sign change of tells you what happens at a critical point.
The Second Derivative: Concavity
The second derivative gives information about the way the graph bends.
When is positive, the graph is concave up. When is negative, the graph is concave down.
A useful way to visualize this is to think about the slopes. On a concave-up graph, the slopes are becoming more positive as you move from left to right. On a concave-down graph, the slopes are becoming more negative.
Inflection Points
An inflection point is a point where the concavity actually changes. It is not enough for the second derivative simply to equal zero.
Therefore, when sketching a graph, look for locations where the sign of changes from positive to negative or from negative to positive.
A point where is a candidate for an inflection point, but you must verify that concavity actually changes. For example, has at , but the graph is concave up on both sides, so there is no inflection point there.
A Reliable Sketching Procedure
When an AP Calculus problem gives you derivative information and asks you to sketch or describe the graph of , use the information in a consistent order.
- Mark important x-values. Start with endpoints, critical points, zeros of the derivative, discontinuities, and any points where the derivative changes behavior.
- Determine increasing and decreasing intervals. Use the sign of .
- Identify local extrema. Look for sign changes in .
- Determine concavity. Use the sign of .
- Identify inflection points. Look for actual changes in concavity.
- Use known function values. If you are given points such as , plot them before connecting the graph.
- Connect everything smoothly. The final curve should respect every piece of derivative information simultaneously.
Worked Example: Reconstructing a Graph
Suppose a function satisfies the following:
- for
- for
- for
So even without knowing the formula for , you know that the graph must rise, reach a local maximum, fall, reach a local minimum, and then rise again.
Worked Example: Adding Concavity
Suppose has the following behavior:
- on
- on
- on
- on
How the Pieces Fit Together
There are four especially useful combinations to recognize.
Reading a Derivative Sign Chart
AP Calculus problems often give derivative information in a sign chart rather than as a formula. The key is to translate every sign into a visual instruction.
For example, imagine the following pattern:
Using Function Values to Place the Graph
Derivative information tells you the shape of the graph, but it does not necessarily tell you exactly where the graph sits vertically.
Suppose you know that and that is positive for all relevant values of . You know the function is increasing, but you still need the point to anchor the graph at a specific location.
This distinction is important:
- Derivative information tells you the graph's behavior and shape.
- Function values tell you where the graph is located.
- Both together allow you to make a much more precise sketch.
Common Mistakes
tells you that the function is increasing. It does not mean that itself is above the x-axis. A function can be increasing while remaining entirely below the x-axis.
A critical point only identifies a location where a local maximum or minimum might occur. Check whether changes sign.
You need a change in concavity. If stays positive on both sides, the graph remains concave up and there is no inflection point.
A sketch is not correct merely because it "looks reasonable." Every section must agree with the stated increasing/decreasing and concavity information. If is positive, your graph cannot be decreasing on that interval.
Practice Problems
Suppose is positive on and negative on . What happens at ?
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Suppose is negative for and positive for . What type of critical point occurs at ?
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A function satisfies on an interval. Is the graph concave up or concave down?
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A function has at , but is positive on both sides of that point. Is there an inflection point?
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Suppose is increasing on , decreasing on , and increasing again on . Describe the locations of the local extrema.
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