Interpreting Derivatives from a Velocity Graph

The Big Idea: Velocity Is Already a Derivative

Many AP Calculus questions give you a graph of an object's velocity rather than its position. That may seem like you're missing information, but a velocity graph already tells you a great deal about the object's motion.

If represents the object's position, then velocity is its derivative:

And acceleration is the derivative of velocity:

So when you are given a velocity graph, the slope of that graph is telling you about acceleration. The height of the graph is telling you about velocity.

This distinction is one of the most important things to recognize on the AP exam: height tells you velocity; slope tells you acceleration.

What the Height of the Velocity Graph Tells You

At any particular time, the vertical coordinate of the velocity graph gives you the object's instantaneous velocity.

The sign of the velocity tells you the object's direction of motion.

• If velocity is positive, the object is moving in the positive direction.
• If velocity is negative, the object is moving in the negative direction.
• If velocity is zero, the object is momentarily at rest.

For example, if a velocity graph has a height of at , then:

The object is moving in the positive direction at that instant.

What the Slope of the Velocity Graph Tells You

The slope of a velocity graph is the derivative of velocity. Since the derivative of velocity is acceleration, the slope tells you the object's acceleration.

This gives you a very useful graphical rule:

• Positive slope → positive acceleration.
• Negative slope → negative acceleration.
• Zero slope → zero acceleration.

Notice that acceleration is determined by the slope, not by whether the velocity itself is positive or negative.

Velocity and Acceleration Are Not the Same Thing

A common AP Calculus mistake is to look at a velocity graph above the horizontal axis and conclude that acceleration must be positive. That's not necessarily true.

The object can have positive velocity while its acceleration is negative. All that matters for acceleration is the slope of the velocity graph.

This is an important distinction. Positive velocity means moving forward; negative acceleration means velocity is becoming smaller.

How to Tell Whether an Object Is Speeding Up or Slowing Down

Speed is not the same as velocity. Speed measures how quickly an object is moving, without regard to direction.

An object is speeding up when its velocity and acceleration have the same sign. It is slowing down when velocity and acceleration have opposite signs.

On a velocity graph, this becomes a simple graphical test:

• Above the axis + increasing: positive velocity and positive acceleration → speeding up.
• Above the axis + decreasing: positive velocity and negative acceleration → slowing down.
• Below the axis + increasing: negative velocity and positive acceleration → slowing down.
• Below the axis + decreasing: negative velocity and negative acceleration → speeding up.

The key is to compare the sign of the velocity with the sign of the slope.

Worked Examples

When Is the Particle at Rest?

A particle is at rest when its velocity is zero. On a velocity graph, look for where the graph crosses or touches the horizontal axis.

But being at rest at one instant does not necessarily mean the particle stops permanently. It may immediately begin moving again.

To determine whether the particle actually changes direction, look at what happens to the sign of velocity on either side of the zero.

• Positive velocity → zero → negative velocity: the particle changes direction.
• Negative velocity → zero → positive velocity: the particle changes direction.
• Positive velocity → zero → positive velocity: the particle stops momentarily but does not change direction.
• Negative velocity → zero → negative velocity: the particle stops momentarily but does not change direction.

What the Shape of the Velocity Graph Tells You About Position

Since velocity is the derivative of position, you can also use a velocity graph to determine whether the position function is increasing or decreasing.

Therefore:

• If the velocity graph is above the axis, position is increasing.
• If the velocity graph is below the axis, position is decreasing.
• If velocity is zero, the position function has a horizontal tangent.

You can also determine the concavity of the position function because acceleration is the second derivative of position.

• Positive acceleration → position is concave up.
• Negative acceleration → position is concave down.
• Zero acceleration → the velocity graph has zero slope.

A Complete Graphical Interpretation Strategy

When an AP Calculus problem gives you a velocity graph and asks you to interpret the motion, use this sequence.

1. Look at the height of the graph. This gives velocity.
2. Look at whether the graph is above or below the axis. This tells you the direction of motion.
3. Look at the slope. This gives acceleration.
4. Compare the signs of velocity and acceleration. Same sign means speeding up; opposite signs mean slowing down.
5. Look for zeros of velocity. These are moments when the particle is at rest.
6. Check the sign on both sides of a zero. This determines whether the particle changes direction.
7. Use the sign of acceleration to determine concavity of position.

Common Mistakes

Practice Problems