The figure shows the graphs of four functions. One is the position function of a car, one is the

The figure shows the graphs of four functions. One is the...

Key Concepts

Position function

This function describes the car's location over time. Its graph often appears smoother and less steep compared to its derivatives, and its local slopes indicate the car’s instantaneous velocity.

Velocity function

Representing the rate of change of position, the velocity function's graph is the derivative of the position function. It shows how fast and in what direction the car is moving, with its sign indicating direction and its magnitude indicating speed.

Acceleration

This function represents the rate of change of velocity, or the second derivative of the position function. The acceleration graph indicates how quickly the car's speed changes, with positive values corresponding to an increase in speed and negative values indicating a decrease.

Jerk

Jerk is the rate of change of acceleration, making it the third derivative of the position function. Its graph captures how smoothly the acceleration changes, with larger magnitude implying more abrupt changes in the car's motion.

Transcript

00:02 So for this problem, we are given four graphs, each representing the change over time of one of the following quantities.

00:09 Position, velocity, acceleration, and jerk.

00:15 And we're asked to identify which function represents which quantity.

00:20 So the most important thing to understand for this problem is how these quantities relate to each other.

00:27 Starting with position, velocity is the rate of change of position.

00:31 So velocity is the first derivative of position.

00:36 And if we wanted to get from position to velocity, we would take the derivative with respect to t.

00:44 Likewise, acceleration is the rate of change of velocity.

00:49 So to get from velocity to acceleration, we would take the derivative with respect to t.

00:55 And jerk is the rate of change of acceleration, so to get from acceleration to jerk, we would take the derivative with respect to t.

01:04 So one strategy to approach this problem is to start with the most complicated graph, in this case that would be a, and then work backwards.

01:17 So if we start with a, we see that a is increasing up to about here, then decreasing, then increasing after about here.

01:29 If we think about the properties of derivatives, we would expect the graph of the derivative of a to be positive when a is increasing and negative when a is decreasing.

01:44 So the graph of the derivative of a would need to be positive up to this point and then negative after that point.

01:53 And if we look at the graphs of b, c, and d, none of those graphs have that property.

02:00 So we have to conclude that the graph of the derivative of a is not represented here.

02:07 And if we look at the quantities that we're trying to match up to the graphs, we do have one quantity which we would not expect the graph of its derivative to be represented, and that's jerk, which means that a must be jerk...

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