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

Name: Problem 52, Chapter 2: Calculus: Early Transcendentals
Uploaded: 2017-12-12T01:09:46-08:00
Duration: 3 min 22 s
Channel: Daniel Jaimes
Description: The figure shows the graphs of four functions. One is the position function of a car, one is the velocity of the car, one is its acceleration, and one is its jerk. Identify each curve, and explain your choices.

The figure shows the graphs of four functions. One is the position function of a car, one is the velocity of the car, one is its acceleration, and one is its jerk. Identify each curve, and explain your choices.

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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

00:01 This is problem number 52 of the stewart calculus 8th edition, section 2 .8.

00:05 The figure shows the graphs of four functions.

00:09 One is the position function of a car.

00:11 One is the velocity of the car.

00:12 One is its acceleration.

00:14 And one is its jerk.

00:16 Identify each curve and explain your choices.

00:19 So here we have our graph where it shows all four curves, a, b, c, and d.

00:24 And we're just going to start, let's say, with curve d and see what we can determine.

00:31 From discussing the derivative of each function.

00:37 So if we look at curve d, it is always increasing.

00:43 If this curve is always increasing, its derivative must only be above the x -axis, so only positive.

00:52 And we see that this is only true of graph c.

00:55 This curve a and b, they have negative parts, meaning that there's no way that a curve a and curb b can describe the ever -increasing derivative of d, the ever -increasing nature, behavior of d.

01:13 So curb c must be the derivative of curve d.

01:16 Curve c itself is all increasing up until it gets to this point here, where it maxes out, small minimum, maximum.

01:25 Afterward, it just slightly decreases...

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