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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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Graph d is position, c is velocity, b is acceleration, and a is jerk.

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 8

The Derivative as a Function

Limits

Derivatives

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Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

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The figure shows the graph…

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The graph shows the positi…

This is problem number fifty two of the Stuart Calculus eighth edition, Section two point eight. The figures shows the grafts of four functions. One is the position function of a car. One is the velocity of the car. One is its acceleration and one of its jerk Identify each curve and explain your choices. So each here we ever graph where it shows all four curves A, B, C and D Ah, and we're just going to start, Let's say with curved D and see uh, see what we can determined from ah discussing the derivative of each function. So if we look a curve D, it is always increasing. If this curve is always increasing, its derivative must only be above the X axis, so only positive. And we see that this is on ly true. I'd grab C because curve A and B, they have negative parts. Meaning there. There's no way that a curving beacon to describe the ever increasing problem derivative of the ever increasing nature behavior Buddy. So currency must be the river of curvy curve. See itself. Ah is all increasing. I've been till it gets to this point here. Where'd maxes out it's from a small minimum maximum. Afterward, it just slightly decreases. So we need a function that's mostly positive. Here we see B is mostly positive. A has negative region here, eyes negative year, which would not correspond with this. You have your FC. So P is mostly positive. And then at the point here, it looks like it lines up perfectly with the maximum embassy and then afterwards speed slightly negative, corresponding to the decrease here after its maximum. So it looks like B is a derivative of seeing. And then we recall against you with the derivative of the So far it seems like everything's pretty consistent. We look at Kirby, it has a maximum here, meaning that its services should cross the X axis here and ese only line that does that. So the bakers must be the derivative of B on DH. Then way know that bee is increasing up until his maximum. So it's derivative. What's that? Positive values, which we see from curry and then afterwards be, is always decreasing. So afterwards, the derivative of bi must have all negative values, which is true here, shown after a crosses the X axis. So Amos Peter being, uh and then be well was also the derivative. CNC is the dirty. It's on order. He must be the position function of the car since it's, uh, the first function is the position. Junction sees the velocity Earth See the velocity function. Since it's the start of See, He's the acceleration functions, it's it's derivative of acceleration and athe jerk function. Jerk, meaning the derivative of acceleration Turkey is that, well term used to descriptive, a derivative of acceleration. And so all of these curves, all of these choices I were confirmed by investigating and analyzing the graph.

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