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

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03:43

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 8

The Derivative as a Function

Limits

Derivatives

Oregon State University

Harvey Mudd College

University of Nottingham

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…

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The graphs of three positi…

So in this problem were given a graph That has three lines on it, a BNC. And we're told that one of these is positioned function of a car. One is the velocity of the car and one is the acceleration and were asked identify each curve and explain your choices. All right. Well, first of all we know that the acceleration is the derivative of the velocity and the velocity is a derivative of the position. And so if we think about this for mitt the derivative meaning the slope of that curve. Well, if we look at it for a second as C starts to climb directly, Starting out here at zero and see starts to climb linearly across here, you notice that be shows that linear acceleration happening right then. See turns upward across this interval here he turns up and then starts to decrease on this last interval. So it really looks like that. Be B is the slope or derivative of C. Then if we look closely again as B is climbing A is climbing even faster. So it looks like the slope of the derivative of be going on. And then as B starts to approach the top of the curve in this section here we see that we hit a maximum and start to decrease then right. Which is what the slope does until it starts to go negative right? And then becomes less and less negative out here toward the end. So it looks like the A. Hey, is the slope of B. So therefore see is the position of the car, B is the velocity and C is the acceleration of the car.

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