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Sketch the graph of the function defined in the given exercise. Use all the information obtained from the first derivative.$f(x)=x /\left(x^{2}+1\right)$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 2

The First Derivative Test

Derivatives

Missouri State University

Campbell University

University of Nottingham

Idaho State University

Lectures

04:40

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.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

01:09

Sketch the graphs of the g…

01:37

01:15

Sketch the graph of the fu…

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01:18

let's get to the graph of the function to find in the given exercise. We want to use all the information obtained by the first derivative. This can be a five ax equals x. Okay, Whatever X squared plus one. It's really great this and we wanna um sketch the graph so We know that for example this is going to have a Y intercept or a horizontal ass instead of like 10. There's not going to be a very classic code. We also see that if we use the derivative function, We'll get a local maximum metrics equals one. We'll also get an inflection point at X equals zero. And then we also have some other important values but these are all manifest himself in the graph that we see here The same thing with the 2nd derivative graph. We see the inflection point as well as the changes in connectivity. From concave down to come keep up and vice versa in other areas. So it's going to be our final answer.

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