Graphical link between function and derivate

In summary: If the value of the function is negative, the slope is downward. If the value of the function is positive, the slope is upward.In summary, the slope of the derivative of a function can be positive or negative depending on the value of the function at that point. The relationship between the function and its derivative is that when the function is increasing, the derivative is positive, when the function is decreasing, the derivative is negative, and when the function is changing direction, the derivative is zero.
  • #1
mad
65
0
Hello all,
first, excuse my english I don't speak it very well

I have a problem. We have two sheets. One are graphics of functions, and the other are graphics of the derivate of those function. Now my problem is I don't know how link a graphic of a function to the graphic of its derivate. I know that, for example, y = x^2, for ]-oo, 0[ , that the slope (sp?) will be negative. So why , on the graphic of the derivate which is y=2x, is the slope positive? How can I associate a function to its derivate?
thanks a lot
 
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  • #2
If you don't understand what I'm asking, here is an exercice exactly like the one I'm talking about.
http://gmca.eis.uva.es/wims/wims.cgi?lang=es&+module=U1%2Fanalysis%2Fderdraw.en

Choose degree 3 or more and it asks to draw its derivate graphic. But I don't know how
 
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  • #3
The gradient function of y = x^2, that is: y = 2x, is negative for values of x less than zero. So, although the slope of the gradient function is positive, you can see that value of the gradient function at say, x = -2 still gives the slope of y = x^2 at x = -2.

The slope of the gradient function would only be negative if the original function was y = -x^2.

It helps to plot the two graphs y = x^2 and y = 2x above and below each other, and matching respective x values on both, to get a feel for what's happening in the gradient function.
 
  • #4
The relationship is:

When the function is increasing, the derivative is positive

When the function is decreasing, the derivative is negative

When the function is changing direction, the derivative is zero

So, in the case of [tex] f(x)=x^2 [/tex] from [tex]-\infty\rightarrow 0 [/tex] the function is decreasing and the derivative is negative. At the point (0,0) the function changes direction, so the derivative is zero, and from [tex] 0\rightarrow\infty [/tex] the function is increasing so the derivative is positive.
 
  • #5
Although the slope of 2x is postive, the value of the function is negative.

It is the value of the function which you must be concerned with.
 

FAQ: Graphical link between function and derivate

What is a graphical link between a function and its derivative?

A graphical link between a function and its derivative is a visual representation of the relationship between the two. It shows how the slope of the function changes at different points, and how it relates to the values of the derivative at those points.

How can the graphical link between a function and its derivative be represented?

The graphical link can be represented using a graph, with the function plotted on the x-axis and the derivative plotted on the y-axis. The points where the slope of the function changes correspond to the points where the derivative crosses the x-axis.

What does the slope of the function tell us about the derivative?

The slope of the function at a particular point is equal to the value of the derivative at that point. This means that the slope of the function can give us information about the rate of change of the function at that point.

How does the graphical link between a function and its derivative help us understand the behavior of the function?

The graphical link allows us to see how the function is changing at each point, and how the derivative is related to that change. This can help us understand the behavior of the function, such as whether it is increasing, decreasing, or reaching a maximum or minimum value.

Why is the graphical link between a function and its derivative important in calculus?

The graphical link is important in calculus because it helps us understand the relationship between a function and its rate of change. This is crucial in many applications, such as finding maximum and minimum values, determining the behavior of a function, and solving optimization problems.

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