graph of functions and derivatives - Tilted Forum Project Discussion Community

khuct · 11-11-2004, 10:57 PM

I can't seem to identify some of the first and second derivatives graphs and the function graph it came from. I was told that if I wanted to identify a derivative graph, then I will have to look at the functions optimizing points. If the function graph goes to a negative slope after it's optimizing point, then the derivative graph around that area is negative relative to the X-axis. If the function is a positive slope after the optimizing point, then the derivative graph around that area is positive relative to the X-axis. But that doesn't seem to work for some graphs. And in addition, how do I identify the second derivative if given 3 graphs (function, first derivative, second derivative). Yeah a little confusing, but I"m not too good at explaining math. Thanks in advance

Slavakion · 11-12-2004, 03:49 AM

The first derivative graph will be positive wherever the slope of function f is positive. Always, unless there's some exception I haven't learned.

The second derivative graph will be positive wherever the 1st deriv is pos.

To identify the second derivative, I'd say try to find the "simpler" graph.

I'd explain better, but I'm out the door.

manalone · 11-12-2004, 11:23 AM

What Slavakion said, more or less.
The graph of the derivative of a function will follow it slope. It's tough to visualise, I find it easier if I do the d/dx and visualise the derivative function.

Slavakion · 11-12-2004, 11:34 AM

Okay, I'll try a little better this time.

For a function, its derivative equals its slope.

In this graph, you see the function x^3. Roughly. Yes, it's a paint job. You can see that as x approaches zero from both sides, the slope gets smaller until it actually equals zero. But the whole time, the slope is positive (except zero, cuz zero isn't positive or negative).

This is the graph of the derivative, x^2. You can see that it is positive the whole time except when x=0. There, the y value becomes zero.

Hope this helped...

molloby · 11-12-2004, 12:06 PM

With the second derivitive, remember that it is simply the derivitive of the derivitive, so find the derivitive and look for the second derivitive using the same method.

fckm · 11-12-2004, 01:30 PM

LOL. Is that MS paint?!

Stompy · 11-12-2004, 03:09 PM

The red line is x^2.
The blue line is 2x (derivative of x^2)

Use a straight (vertical) line and move it along the X axis. Start at, say, x = -2. X on the red graph is 4, but x on the blue graph is -4. The slope of x^2 at x = -2 is -4.

Notice as you move the line from x = -2 to x = 0, the slope (blue line) of x^2 INCREASES to 0. At x = 0, there is no slope on x^2, therefore 2x = 0.

Move the line to the right to x = 1.

x^2 = 1, but the slope is 2.

Move to x = 2. When x = 2, the slope of x^2 is 4, but notice the lines intersect because the slope is also 4.

Now, let's look at the derivative of 2x, which is a constant 2.

No matter what x equals, the slope of 2x is always 2. That's why there's a straight line at 2. x could be 293849238492, but the slope of 2(293849238492) is still 2.

Now put all 3 together:

Stompy · 11-12-2004, 03:23 PM

If you wanna see another example...

When x is negative and gets closer to about .75, the slope is positive, but is getting SMALLER (which is why the yellow graph shows a decreasing answer as x approaches .75).

Once x hits .75, the slope is 0 and begins to decrease, but then starts to level out again as x gets closer to 0 (which is why from .75 to roughly 0, the yellow graph decreases, then increases).

After that, the slope does nothing but increase, as evident by the yellow graph.

I forgot to mention earlier: Third derivative graphs determine concavity of the first equation (either concave up or down, which is the "U" or upside down "U" shape the graph takes)

If you want to test these out on your computer, you can use the program I used to make these graphs (WinPlot, it's free): http://math.exeter.edu/rparris/winplot.html

Once you load it, hit F2 to load up a blank graph, then F1 to add equations to it. You can change the zoom and everything to give you a better idea of how these graphs work together.

Slavakion · 11-12-2004, 05:01 PM

Quote:

Originally Posted by fckm

LOL. Is that MS paint?!

Quote:

Originally Posted by Slavakion

In this graph, you see the function x^3. Roughly. Yes, it's a paint job.

Yeah, I need to get a copy of Maple or somesuch software.

fckm · 11-14-2004, 08:22 AM

try bittorrent.
that's where i got my matlab.

Slavakion · 11-14-2004, 10:43 AM

Yeah, just got Maple 9.5. I don't know how to describe it... in a good way.

11-11-2004, 10:57 PM	#1 (permalink)
khuct Upright	graph of functions and derivatives I can't seem to identify some of the first and second derivatives graphs and the function graph it came from. I was told that if I wanted to identify a derivative graph, then I will have to look at the functions optimizing points. If the function graph goes to a negative slope after it's optimizing point, then the derivative graph around that area is negative relative to the X-axis. If the function is a positive slope after the optimizing point, then the derivative graph around that area is positive relative to the X-axis. But that doesn't seem to work for some graphs. And in addition, how do I identify the second derivative if given 3 graphs (function, first derivative, second derivative). Yeah a little confusing, but I"m not too good at explaining math. Thanks in advance

11-12-2004, 11:23 AM	#3 (permalink)
manalone Once upon a time...	What Slavakion said, more or less. The graph of the derivative of a function will follow it slope. It's tough to visualise, I find it easier if I do the d/dx and visualise the derivative function. __________________ -- Man Alone ======= Abstainer: a weak person who yields to the temptation of denying himself a pleasure. Ambrose Bierce, The Devil's Dictionary.

11-12-2004, 03:09 PM	#7 (permalink)
Stompy Banned from being Banned Location: Donkey	The red line is x^2. The blue line is 2x (derivative of x^2) Use a straight (vertical) line and move it along the X axis. Start at, say, x = -2. X on the red graph is 4, but x on the blue graph is -4. The slope of x^2 at x = -2 is -4. Notice as you move the line from x = -2 to x = 0, the slope (blue line) of x^2 INCREASES to 0. At x = 0, there is no slope on x^2, therefore 2x = 0. Move the line to the right to x = 1. x^2 = 1, but the slope is 2. Move to x = 2. When x = 2, the slope of x^2 is 4, but notice the lines intersect because the slope is also 4. Now, let's look at the derivative of 2x, which is a constant 2. No matter what x equals, the slope of 2x is always 2. That's why there's a straight line at 2. x could be 293849238492, but the slope of 2(293849238492) is still 2. Now put all 3 together: __________________ I love lamp. Last edited by Stompy; 11-15-2004 at 12:24 PM..

11-12-2004, 03:23 PM	#8 (permalink)
Stompy Banned from being Banned Location: Donkey	If you wanna see another example... When x is negative and gets closer to about .75, the slope is positive, but is getting SMALLER (which is why the yellow graph shows a decreasing answer as x approaches .75). Once x hits .75, the slope is 0 and begins to decrease, but then starts to level out again as x gets closer to 0 (which is why from .75 to roughly 0, the yellow graph decreases, then increases). After that, the slope does nothing but increase, as evident by the yellow graph. I forgot to mention earlier: Third derivative graphs determine concavity of the first equation (either concave up or down, which is the "U" or upside down "U" shape the graph takes) If you want to test these out on your computer, you can use the program I used to make these graphs (WinPlot, it's free): http://math.exeter.edu/rparris/winplot.html Once you load it, hit F2 to load up a blank graph, then F1 to add equations to it. You can change the zoom and everything to give you a better idea of how these graphs work together. __________________ I love lamp. Last edited by Stompy; 11-15-2004 at 12:25 PM..

11-12-2004, 03:49 AM	#2 (permalink)
Slavakion Mjollnir Incarnate Location: Lost in thought	The first derivative graph will be positive wherever the slope of function f is positive. Always, unless there's some exception I haven't learned. The second derivative graph will be positive wherever the 1st deriv is pos. To identify the second derivative, I'd say try to find the "simpler" graph. I'd explain better, but I'm out the door.

11-12-2004, 11:34 AM	#4 (permalink)
Slavakion Mjollnir Incarnate Location: Lost in thought	Okay, I'll try a little better this time. For a function, its derivative equals its slope. In this graph, you see the function x^3. Roughly. Yes, it's a paint job. You can see that as x approaches zero from both sides, the slope gets smaller until it actually equals zero. But the whole time, the slope is positive (except zero, cuz zero isn't positive or negative). This is the graph of the derivative, x^2. You can see that it is positive the whole time except when x=0. There, the y value becomes zero. Hope this helped...

11-12-2004, 12:06 PM	#5 (permalink)
molloby Tilted Location: Sydney, Australia	With the second derivitive, remember that it is simply the derivitive of the derivitive, so find the derivitive and look for the second derivitive using the same method.

11-12-2004, 01:30 PM	#6 (permalink)
fckm Insane Location: Ithaca, New York	LOL. Is that MS paint?!

11-14-2004, 08:22 AM	#10 (permalink)
fckm Insane Location: Ithaca, New York	try bittorrent. that's where i got my matlab.

11-14-2004, 10:43 AM	#11 (permalink)
Slavakion Mjollnir Incarnate Location: Lost in thought	Yeah, just got Maple 9.5. I don't know how to describe it... in a good way.