How do you answer these: (1 Viewer)

[jamesfirst]

When it asks you "when does the curve start increasing?"

or "what value of x does the curve start increasing?"


is it referring to the gradient of the curve??? how do you mathematically show that it's increasing ??

 

[SpiralFlex]

I was just doing an exercise about this. Can I have an example?

Do you mean this,

Show that is increasing for all values of .



Therefore for any value of , it is positive.

Since the curve is always increasing.

 

[jamesfirst]

Yeah like those.

Do you have any harder examples ??? like a harder curve ?

 

[SpiralFlex]

Yes. What's your definition of a harder curve?

 

[laplace11]

Yes. What's your definition of a harder curve?

Weierstrass function

 

[clissold]

When it asks you "when does the curve start increasing?"

or "what value of x does the curve start increasing?"


is it referring to the gradient of the curve??? how do you mathematically show that it's increasing ??

The curve is increasing when the first derivative is greater than zero..

Therefore, no matter what the equation is, just find the first derivative, make it greater than zero, and solve for x..

 

[jamesfirst]

okay. Now I got this covered

what happens if second derivative is 0. what does that show ?

 

[K4M1N3]

First Derivative = gradient function
Second Derivative = concavity function

when solving the second derivative when = 0 you are finding when the concavity changes, ie a point of inflexion.

 

[sinophile]

First Derivative = gradient function
Second Derivative = concavity function

when solving the second derivative when = 0 you are finding when the concavity changes, ie a point of inflexion.

i thought it only suggested it was a point of inflexion, and you have to check concavity on both sides to see whether it really was.

then again, its been two years since i did this topic and im the type of guy who crams everything the night before the exam and forgets it the moment i put my pen down at the end

 

[K4M1N3]

Yeah you are right, it does only suggest points of changing concavity. To actual determine inflexion you HAVE TO test points.

I mighta been a little lazy :S

 

[clissold]

okay. Now I got this covered

what happens if second derivative is 0. what does that show ?

If the first derivative > 0 ... the curve is increasing
If the first derivative < 0 ... the curve is decreasing
If the first derivative = 0 ... the curve is stationary

If the first derivative = 0 and the second derivative > 0 ... minimum stationary point, hence the curve is concave up
If the first derivative = 0 and the second derivative < 0 ... maximum stationary point, hence the curve is concave down
If the first derivative = 0 and the second derivative = 0 ... point of inflexion only if concavity changes (you must show that it changes)

Hope this helps