deriative notation question

[Farcus]

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I was wondering if df ' (x)/d(x) is a form of notation for second derivative, I know all the other ones i just never seen this one before, i would of thought 1st derivative otherwise.

also to find the concavity of a function, you find the f ''(x) then set it to 0 and then set up a pie chart and choose numbers within the intervals of each number at y=0 and plug into f ''(x) and you'll find a +/- number which'll show concavity right? Sort of rusty on this.

 

[Tatodotcom]

I'll answer the second question first....Yes you are right. Take the derivative twice and set it eqaul to zero..pick easy numbers untill you see a sign change (- to +) or vice versa. This will give you the points of concavity.

To answer your second question. That is just another way of saying take the derivative(with respect to x). Same as f ' (x).

I made an A in calc I and II.....but i myself am rusty on this but im pretty sure the info i gave you is correct(but im not perfect) 🙂 .
Hope this helps
T

 

[ffpickle]

Yes, df'(x)/dx is 2nd derivative of f(x) since f'(x) is first derivative

ie. f''(x) = df'(x)/dx

 

[ffpickle]

not too sure what you guys are talking about wrt concavity..

can you point me somewhere (kaplan, cliffs, barron's?) where i can read up on that? something brief and easy to understand maybe?

 

[Farcus]

its not in kaplan but i'm 70% i saw it on january. Its related to second derivative test.

 

[ffpickle]

ok cool, i just wiki'd it so let me explain what my understanding is, b/c it differs from what is stated above.

You *do not* set f''(x)=0
Set f'(x) = 0 and solve for the intervals which satisfy this.

plug a number within each of these intervals into f''(x) and check to see if f''(x)<0 (concave down, local maximum) or f''(x)>0 (concave up, local minimum).

according to wiki, "The points that separate intervals of opposing concavity are points of inflection."

keep in mind that you can't do these questions if the power of the function is too high (cubic or higher i think???)

if you provide an example, that would be awesome, we could work through it...

 

[Farcus]

dude you're talking not about concavity then... you're talking about whether the function is increasing or decreasing and that is by using the first derivative and then find the critical numbers and set values. That is SIMILAR to second derivative test except it will not TELL you the concavity. I don't know what the wiki said but I know this

 

[ffpickle]

can you be more specific?

you stated that we are to set f''(x)=0, when in fact we are to set f'(x)=0 and check if f''(x) is >0 or <0 for concativity.

 

[Koiv72]

Ok, let me see if I can clear this up. I'll try to give you the whole nine yards on how to sketch a graph from a function, f.

1) Domain - you just need to figure this by looking at the f function.

2) Intercepts - set 0 to x to find y-int. Set 0 to y to find x-int.

3) Symmetry - odd or even symmetry. Put (-x) in for x. If f(-x) = -f(x) [negation of f], it's odd. If f(-x) = f(x) [the same f], it's even.

4) Asymptotes - take limits of f(x) (original function) to see if they go to + or - infinity. If they both go to different infinities, then it's no horizontal asympt.

5) Intervals of Increasing or Decreasing - Take first derivative. Where f'(x) does not exist (DNE) means cusps or vertical tangents. You need to take lim of f' where x->#DNE. If they both go to different infinities, then it's cusp. Where f'(x) = 0 means it's within domain and you can plug numbers in the f' between critical #'s to see changes in signs. If ---, then f is decreasing, and ++++, then f is increasing.

6) Local Max or Min - the number/s between sign change of the f' determines local max or min of f. So if ---- 7 +++++ means f is decreasing until 7 and increasing after 7. 7 is the local min.

7) Concavity and Inflection Points - Take second derivative. You don't have to find where f'' DNE because it's not important to find the behavior of f' graph. However, f''=0 gives you #'s to compare sign changes. Signs in f'' give you concavity and inflection points of f function. So if ---- 2 ++++ means f is concaving DOWN until 2 and concaving UP after 2. 2 is the inflection point.

Mind you, the 7 and 2 were numbers along the x-axis (after putting f' or f'' equal 0). To find the actual height, or the y value to complete a coordinate, you'd need to plug 7 and 2 in the f function.

8) Graph the function based on the seven guidelines above.

 

[Farcus]

well for asymptotes there's a easier method for vertical/horizontal/slanted ones. But yeh everything is right and thus second derivative test = concavity not the first derivative .