Finding Critical Numbers - Example 1 Involving a Rational Function (Fraction)

In this video we look at the calculus concept of finding a critical number for a function.
In this case we look at the function f(x) = x^2 / (x^2 - 9) and try to find the critical numbers of the function.
Recall that to find the critical number we first find the derivative of the function and then find those values of 'c' so that f '(c) = 0 or f ' (c) = undefined.
Notice that to find the derivative of this particular function we must first use the quotient rule.

#calculus #derivative #quotientrule #critical #domain #rationalfunctions #maximum #minimum

[Domi]

The fact you made this video in 2008 and are still answering people's questions makes me feel a little better about humanity. I too think i learned more in this four minutes than i did in my last week of calc.

[patrickjmt]

i did not want any kids, and now i have millions : )

[TCOHE]

what you have taught me in less than five minutes saved me from a day of class in college calculus, you are the man!

[othmanmohammed]

a video from 14 years ago is still better than how my professer taught me, you are the GOAT

[princeist]

Mr. Patrick : do you know I press Like before I look to the video. The only thing I can do it for you. Simply, you are amazing guy

[balachap]

Late night studying for a calc test and this video was a blessing. Thank

[patrickjmt]

correct! all critical numbers for polynomials will come from setting the derivative = 0. nothing makes a poly undefined!

[FleetingMoons]

so i decided to take calculus over the summer so I could get ahead in the fall semester, and your videos have seriously helped me so much. Thanks a bunch for posting these and explaining them so well. (:!!!

[patrickjmt]

if you find values that make the derivative zero or undefined but those same values are NOT in the domain of the original, those numbers are NOT considered critical numbers.

[RadoAller]

WOW, , , this is better then my 50min lecture, , , with a crazy maniac teaching to the board!!

[noartificialflavor]

i wish i had discovered these videos earlier in the semester! Thankyou so so so much!

[jadatassi]

I think I have learned more calc here than I have in class. Thank you so much

[diana]

Stating the domain before the problem is solved would definitely help. Towards the end, I was a little confused because f'(c) is undefined, making it a critical number, but then you stated it wasn't within the domain (we are unaware of what the domain of the original function is).

[patrickjmt]

ha! i am glad you think so.... some people might disagree with the latter statement : )

[patrickjmt]

@iixsarah the values also have to be in the domain of the original function. this is the part no one remembers!

[patrickjmt]

answers to what though? we are just sketching! and when sketching, you would want to know where the function is undefined, or your graph would be wrong

[iveseenspikynorman]

Thank you! Thank you! Thank you! You've accomplished what my AP teacher, and tutors/friends could not!

[Keninho]

God bless you dude. I always check your vids for reference to see if I'm doing things right.

[boxhouse]

@MultiMoe91
It simply means there isn't an answer for it. You can't divide by 0.

For the reasoning, just go back to what dividing really means. It means having a certain amount of stuff, and sharing it equally amongst a certain amount of people. You can't really share equally amongst 0 people, so dividing by 0 is undefined.

[patrickjmt]

@brittnibeers you should read the definition a bit closer for critical numbers.