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

[@Domi39]

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.

[@richb.307]

Truth.

[@i3ulletstorm16]

eleven years later and this still applies

[@kayzong9260]

2022

[@thanquangphuongk173]

@@kayzong9260 2022 too :D

[@patrickjmt]

And here I still am... :)

[@patrickjmt]

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

[@michaelbanks1000]

Epic

[@MegaBev]

Have my babies

[@hahahahahahhahaha6]

nahhhh have mine

[@sarahisabellastein920]

Still saving grades 10 years later

[@othmanmohammed4385]

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

[@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!

[@tranvo1846]

You save lives do you know that

[@108mtsan]

yea he saves lives. Because people who fail their maths end up suiciding and patrickjmt prevents people from failing their maths

[@balachap9848]

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

[@prince10ist]

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

[@patrickjmt]

ha, thanks so much ;)

[@kurtebirde6385]

Lol! Heck ya "Brother" , rips off shirt and breaks table in the exam hall

[@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]

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

[@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.

[@diana6921]

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).

[@hielenaug]

Thank you Patrick, I don't know what I would do without you! you are the greatest at explaining ! love you!

[@patrickjmt]

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

[@wizikitheguru]

What do you do for a living? I really appreciate these videos! Thanks for making time in your probably busy schedule to post these :)

[@patrickjmt]

@moonira0209 the domain of the function... videos on that too : )

[@iveseenspikynorman]

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

[@patrickjmt]

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

[@Keninho21]

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

[@RadoAller]

WOW,,,this is wayyyyy 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!

[@jadatassi9388]

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

[@alenchavez1537]

Wow! You just explained a whole lesson in 4 min. Thats Crazy! Seems so simple now!!

[@patrickjmt]

Glad you think so!

[@1RonixWake]

I have a take-home quiz and this is exactly one of the problems. THANK YOU!!!!!!! I love my life..

[@patrickjmt]

@persianpride93 they teach you tricks on how to do it in school if you take education classes

[@boxhouse23]

@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.

[@NaBiSc0DiSc0]

With your extensive knowledge you'd do extremely well in the quantitative finance field, although I sense you've found a higher calling: education. And for that, I'm deeply impressed; you're a far better man than most. People like you change the world for the better. Thank you!

[@statik8460]

You’ve taught me more in 5 min than my ap calc teacher in a week

[@EmmaTheHurdler]

So helpful, way better than my teacher. I can understand this easily and it's very clear.

[@ccase2k]

you should become a professor or a junior professor, I love your videos. You've really helped me so much! Thanks Patrick!

[@oof9701]

An absolute Angel. I'll think of you when I pass my test.

[@SimplifiedQuestion]

I love your videos! You make my theoretical mathcourses so much easier! thanks alot!

[@pascualrosario8633]

lol my professor took forty minutes to explain this topic. I literally watched for 2 mins, and got it. Thanks dude.

[@schoolissoboring1]

Your videos are amazing! Its clear,precise, and fast. My professors literally wrote an essay on this and proofs. -.-"

[@KonKonParty]

'A zero or discontinuity of the derivative is called a critical value of a the function.' There is an example of y = 1 / x(x-4) and states the critical values as 0,2 and 4. But according to you, and a lot of other sources it states that if it is not included in the function's domain; ie. being undefined, it is not a critical value. I just wanted to know which one was which, as I am confused as whether 2 is the only critical value, or whether 0 and 4 are also considered a critical value.

[@patrickjmt]

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

[@patrickjmt]

@Andyterranbase1 thank you for pointing out the obvious.

[@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

[@genegene009]

these things give me headaches. -_- THANKS FOR THIS SIR! helped alot. got my third departmental exam today, still reviewing with you. :))

[@TheDrDewman]

-3 and 3 are not critical numbers because the critical numbers have to be defined in the original equation to be considered in its given domain, unless the domain is explicitly stated. Plugging 3 into the original equation would create an undefined case - the undefined or zero case we're looking for (to actually locate the critical numbers) is for the derivative, not for the original. Does that make sense?

[@kkamous7278]

Patrick I am thanking from my heart for this video. you're great

[@patrickjmt]

@TCP2315 nope

[@keuriqueen12]

Hey, first of all, I just wanna say that your videos are awesome! They helped me loads whenever I fell asleep in my calc class. =] But, in this video the critical numbers SHOULD include positive and negative 3 along with 0. In my calc book it says that the Definition of Critical Number is: "Let f be defined at c. If f'(c)=0 or if f' is undefined at c, then c is a critical number." Let me know if I misunderstood this whole thing. Thanks~

[@SPBigJoe]

I love your videos they are a great help when my professor doesnt explain something well and rushes over it. I was just wondering, isnt it faster to use the chain rule to differentiate the function rather than the quotient rule, or are you just using the quotient rule in case viewers dont understand the chain rule?

[@enigma3432]

please ignore my previous comment had an idiot moment. I didn't consider that 3 would make the original function undefined :) thank you so much for the videos. they are SO helpful!

[@nataliechengstudio]

Question: So does it mean that points of asymptotes and holes are not considered critical numbers since they do not exist in the original function, while jump discontinuity is considered a case of critical number (UND) because it exists in the function???

[@zedrickrodriguez1311]

mr. patrick thanks for this video, really i do, it is very helpful for students like us who r having a hard time in this topic. our professor doesn't know how to explain this topic very well . so thanks again your teachings are easy toget :D

[@manjuchowdhury]

Oh very good. Helps you learn concept very fast and easily

[@mohamednuskhan7526]

It's helped me after 12 years

[@hahahahahahhahaha6]

no

[@patrickjmt]

yes, you left out the part that to be a critical number, that value has to be in the domain of the original function, so +3 and -3 SHOULD NOT, SHOULD NOT be called critical numbers... so read your def'n a bit closer : )

[@reedOsama]

@iixsarah the derivative f '(x) needs to be undefined but if you plug 3 and -3 into the original equation f(x) it's also undefined which means it doesn't exist on the graph... at least that's how I understand it.

[@alexnazari1452]

why do you find the number that makes it undefined only to say its not in the domain because its undefined, is there a number that you solve the denominator equal to zero and it doesnt make the original equation undefined?

[@methamphetamememcmeth3422]

He found it to exclude it from the domain of the function.

[@PreciousDesi71]

So I have the problem F(x)=x^3-12x^2. The critical numbers I found were 0 and 8. The Quiz I took says those two are the correct values. But as you mentioned, you said we have to plug it back in to make sure it gives us a 0, and when I plug in 8 it does not give me a zero. So why is it still a critical number?

[@kyzuno5066]

You clutch my homework, thank you sir!

[@patrickjmt]

no problem!

[@jennaXhye818]

Patrick, your videos never fail to help ♥ cheers to passing calc! (:

[@ThatEmoKidd00]

So a critical number can be undefined AS LONG as it is in the domain of f(x)?

[@kart1648]

Yes

[@A_Box]

Kartikeye Shrivastava Do you know an example of a function with that characteristic?

[@TheGeneralThings]

ThatEmoKidd Any function with a cusp or a sharp point. Note that the derivative at the cusp is undefined, not the actual point itself on the domain of the function.

[@hafeezhamama9580]

Mr. Patrick, what about the numerator of rational functions? Do the zeroes of the numerator mean anything when looking for critical numbers?

[@enigma3432]

Patrick, you said is -/+3 is not a critical number because it's not in the domain. But you didn't mention anything about that function example having a domain... So I'm a tad confused by that.

[@happymake]

umm your handwriting is perfect and thank you so much!

[@andreanwosu]

I had a question in my math textbook it was to find the critical numbers of the function for f(theta)=2cos(theta)+sin(theta)^2, what would be the critical numbers for this

[@arizonaicetea92]

Patrick what if you are trying to find the maximum and minimum and there is no interval written in the problem... how do you get the max&min after getting it's critical points?

[@patrickjmt]

well, what is the domain of a function?

[@luffmanjestes9451]

X equals positive 3 or negative 3 are still critical numbers... because it makes the function undefined. Critical numbers exist at x = c when f'(c) equals 0 or is undefined. It's important to know where the function is undefined to determine which type of discontinuity exists there.

[@davidazer2171]

Thank you Patrick!

[@cc-ce8wn]

Patrick- Do you know what I'd do if I wanted to then find concavity? The second derivative doesn't involve an x so how can I find where it concaves up/down?

[@reyrey389]

question: if you have a function with a hole(point discontinuity) that has an undefined slope (vertical tangent), i know the point is not a critical hence its not in the domain, but if concavity were to change at that point, would it still be a point of inflection? thank you.

[@MrAWD-ox3gk]

God bless you brother.

[@MetallicaRepublic]

@patrickJMT what do you mean by not in the domain of the original?

[@judeokoroafor1423]

You have shown me mercy as you have proved to be my medicine... !

[@dizzy4u]

is there a difference between critical value and critical numbers? in my text book a critical value is the value of f(c) but in my worksheet they are asking for critical numbers and you are saying it is the value of c.

[@Nasserrvn]

Hello Patrick, I have a question please, when you found the domain of the original function, you sat the numerator and denominator of x to be zero. However, in other video you only used the denominator to find the domain so how?

[@InBeforTheLock]

Because they are non-permissible values. See how plugging +3 or -3 in the denominator of the original function evaluates to 0. You can't divide by 0, so its a NPV ie x cannot be +/- 3.

[@carbon273]

EXACTLY WHAT GOT ME CONFUSED HERE

[@nicknamepeach]

hi. you've been a big help to me. i was wondering do you have videos about inflection point? it's just that i can't find any, or may be there is but what are the keywords to be used to might actually find it? btw, thank you so much. you are my hope in passing calculus. :)

[@MileeWilson]

Where do you plug in the X value to find the critical point.

[@sakuranbo888]

cuz, if u graph it out, u will c that at 3 & -3 there are asymptotes, but the slope doesnt change.

[@mmeccafork]

can we find all the defined critical numbers just by looking at the critical numbers of the first derivative?

[@shaylaakers3493]

Love your videos so much.. very helpful! Safe to say you helped me pass College Algebra lol.

[@DanTheShotokanKid]

do you have any videos on Critical Points and Extrema solving for x. Because I'm in a charter school and i looked at the daily quiz and 1st question says "solve for x" and i'm like "wtf...i thought we were graphing!"

[@melisajahiu7194]

Do we have critical points if the numerator is a number after we derivate the function? So the numerator doesn't have an x!! Eg. 8/(x+5)^2 ... what can we do in that case?

[@patrickjmt]

i now make math videos for a living! ;)

[@patrickjmt]

and now i dont!

[@brco2003]

Does this mean that any rational polynomial doesn't have a critical number, c, such that f(c) is undefined?

[@Truck--kun]

Praise the internet for free good teaching compared to expensive college with bad professors

[@Element324]

@venomrx it seem as if you will have to use chain rule with this problem and product rule!

[@setheps]

the on thing i dnt get is why do you solve the denom to equal zero if a number divided by zero cant be in the domain

[@GoBaltimoreOrioles]

do you have to simplify the derivative down?

[@ImC4Chris]

I thought the undefined made it the critical number. Da fuck

[@vichiabary]

how a lovely teaching style!!!!

[@treegerl]

you're a life saver! thank you soooo much

[@patrickjmt]

@sueishungry your statement confuses me

[@bubblesxo]

thanks watching all ur vids 2x speed to study for my final huge help

[@moosegoose1282]

hey kind sir, is there a way to know if an equation have critical numbers without solving it first? thanks.

[@sarazuu]

for x=+/- 3 it is undefined so why is it not a critical value if you stated at the start that a value that makes the function undefined is a critical value?

[@ASunDr]

Why do you only set the numerator of the first derivative to equal zero? What if the numerator of the first derivative has no variable?

[@brendhaanilus3199]

The quotient rule is f(x)1 times g(x) - f(x) times g(x)1/ g squared. And you placed g before f which isn't right

[@happymake]

wait, you said c is a critical number if it makes the derivative function undefined, then why are 3 and -3 are not critical numbers when they make the derivative function undefined in this case?

[@ortheechoudhury9541]

where do they state the domain of the original function.. how do u know +/-3 arent in the domain

[@CHARACTERK19]

@patrickJMT how can i know that c is in the domain?

[@tzghw7]

If we were dividing by 0, it would be undefined. but what does that represent on this problem?