Graphing Functions and Their Derivatives

[@Thaumius]

I learned concavity by thinking of faces. If the second derivative is +, happy face (concave upwards) If the second derivative is - sad face or frown (concave downwards)

[@brucedienst7553]

Thaumius thank you

[@momowon8611]

You help me guy..thank u..😄

[@ddahstan6876]

Nifty! Will keep it as my mantra for future use. "Concave-up" Cheer!

[@NoActuallyGo-KCUF-Yourself]

That's how I teach it to my students: smiley face or frowny face curves.

[@lyricass7810]

Thanks man ♂️😂

[@wanjirurandolph242]

i love the way this was explained having real world examples like the ball makes this easier to remember

[@NilEoe]

It's impressive how a few youtubers are doing so much good in helping thousand (if not more) students around the world thanks to how much better they are at explaining these concepts (plus, the video format is in my opinion better than a traditionnal lecture for this, in fairness to uni lecturers). You have a great outcome on the world :)

[@critz6768]

Learned more here, than school. Thanks.

[@ahappyimago]

This video needs 100x more views!!

[@abbyelizabeth3748]

This was SO helpful. Thank you!!!

[@atp9771]

Thank you so much for explaining this step by step; I can never keep up with my teacher because he’s always referencing things rather than explaining them

[@trieuhuyvan5020]

chào 🥰

[@DarinBrownSJDCMath]

7:52 Second derivative = 0 does not imply inflection point. But if the graph of the second derivative crosses the x-axis, then you will have an inflection point. You did mention that later, but the change of sign in f'' is necessary.

[@angelisvegan5826]

Yeah

[@theastuteangler]

Crossing the x-axis is called a zero, is it not?

[@DarinBrownSJDCMath]

@@theastuteangler A zero is just a root, but usually when one says a graph "crosses the x-axis", we mean it also switches sign from positive to negative or vice versa. There are zeros where the graph does not cross the x-axis, the simplest example being f(x) = x^2 at the origin. In general, whether the graph of a polynomial crosses the x-axis at a zero depends on what's called the "multiplicity" of the zero, which is the highest degree of the corresponding linear factor that divides the polynomial. Odd multiplicity, it will cross, otherwise nudge. For example, f(x) = (x - 2)^3 * (x - 5)^4 * (x + 6)^7 the graph crosses the x-axis at 2 and -6, but nudges the axis at 5.

[@NoActuallyGo-KCUF-Yourself]

There can also be infections where the second derivative is undefined. I'll assume Dave brings those up in a later episode.

[@DarinBrownSJDCMath]

@@NoActuallyGo-KCUF-Yourself Good point! A nice example is f(x) = x*ln(abs(x)) - x at the origin.

[@smartervilleonroute66]

Best explanation in my 56 years!

[@Kiky_MedPhysicist]

Thank you sir for your dedication and for making this free! 🙏

[@aranichowdhury7698]

I have been looking for a video like this and I finally found it! Thanks 🙏🏻

[@rafurafu9645]

i love the pop quiz after the video. makes me feel 20 IQ points smarter! thanks

[@user-mn5op9tg3x]

Btw if anyone's wondering you approach the graph from left to right when determining if it's increasing or decreasing. Think of DRY MIX D = dependent variable R = responding variable Y = graph information on the vertical axis M = manipulated variable I = independent variable X = graph information on the horizontal axis Most graphs show the independent variable moving from left to right.

[@abubakaryrajabu8166]

Уууууциу

[@ArnavGarg-rl6qh]

Yoooooooo i was struggling with physics graph and now i finally understand it 🤯🤯🤯 thank you so much

[@ericfricke4512]

Thank you for your CONCISE explanations!

[@annyan904]

i must say hats offf professor.......i am starting to love calculus

[@user-yr3uj6go8i]

Same. Calculus seems so much fun now.

[@archerdev]

Thank you so much Prof. Dave. Science bless you man.

[@mauroymgch339]

Wonderful! Can't be better explained! Thanks for sharing!

[@KaranGupta-p6p]

Prof. Dave I would suggest making a position graph when we consider the x-axis as time totally in positive x-axis as it seems really hard to correlate time in a negative direction even if it's a position graph, I would suggest shifting this graph totally into a positive x direction.

[@SurajKumar-ey2et]

That's what I was thinking...

[@dima_math]

10:48 It is not odd function

[@josephshaff5194]

Thanks Dave. . lol Book says if f crosses x-axis it makes an I.P. on f". I see they labeled the T.P.s of f as I.P.s on f'. Then they said where f crosses x axis those are I.P.s on f" lol I'll be ok. Did the section 2x going for 3rd but now I have to make my case. It's only this one problem it always looked weird. Thanks for the vid cleared things up.

[@giuseppecalvi6289]

Hi Dave, very nice video!! Many thanks for posting it! However, please note that (x^3 - 12* x +1) is NOT an ODD function.

[@carultch]

It is an odd function, if you translate it so its inflection point is at the origin.

[@NoActuallyGo-KCUF-Yourself]

Shouldn't there be a second-derivative test or other to confirm change in concavity before assuming an inflection point? For example, d²/dx² of x⁴ is 0 at x=0, but there is no inflection there.

[@jugg3rnauthd439]

on the first example how is the derivative positive at the top left of the Cartesian plane if it descends and its in the negative coordinates...

[@ProfessorDaveExplains]

because the original function is increasing throughout the first quadrant! if it is increasing it has a positive rate of change, it doesn't matter whatsoever that the values of the function are negative. if it is increasing, the derivative is positive.

[@DarinBrownSJDCMath]

10:45 Whoa there, not every cubic function is odd!! In fact, this function f(x) = x^3 - 12x + 1 is not an odd function. I think you mean to say it has odd degree which does determine its end behavior.

[@science-y9209]

Yup that's what he meant.. he needed to rephrase that sentence. .. Nevertheless... mathelectuals like yourself should understand or point that out

[@victorpayne1731]

If you follow this exact playlist, and the preceding video on Graphing Algebraic Function (what Dave referred to), he explicitly discusses predicting end behavior based on the leading coefficient of a polynomial, and calls them "odd" or "even" for simplicity. I'm assuming you're talking about symmetry, which is a fair point when it comes to the names, but this shouldn't be confusing for anyone using Dave's material.

[@carultch]

x^3 - 12*x + 1 is not an odd function relative to the origin, but it is an odd function relative to its own inflection point. And if you translate the graph so the inflection point moves to the origin, it will be an odd function. All cubics are odd functions, if you are flexible to translate the graph so the inflection point is at the origin.

[@DarinBrownSJDCMath]

@@victorpayne1731 If that's what he did in a previous video, it's a very poor choice of terminology likely to confuse students later. "Odd function" has an accepted definition and is not equivalent to "odd degree polynomial". If a student has absorbed this mistaken terminology, they will have to unlearn it in the future.

[@DarinBrownSJDCMath]

@@carultch You're correct, and it's a very specific property of cubic functions based on depression. It's not something that carries over to higher degree polynomials.

[@ashamselectronics]

Thanks for you👍

[@BeautyChannel4]

Thank you so much for this

[@MiltosPol-qn3zh]

Professor, whats the difference between f(x)=x^2 and y=x^2???

[@ProfessorDaveExplains]

nothing really!

[@carultch]

Notation choice. f(x) = x^2 means we are defining a function f, and its input is x, and it equals x^2. y=x^2 means we are defining a relationship between the variables y and x, and that relationship is y=x^2. The function notation with the (), and commas if there are multiple inputs, is just a way to remind ourselves that the function depends on those variables as inputs.

[@nisarmasroor7961]

Very informative lesson thank you for sharing video.....

[@dijilap7313]

Thanks😊

[@codexcodexcodex]

I'm only 14 yet thanks to this yt channel, I now learned calculus. 👍👍😄

[@johnpro2847]

Thanks Dave ..i might take the mensa test instead...might be easier.

[@carloselfrancos7205]

Thanks for your videos ^^ You've done great work :D And I EVENTUALLY understood derivation xD (yay)

[@MP-cv6if]

quite a well made video

[@majdihassan647]

it took me days to sort this out on my own back 1995 in the UNI.

[@itzgoku]

Must have been really tough for you

[@ejsafara456]

hi dave ^^ i mainly know you from debates, but holly hell ur good at explaining math too! :D

[@rafaburdzy449]

Thanks

[@hhl-rh9xp]

appreciate it very much

[@momtazahmed5079]

Thank you!

[@ArfatXeon]

"Painful mental gymnastics..." lol

[@frozen_jellyfish4353]

thank you

[@tGoldenPhoenix]

Done.

[@aselim20.]

I wrote it.

[@satbirsingh7269]

Is derivative means a small substance or small part of any object??

[@ProfessorDaveExplains]

not in math!

[@NoActuallyGo-KCUF-Yourself]

Generally, a derivative is anything derived from something else. The derivative of a function is derived by differentiation. A small substance or part of something can be derived by excision or other extraction and be called a derivative of that object. But that is not the derivative of a function.

[@lengbo1036]

value of time can be -ve?

[@carultch]

Yes. That just means that the point in time is before the instant in time when we've arbitrarily decided that time=0. One example where you see this, is the BCE / CE notation for identifying years. There was a year we decided would be 1 CE. All the CE years that follow, are positive, and the BCE years that came before it, are negative. For historical reasons, there is no year zero. The de-facto year zero, would be what we call 1 BCE, and if you tried to use our modern number line to keep track of the years, all the numbers for the BCE years would be off by 1, such that 2 BCE would be the year -1. If you are accustomed to these being called BC / AD, it still is the same calendar, just with names that are meant to be culturally neutral. Another example where you see this, is the "t minus 30 seconds" that we say before an event will occur. This means the time is -30 seconds, and something of interest will happen 30 seconds later. Like a rocket launch.

[@AimingShileld]

Derivetive I rate of change of scientific

[@anastasiaanautodidact9856]

finally, my differential existential crisis is over .

[@mostafaabdelrauof3185]

Can this be applied on a third degree equation !!!!! ?

[@mostafaabdelrauof3185]

Sorry I asked the question before half of the video 😄😅🤣

[@trieuhuyvan5020]

good!! 😍😍😍😍

[@science-y9209]

5:00 how is that analogy even correct 🤔.. I mean.. that graph is constantly decreasing but the speed of the ball increased at some point after decreasing..

[@ProfessorDaveExplains]

but in the negative direction

[@science-y9209]

@@ProfessorDaveExplains what does x axis represent?? Y axis represents velocity right?

[@science-y9209]

@@ProfessorDaveExplains wait a sec .. isn't that graph supposed to represent something decreasing at a constant rate.. 🤔if that's the case then how do you make it show the increase in the velocity (even in the negative direction)..

[@ProfessorDaveExplains]

Rate of change of velocity is acceleration, which is indeed constant.

[@science-y9209]

@@ProfessorDaveExplains isn't the whole position time graph supposed to be in the 1st quadrant

[@MeiziVu]

nice

[@monak4854]

✅

[@Zeraley]

Sell this intro ughh my Eyesss! MY EYEEES >:C

[@aditichaudhari2904]

has anyone ever told you that you look like jesus

[@ProfessorDaveExplains]

every day.