Calculus 5.2b - The Notation

[@alexkasimir9931]

Amazing! such a beautifully simplistically detailed explanation.! A pedagogical work of art ! THANK YOU!

[@PaulFSmith]

Perfect! Extremely well explained without a single mistake.

[@leemonastesse84]

Derek, Thank you for uploading your videos! I have watched them all semester and have helped me tremendously in my Cal 1 class. I have never had an instructor be so clear and thorough as you are! Thank You!

[@achannel1818]

Final year Engineering student here and this video is helpful for me. Going back to basics is worth it sometimes

[@kennykepra1695]

The best explanation I never got Thank you very much.

[@Abdulrahman-uk9ny]

I never saw an explanation like this at high school and university.

[@hikida9257]

Thank you from Ethiopia

[@funmaster5249]

Thank you so much, this cleared up so much for me. Elegantly explained and some really cool concepts discussed here.

[@Sins68]

Neat, informative, and entertaining. Couldn't have been any better, thank you for these videos.

[@toakleaf]

This an unbelievably elegant video and explanation. Thank you.

[@kotpx3]

I love the style of this video! Makes it really interesting to watch.

[@derekowens]

Thanks very much. I enjoyed making that one.

[@johnf4388]

I bet calculus is easy but people who understand it suck at explaining it. Math has a bad reputation for this reason. If only teachers could take the time to explain the basics properly for people to have good foundations im sure students would love and enjoy math. Thanks for this video, 100% clarity

[@Maxxx20]

wow amazing explanation. Makes it so clear. Thank You Derek.

[@ian.ambrose]

Thank you father.

[@linpeng8149]

Very lucid explanation! Thank you!

[@TheDarkerPath]

Outstandingly clear teaching :) Thanks so much.

[@arutyundemirchyan4960]

Thanks you, great vid . Answered a lot of my questions 👍👍👍

[@Francis-zk3io]

You gave me hope

[@Bripirate]

So would dx really just represent a line from the x axis to the curve and that we're actually summing the areas of an infinite number of lines between the two points on the x axis? Although that logically implies that a line has an area which doesn't seem right to me.

[@Christian-nc2sf]

DOPE! Math is my favorite subject and you helped my outtt

[@honestman276]

Thanks. From Bangladesh.

[@derekowens]

Thanks for watching! I'm glad it was helpful.

[@iqraazam9380]

Amazing!

[@8trim550]

I miss early 2010s 😭😭😭

[@derekowens]

I do as well. Things have changed so much since then that it seems like a long time ago. But 2+2 still equals 4.

[@matthewhubbard1672]

superb!

[@gary3ward]

The tiny little segment is called a secant.

[@ethanksiazek5548]

Great stuff

[@dwaipayandattaroy9801]

You mean the point that needs its slope value expressed cant be found exactly by rise/run value . Why dont we then keep the point in between two spots of equidistance from both sides and then find the rise/run value, better take the rise /run of 3 consecutive points ( start middle and end) and average out the results

[@derekowens]

Yes, that will definitely work. That is one of the methods for approximating the slope. It will work well in most cases, but can fail in some situations, such as near an asymptote or a discontinuity. The method you describe is valid. The key idea, though, is that that method only gives an approximation, where the techniques of calculus can give us an exact answer.

[@83jeffa]

the existential crisis is if there are a huge sum of infinitely small strips then there is no area at all.....

[@derekowens]

+j3ffr3y If there are an infinite number of infinitely small things, then the total is either zero because they are infinitely small, or infinity because they are infinite in number, or else some finite value between 0 and infinity. The interesting problems of this nature are the ones for which the answer is a non-zero finite value, and these are the problems that calculus allows us to solve.

[@ThomasDeLello]

He doesn't use the term "tangent line" in describing dy/dx which is simply the slope or gradient of a "tangent line" to a curve where Delta ( differential ) approaches Zee...ROH...! A "tangent line" is simply the one and the only line that makes contact with a curve of a function in only one place. A "function" meaning that for every value of 'x' is one value for 'y' in a Cartesian system.

[@fredoaks6375]

dy/dx, not dx/dy.

[@aok1425]

thank you!

[@chestermicek]

I took three Calculus courses and got a C, B, & an A, but not a single PhD instructor had the verbal capability to explain exactly what f(x)dx meant. The C was at an Ivy League University and the instructor was an inarticulate savant who could do one proof after another, but couldn't write or speak to anyone but another freak like him. We actually brought cameras to class to photograph his proof de jour before he erased it to write another one down. That geek ruined students' lives because he sent us into Economics 101 & 102 without any knowledge of what the fu@k we were doing or why. Most of us were 18-years old when we ran into this fool, but I was 17 & wasn't raised to question authority. That changed. You might say, you should have taken the bull by the horns, but you've never been in a classroom where a 96/100 got you a B on an exam. That was in an atmosphere where the damn Economics Professor couldn't do Calculus, but tested us as if he could. In other words, our Calculus teacher couldn't do Economics, Chemistry, or Physics and our Science Teachers couldn't do Calculus, but we were expected to do & graded as if we could do both. It's not unusual for a math whiz to be totally inarticulate and unable to relate to someone whose discipline requires them to not only be fluent in English, but also able to explain how Calculus is the real introduction to abstract thought. By that I mean, a Guns & Butter graph using money & time is analyzed the same way a foot x foot f(x)dx is: you use the integral & the differential. Anyway, that's my way of framing a "thank you'.

[@derekowens]

Hello, Chester, and thanks for the note. I'm glad you liked that explanation. You can thank an older gentleman named Wayne Murrah, who taught me calculus back when I was in school in the 1980's. He did a great job and it stuck with me. What I put in to my explanation was essentially the explanation I received from his class. It was mind-expanding, and it changed the way I see physical things and processes in the world.

[@danielmaluenda9731]

9:05 that graph in the bottom right looks like pi. Coincidence?!? I think not

[@derekowens]

Haha! Yes, it does look like Pi. Actually my Calc III professor in college used to always make his generic f(x) graphs in that shape. I think I picked that up from him.

[@danielmaluenda9731]

Derek Owens same here actually, she makes it always look like that haha. Great job on the video, it did help me quite a bit.

[@arutyundemirchyan4960]

Perfect

[@teddy9267]

That is George W Bush's voice.... something his doing after retirement. Great guy, who new🙂

[@gentlemandude1]

I thought that I was the only one who thought that.

[@ayushshah4649]

genius

[@dwaipayandattaroy9801]

| | S |__________ the S originates from x axis and stand tall . Then its expression is who given as { f (y) Dy ??

[@dwaipayandattaroy9801]

So for slope estimation the equation is rise/ run and for horizontal wave its length times width expressed as f(x) instead of y and asusual × for run. Now for the third hraph , summation of f(x) times Dx . And then for rest of the rectangles what shall i do ? Sum up the value of f(x)times Dx four times , since there lays 4 rectangles

[@alexkuligowski4092]

For math teachers who don't like this being presented this way... please stop teaching

[@youyoumu]

It must be f(xi) not f(x)

[@tonysmith2203]

not good

[@mononix5224]

If you use infinitesimals you have to at-least use the standard part function, else it's just cheating man xP

[@derekowens]

Can you elaborate on what you mean by standard part function? TIA.

[@mononix5224]

@@derekowens I most certainly can, I will keep it simple though, because it is quite easy to find more information on the internet. Suppose we let h* be an infinitesimal, which would typically be on the hyperreal number line (within non-standard calculus), and we use our typical definition of the derivative, but replacing the limit with said infinitesimal. What would happen is that we would take our real function f(x) into the hyperreals, because we define f'(x)* = ( f(x + h*) - f(x) ) / h*. Now f'(x) => f'(x)* is in the hyperreals, so to go from the hyperreals back into the standard real numbers we would take the standard part, which is the real number infinity close to our infinitesimal, so f'(x) = st( ( f(x + h*) - f(x) ) / h* ). Then we have the real derivative f'(x) while being more mathematically rigorous and without setting all the limit lovers off ;P Still a great video though ^^ PS I wrote this on a phone so I don't know how bad it looks ;3