Dear Professor Leonard: There is nothing that I could say about you that has not already been said by others. You are a rare breed of educators that is able to reach the majority of the people who never thought they had the ability to learn math. You have proven that anyone can learn math, given the right conditions. You are precious to so many of us. We are grateful for your talent; and honored to be in your company. Let me just share with you a few qualities that I am able to pick from your lessons that made you so effective: 1) The tone in your voice is so low key. It does not look down on your viewers. 2) Your sentences are short and simple. You do not use convoluted sentence structures to explain math concepts. You have this unique ability to just use the right amount of words to explain complex ideas. 3) You use layman's language to explain your point. 4) You repeat the points over and over and over to drive home the concept, using different scenarios each time. 5) your whiteboard presentations are very effective 6) you use examples that can be understood by the majority. Thank you so very much for everything. By the way, for those of us that are just following this lecture for fun, what textbook(s) and workbook(s) do you recommend, or do you use?
I never got to tell you this. But i passed multivar calculus (after bitterly failing it) some years back. Next summer I will have my masters degree in energy engineering, thank you so much. You videos matter to people.
I was never interested in math, I thought that is is hard and never quite understood concepts behind it, I always "brute-forced" the way through my calculus class by learning the patterns just to pass the class, until I found your channel, now I understand all concepts and the math is fun for me. And I have the motivation to learn math because with it I can create cool stuff like neural networks. I just want to thank you for your series about math, they mean a lot and they help a lot of people.
I honestly would have given up on taking Intermediate Algebra/College Algebra had it not been for this videos. Thank you Professor Leonard, you made math easy and understandable for me,
taking college precalculus for the third time second complete... YOU make me understand this so much better than previous professors. You don't "dumb" it down you simplify it to get straight to the information. :) thanks Professor Leonard.
Chugging along with the lectures and trying to watch the entire video playlist. My ti84 Graphing calculator arrived today so I am excited to try it out.
9 videos in this series and I am starting like like math! Thanks Professor Leonard, wish colleges were plagued by professors that are as good and care as much as you do!
Respected Leonard !!! You are doing an awesome job of teaching maths in such an interesting and captivating way......I never have come across a teacher like you in my whole academic career. Superb!! Thanks alot for sharing these lectures .
Professor Leonard, thank you for a well explained lecture on increasing and decreasing functions. I never really understand this topic in precalculus over the years, however your explanation is excellent from start to finish. Practice is the key to understanding the material.
because i dont have much money, so i become patreon for 1 dollar a month. but i dont skip any of the ads coz i know that will increase your payment from TH-cam.
As the input (x-value) increases, the output (y-value) decreases. That's what decreasing means. You go left to right checking whether the graph is falling (decreasing) or climbing (increasing). Two years late, but hope that clears it. How are you doing now? Did you master Precalculus? Passed the class/test/exam? I hope you're doing alright.
Can someone clear up some confusion I have? I've seen this a few times already.. near the end of the video when he is doing that last graph.. he assumes it to be a vertical asymptote. How can we assume that? The output of the function could very well pass 0 and go into the negatives right? we wouldn't know. Is there some fundamental knowledge of what the arrows mean on a line that I am missing? pls help :)
Dear Bro or Sis! u have a genuine question, however it needs some base. Keep reading. sir didn't assume it a vertical asymptote, it actually is. yes we truly have some basic knowledge for the arrow notations: this arrow shows that the graph is approaching the y axis, but it is not( don't get tensed, you'll get it) . Let me simplify what am tryna' expressing, try feeling it! what is a vertical asymptote? well, it's an *imaginary* line that cannot be crossed, but can be approached, mark it, can't be crossed but can be approached, just like u approach infinity. Now, the graph that we have is going to negative infinity vertically, but what will be the input or the x value for that? ( Now this is the point where u might have got a slap, open your eyes wide and read it with ur backbone as straight as possible) well, we can possibly observe that there is no y-intercept for x=0, so what does this reflect? again, WE CAN OBSERVE THAT THERE IS NO Y-INTERCEPT AS THE GRAPH ENDS WITH AN ARROW AS IT'S TAIL, SHOWING THAT "BYE!! MEET YOU AT INFINITY" , so what does this reflect? this reflects that for some x value the y value will be negative infinity, such that the x value is not negative ( READ ALOUD: the x value will not be negative because there is no y intercept, on a possible y value before infinity) so, it is your intuition that if, for example, x=1 and x=0.5 , then from the graph, we can see that for lesser input values, the output is approaching (learn this word by heart, thank me later) the y axis, so clearly at 0.5 that graph will be closer to the y axis. Now, at the same time , the y value for 0.5 will be closer to negative infinity, compared to the y value of 1, similarly 0.2 will be even closer to neg. inf. and 0.000000000000001 will again be closer to infinity than any of the above, similarly if x=0, will be negative infinity itself. Hence, we consider this a vertical asymptote because here the y value will be approaching negative infinity , which is impossible, and hence for this graph putting 0 is also approachable not reachable. hence it is a vertical asymptote. Sorry to make you mad. Read this word by word and you'll get it.
@@anamikajha7210 Hi thanks for the response, I appreciate it, however I'm not sure it answers my question. I think you are just explaining what an asymptote is, which I'm familiar with, but what I don't understand is how that is represented in his graph. Yes, I see that the arrow is pointing down to negative infinity, but there's no equation for the function so how can we assume that it never crosses x=0? I'm sure it's just a quick representation of a specific function, but it seems to me that we can't actually know (from the representation of the graph alone and not any further assuming) whether or not that line passes x=0 or not. Does my question make sense?
@@alistaircornacchio5727 Yes, I was wondering about the same thing. I think the equation for the graph would actually make clear that it has a vertical asymptote, but it is not given here. Let me know if you have come across other satisfying explanations.
(-infinity, infinity) would mean x goes towards -infinity and as you can see from that graph it never goes lower than 0, it just gets really really close to 0 but never reaching it (that's called an asymptote) which is why we have open parantheses (0, infinity)
I've been doing some extra reading about increasing and decreasing functions, and it seems to be the case that there are functions that can be classified as "strictly increasing" and functions that can be classified as just "increasing". Basically the best way I can define the strictly increasing relationship of a function is that it is a one-to-one (injective) function, that for every input x it holds the following condition - f(xₙ) < f(xₙ₊₁). However, for just increasing functions, the following condition holds true for all inputs x - f(xₙ) ≤ f(xₙ₊₁), basically meaning that a constant can be classified as an increasing function. Am I completely off base here?
i missed 8weeks of differential equations lecture due to injury from car accident. but still got full mark for my midsem exam. all thanks to you prof.
Great Job!!
@@ProfessorLeonard Sir you are great , I wish If I could find you 3 years earlier .
Dear Professor Leonard:
There is nothing that I could say about you that has not already been said by others. You are a rare breed of educators that is able to reach the majority of the people who never thought they had the ability to learn math. You have proven that anyone can learn math, given the right conditions. You are precious to so many of us. We are grateful for your talent; and honored to be in your company.
Let me just share with you a few qualities that I am able to pick from your lessons that made you so effective:
1) The tone in your voice is so low key. It does not look down on your viewers.
2) Your sentences are short and simple. You do not use convoluted sentence structures to explain math concepts. You have this unique ability to just use the right amount of words to explain complex ideas.
3) You use layman's language to explain your point.
4) You repeat the points over and over and over to drive home the concept, using different scenarios each time.
5) your whiteboard presentations are very effective
6) you use examples that can be understood by the majority.
Thank you so very much for everything.
By the way, for those of us that are just following this lecture for fun, what textbook(s) and workbook(s) do you recommend, or do you use?
I never got to tell you this. But i passed multivar calculus (after bitterly failing it) some years back. Next summer I will have my masters degree in energy engineering, thank you so much. You videos matter to people.
Great job!!!
@@ProfessorLeonard why thank you. It was your videos on the subject that finally made the subject click!
How is it going now? it's been 4 years
I'M FROM South Africa AND WOW YOU CHANGING LIVES THIS SIDE WITH YOUR MATHS LECTURES. THANK U AND GOD BLESS!!!!
I was never interested in math, I thought that is is hard and never quite understood concepts behind it, I always "brute-forced" the way through my calculus class by learning the patterns just to pass the class, until I found your channel, now I understand all concepts and the math is fun for me. And I have the motivation to learn math because with it I can create cool stuff like neural networks.
I just want to thank you for your series about math, they mean a lot and they help a lot of people.
@xxyyzz There's some cool Udemy Courses available using Python!
I honestly would have given up on taking Intermediate Algebra/College Algebra had it not been for this videos. Thank you Professor Leonard, you made math easy and understandable for me,
taking college precalculus for the third time second complete... YOU make me understand this so much better than previous professors. You don't "dumb" it down you simplify it to get straight to the information. :) thanks Professor Leonard.
Chugging along with the lectures and trying to watch the entire video playlist. My ti84 Graphing calculator arrived today so I am excited to try it out.
9 videos in this series and I am starting like like math! Thanks Professor Leonard, wish colleges were plagued by professors that are as good and care as much as you do!
Respected Leonard !!!
You are doing an awesome job of teaching maths in such an interesting and captivating way......I never have come across a teacher like you in my whole academic career. Superb!!
Thanks alot for sharing these lectures .
Professor Leonard, thank you for a well explained lecture on increasing and decreasing functions. I never really understand this topic in precalculus over the years, however your explanation is excellent from start to finish. Practice is the key to understanding the material.
Best math teacher I have ever seen!
because i dont have much money, so i become patreon for 1 dollar a month. but i dont skip any of the ads coz i know that will increase your payment from TH-cam.
Thanks very much!!!
i also recommended your channel to all my junior at my university. they all love it .. thank you professor!!
"Does it make sense to you"
- Prof. Leonard
I LITERALLY LOVE YOU. LIKE OMG MATH IS FUN NOW
So glad you're enjoying it! :)
you make everything is just a common sense. Concept base learning. Thank you super man.
Even English is not my native language but I understand these concepts more better than my teacher.thank you professor.
I feel happy when you upload new videos! Thank you so much professor Leonard!
Professor Leonard... wow thank you, done it again
Wr are waiting for you to continue this series
Don't let us down please
great videos wish you could make videos on engineering math
the way you relate everything is pretty great
Thanks from SA
Thanks for the great lesson, sir.
You can't know how much I love your videos please keep doing themm😭💕💕
absolutely amazing
You give us motivation for both mathematics and gymming ..XD
Thank you!
Very well said. Good video!
So to sum it all up. Whenever the gradient > 0 the graph is increasing. And when the gradient < 0 the graph is decreasing.
Thanks sir
دەستت خۆش❤
Thanks professor Leonard. A quick question, is PreCalculus the same as Advance Functions course? Thank you.
THANJ U SO MYCH
! I finally understand!!!!
Workout routine?
This is the easiest topic for me so far
Super sir thank you for your vedios it means a lot for me
Thanks professor Leonard, any recommendations on a textbook for a self learner?
For pre calculus
@@danielgarcia228 I.A. Maron
Could you make some videos of linear transformations?.
I LOVE YOU
Hello i am very bad at math and i want to start from the begging what vidoe or playlist should i start from ?
i hope you answer
hows it going
HELP!! At 1:48 wouldn’t it be X1 > X2 FOR DECREASING? Because if x1 is larger the
At e,and x2 is smaller...
As the input (x-value) increases, the output (y-value) decreases. That's what decreasing means. You go left to right checking whether the graph is falling (decreasing) or climbing (increasing). Two years late, but hope that clears it.
How are you doing now? Did you master Precalculus? Passed the class/test/exam? I hope you're doing alright.
complete
day 2
Can someone clear up some confusion I have? I've seen this a few times already.. near the end of the video when he is doing that last graph.. he assumes it to be a vertical asymptote. How can we assume that? The output of the function could very well pass 0 and go into the negatives right? we wouldn't know. Is there some fundamental knowledge of what the arrows mean on a line that I am missing? pls help :)
Dear Bro or Sis! u have a genuine question, however it needs some base. Keep reading.
sir didn't assume it a vertical asymptote, it actually is. yes we truly have some basic knowledge for the arrow notations: this arrow shows that the graph is approaching the y axis, but it is not( don't get tensed, you'll get it) . Let me simplify what am tryna' expressing, try feeling it! what is a vertical asymptote?
well, it's an *imaginary* line that cannot be crossed, but can be approached, mark it, can't be crossed but can be approached, just like u approach infinity. Now, the graph that we have is going to negative infinity vertically, but what will be the input or the x value for that? ( Now this is the point where u might have got a slap, open your eyes wide and read it with ur backbone as straight as possible) well, we can possibly observe that there is no y-intercept for x=0, so what does this reflect? again, WE CAN OBSERVE THAT THERE IS NO Y-INTERCEPT AS THE GRAPH ENDS WITH AN ARROW AS IT'S TAIL, SHOWING THAT "BYE!! MEET YOU AT INFINITY" , so what does this reflect? this reflects that for some x value the y value will be negative infinity, such that the x value is not negative ( READ ALOUD: the x value will not be negative because there is no y intercept, on a possible y value before infinity) so, it is your intuition that if, for example, x=1 and x=0.5 , then from the graph, we can see that for lesser input values, the output is approaching (learn this word by heart, thank me later) the y axis, so clearly at 0.5 that graph will be closer to the y axis. Now, at the same time , the y value for 0.5 will be closer to negative infinity, compared to the y value of 1, similarly 0.2 will be even closer to neg. inf. and 0.000000000000001 will again be closer to infinity than any of the above, similarly if x=0, will be negative infinity itself. Hence, we consider this a vertical asymptote because here the y value will be approaching negative infinity , which is impossible, and hence for this graph putting 0 is also approachable not reachable.
hence it is a vertical asymptote.
Sorry to make you mad. Read this word by word and you'll get it.
@@anamikajha7210 Hi thanks for the response, I appreciate it, however I'm not sure it answers my question. I think you are just explaining what an asymptote is, which I'm familiar with, but what I don't understand is how that is represented in his graph. Yes, I see that the arrow is pointing down to negative infinity, but there's no equation for the function so how can we assume that it never crosses x=0?
I'm sure it's just a quick representation of a specific function, but it seems to me that we can't actually know (from the representation of the graph alone and not any further assuming) whether or not that line passes x=0 or not. Does my question make sense?
@@alistaircornacchio5727 Yes, I was wondering about the same thing. I think the equation for the graph would actually make clear that it has a vertical asymptote, but it is not given here. Let me know if you have come across other satisfying explanations.
Hi. Prof...Why is the union signal placed in intervals of increasing and decreasing ?
I guess because they are sets of intervals
Wouldn't the last graph be (-infinity, infinity) ?
It never touches zero, so why include zero
(-infinity, infinity) would mean x goes towards -infinity and as you can see from that graph it never goes lower than 0, it just gets really really close to 0 but never reaching it (that's called an asymptote) which is why we have open parantheses (0, infinity)
Where are your students?
Keep up the good work
please help me to find the value of x
(e^-x)-x=0
There is no such x x is (pn) such that e to the power of x=0
@@timetobesmart5140 thankyou for your reply but i know X = 0.56714329 approximately but i don't know how to find this
Lalit Kumar oh
Да да, фактических величин.
I've been doing some extra reading about increasing and decreasing functions, and it seems to be the case that there are functions that can be classified as "strictly increasing" and functions that can be classified as just "increasing". Basically the best way I can define the strictly increasing relationship of a function is that it is a one-to-one (injective) function, that for every input x it holds the following condition - f(xₙ) < f(xₙ₊₁). However, for just increasing functions, the following condition holds true for all inputs x - f(xₙ) ≤ f(xₙ₊₁), basically meaning that a constant can be classified as an increasing function. Am I completely off base here?