Cannot thank you enough for these, you have given me hope I can actually pass PreCalc. My teacher could really use several novels out of your library of skills, because they seriously do not explain ANYTHING. I'd been afraid of math and trying for higher education in the STEM field because of people like that, and it is such a relief to see that I'm not stupid and I can understand if someone takes the time to actually explain it.
Prof Leonard- I am taking Diff Eq this quarter online and your videos are catching me up to speed. Thank you so much for everything you do. You've really carried me through calculus and beyond.
Professor Leonard, thank you for explaining and solving Inequalities with Rational Functions. Solving the examples in different ways should be helpful to all students in Precalculus, College Algebra and Calculus.
Thank you so much Professor Leonard, you are the best math teacher ever! I am glad that you are sharing your knowledge with us. I am very grateful for it!
At first glance of this comment, I had the same question! After some looking into, it appears that we're mixing up 2 distinct concepts: a sign change on the input (reflections across the 𝓎-axis) and a sign change on the output (reflections across the 𝓍-axis). For example, when given the function 𝒻(𝓍) = 1/(𝓍-1), we get an odd vertical asymptote at 𝓍=1, and this behaves how we expect. However, changing the *input* of 𝓍 to be 𝓍=-1 won't change the direction of the infinities, but reflects the graph across the 𝓎-axis. To get the same result that you're expecting, we'd need to change the *output* sign, like 𝒻(𝓍) = -1/(x-1). TL;DR, it's the difference between -𝒻(𝓍) and 𝒻(-𝓍). -𝒻(𝓍) is a reflection across the 𝓍. 𝒻(-𝓍) is a reflection across the 𝓎. These two functions are *also* related because if you can set them equal to each other, it means that the function itself is an odd function. This is distinct from having odd end behavior-you can have a function with odd end behavior that isn't odd. An odd function is simply symmetry about the origin. There's a lot of overlapping terms when dealing with math I've noticed, and having a clear picture means finding how each concept relates to the others. I find Leonard does a great job of this, generally speaking.
Hey Professor, I had a question that you may have touched on before So I've been graphing the functions after figuring out the inequalities just to see what the final products are, and I remember that you can find simplified versions of the functions without the vertical asymptotes and whatnot by just actually doing the long division (For example, with the equations starting at 27:38, if you simplified it, you'd get x^2-2x+2, or the parabola that matches over the original function). I was trying to do this with the last equation in the video and I came across an issue, I couldn't get a simplified version without it still being in fraction form. Is this normal for functions with an end behavior of 1/x or did I do something wrong? I was trying to do the long division and ended up getting something weird like (2/x+1/2x...), I've never divided in reverse like that before.
Thank you so much Prof. Kindly advice as to whether you do have any videos( or any site that you can recommend) on ANOVA and any experiment designs lectures?
Hey, can anyone else check for the solution to the ex given at 36:31? I am actually getting something completely different and even on Desmos, im getting: (-inf,-1) U (-1,0] U (1,2) Edit: Its ok, turns out I got the 'right' answer but misinterpreted f(x) >= 0
Cannot thank you enough for these, you have given me hope I can actually pass PreCalc. My teacher could really use several novels out of your library of skills, because they seriously do not explain ANYTHING. I'd been afraid of math and trying for higher education in the STEM field because of people like that, and it is such a relief to see that I'm not stupid and I can understand if someone takes the time to actually explain it.
Update: I got an A, thank you again so much!
@@HunterNapier congrats, im so happy for you 🥹
PROF LEONARD= AN ACTUAL LIFESAVER
Passed online calc thanks to this superman right here. Came to see if you're doing alright and Thank God you are!
Prof Leonard- I am taking Diff Eq this quarter online and your videos are catching me up to speed. Thank you so much for everything you do. You've really carried me through calculus and beyond.
Professor Leonard, thank you for explaining and solving Inequalities with Rational Functions. Solving the examples in different ways should be helpful to all students in Precalculus, College Algebra and Calculus.
I'm hurting for linear algebra right now, my first math class without you and its been painful
i feel it man
@@zayse7682 i know that feeling 😢
Hope you are doing well
Thank you so much for these videos Prof Leonard. Your hard work and knowledge helps a lot of people.
Thank you so much Professor Leonard, you are the best math teacher ever! I am glad that you are sharing your knowledge with us. I am very grateful for it!
Thank you thank you thank you! this helps so much!
I hope you are doing well professor leonard !
46:10 - the horizontal asymptote should be at y=1 (not y=3) because the rational inequality is not in 'normalized' form (
how would the negative odd vertical asymptote produce the same shape as a positive on 36:17?
At first glance of this comment, I had the same question! After some looking into, it appears that we're mixing up 2 distinct concepts: a sign change on the input (reflections across the 𝓎-axis) and a sign change on the output (reflections across the 𝓍-axis).
For example, when given the function 𝒻(𝓍) = 1/(𝓍-1), we get an odd vertical asymptote at 𝓍=1, and this behaves how we expect.
However, changing the *input* of 𝓍 to be 𝓍=-1 won't change the direction of the infinities, but reflects the graph across the 𝓎-axis.
To get the same result that you're expecting, we'd need to change the *output* sign, like 𝒻(𝓍) = -1/(x-1).
TL;DR, it's the difference between -𝒻(𝓍) and 𝒻(-𝓍). -𝒻(𝓍) is a reflection across the 𝓍. 𝒻(-𝓍) is a reflection across the 𝓎. These two functions are *also* related because if you can set them equal to each other, it means that the function itself is an odd function. This is distinct from having odd end behavior-you can have a function with odd end behavior that isn't odd. An odd function is simply symmetry about the origin.
There's a lot of overlapping terms when dealing with math I've noticed, and having a clear picture means finding how each concept relates to the others. I find Leonard does a great job of this, generally speaking.
Hey Professor, I had a question that you may have touched on before
So I've been graphing the functions after figuring out the inequalities just to see what the final products are, and I remember that you can find simplified versions of the functions without the vertical asymptotes and whatnot by just actually doing the long division (For example, with the equations starting at 27:38, if you simplified it, you'd get x^2-2x+2, or the parabola that matches over the original function).
I was trying to do this with the last equation in the video and I came across an issue, I couldn't get a simplified version without it still being in fraction form. Is this normal for functions with an end behavior of 1/x or did I do something wrong? I was trying to do the long division and ended up getting something weird like (2/x+1/2x...), I've never divided in reverse like that before.
Thank you so much Prof. Kindly advice as to whether you do have any videos( or any site that you can recommend) on ANOVA and any experiment designs lectures?
Wanna see where a function is greater or equal than another?
Go look where their difference is greater or equal than zero
Gotta love math ❤
I have a question Prof
U didn't mention wavy curve method?!?
Prof ❤
When is linear algebra coming??
Hey, can anyone else check for the solution to the ex given at 36:31? I am actually getting something completely different and even on Desmos, im getting:
(-inf,-1) U (-1,0] U (1,2)
Edit: Its ok, turns out I got the 'right' answer but misinterpreted f(x) >= 0
July 20 2022
February 27, 2023
March 27, 2023
@@dankster7993 Nice
May 19th, 2024
I wish live long
Moi marocco thank you brofsseur
now thats a successfull life
#edutubeeducation
You need put camera more near whiteboard because we see you from mobile
You still haven't covered trigonometry. This is crazy
Hi crazy
I'm eobard
Lmao this is all for free, he doesn't owe you anything
He did.