BONUS Question: In the step where we had the limit as h approaches 0 of (cos(h)-1)/h , what would the limit have been if there was no -1 term? Make sure to SUBSCRIBE HERE ⬇ so you don’t miss my tips and tricks for your next exam! tinyurl.com/numberninjadave Gear Up for the Dojo! 🥋 Before you enter the Number Ninja Dojo, you gotta look the part! Grab some awesome Number Ninja swag and show your dedication to mastering math. Shop the collection here: tinyurl.com/numberninjaswag (Affiliate Links Below - As an Amazon Associate I earn from qualifying purchases.) Level Up Your Math Skills with These Must-Haves: 📚 * Conquer Exam Anxiety! 😫➡😌 My book, “Hey Anxiety, Thank You!” is packed with proven strategies to help you master calculus, banish test-day jitters, and ace your exams by first taking care of YOU. Get your copy here: amzn.to/3Y2LWKv * Crush the AP Calculus Exam! 💥 This comprehensive study guide will equip you with the knowledge and practice you need to dominate the AP Calculus exam like a true ninja. Get yours here: amzn.to/3N5pjPm * The Ultimate Calculus Companion: 🧮 This graphing calculator was my trusty sidekick throughout college. It handles complex calculations with ease and never let me down. Get yours here: amzn.to/4eBNeRS * Precision Engineering: 📏 For precise measurements in engineering and other technical fields, this ruler is a must-have. It's durable, accurate, and built to last. Grab yours here: amzn.to/4doupRk Behind the Scenes: My TH-cam Toolkit 🛠 I use VidIQ to help create the best possible TH-cam videos for you! VidIQ helps me optimize my videos, research keywords, and understand what viewers are looking for. If you're a content creator, I highly recommend checking it out. You can sign up here (and I’ll get a small commission if you do): vidiq.com/numberninjadave
You need to explain where you get the sinh/h limit result to avoid circular reasoning because it is usually proven by L'hopital's which uses the derivative of sin. If you want to use that limit result without L'hopital, the next best thing is using the geometric argument+squeeze theorem method, so you might wanna explain that upfront, otherwise this feels a bit shady since the limit is not "easy".
It is not proven by LHopitals. LH rule comes from it. I’m aware of the circular reason and cover it in another video. But we need to be mindful that it’s only circular if you are using LH rule on the limit, which I’m not here. I communicated it’s simply a known squeeze limit theorem. Using the formal squeeze limit theorem to prove the limit is outside the scope of this video and would be a bit much to keep the video length like my typical lengths. I appreciate the comments though! I’ll consider making a dedicated video on that squeeze limit theorem result
That's how you get F. If using out of scope with "trust me bro" is allowed for no reason, then it's not even a simple proof - just use Taylor series, where you need no limit, no sin sums, just polynomial derivative.
How about this way of doing it? This requires Euler's formula as a starting point. The standard way to prove Euler's formula, is with Taylor series, which would be circular reasoning. But there are alternatives to Taylor series to prove it, that don't require us to know in advance how it differentiate trig functions. e^(i*x) = cos(x) + i*sin(x) e^(i*x) - e^(-i*x) = [cos(x) + i*sin(x)] - [cos(-x) - i*sin(-x)] By symmetry: cos(-x) = cos(x) sin(-x) = -sin(-x) Thus: e^(i*x) - e^(-i*x) = 2*i*sin(x) Solve for sin(x): sin(x) = -i/2 * [e^(i*x) - e^(-i*x)] Take derivative, using the derivative of e^x, along with the constant coefficient and chain rules: d/dx sin(x) = -i/2 * [i*e^(i*x) + i*e^(-i*x)] d/dx sin(x) = 1/2 * [e^(i*x) + e^(-i*x)] Looks familiar to what we had earlier for sine, except with a plus sign in the middle. Repeat the same process, adding exponential terms instead of subtracting them. e^(i*x) + e^(-i*x) = [cos(x) + i*sin(x)] + [cos(-x) - i*sin(-x)] e^(i*x) + e^(-i*x) = 2*cos(x) Rearrange to make it look like what we have, for d/dx sin(x): 1/2*[e^(i*x) + e^(-i*x)] = cos(x) Thus: d/dx sin(x) = cos(x), QED
@@AM-yk5yd My videos are broken into smaller tips and tricks and not full lectures. I’ll handle deciding what I want to keep in scope of a video. Are you here to actually learn? Because if you’re only here to troll, you can go do that elsewhere and not on my channel. I don’t think you read my responses in full, nor watched the video that answered your question on my channel (that’s your homework to find it). Constructive conversation is great. Trolling gets you removed.
BONUS Question: In the step where we had the limit as h approaches 0 of (cos(h)-1)/h , what would the limit have been if there was no -1 term?
Make sure to SUBSCRIBE HERE ⬇ so you don’t miss my tips and tricks for your next exam!
tinyurl.com/numberninjadave
Gear Up for the Dojo! 🥋 Before you enter the Number Ninja Dojo, you gotta look the part! Grab some awesome Number Ninja swag and show your dedication to mastering math. Shop the collection here: tinyurl.com/numberninjaswag
(Affiliate Links Below - As an Amazon Associate I earn from qualifying purchases.)
Level Up Your Math Skills with These Must-Haves: 📚
* Conquer Exam Anxiety! 😫➡😌 My book, “Hey Anxiety, Thank You!” is packed with proven strategies to help you master calculus, banish test-day jitters, and ace your exams by first taking care of YOU. Get your copy here: amzn.to/3Y2LWKv
* Crush the AP Calculus Exam! 💥 This comprehensive study guide will equip you with the knowledge and practice you need to dominate the AP Calculus exam like a true ninja. Get yours here: amzn.to/3N5pjPm
* The Ultimate Calculus Companion: 🧮 This graphing calculator was my trusty sidekick throughout college. It handles complex calculations with ease and never let me down. Get yours here: amzn.to/4eBNeRS
* Precision Engineering: 📏 For precise measurements in engineering and other technical fields, this ruler is a must-have. It's durable, accurate, and built to last. Grab yours here: amzn.to/4doupRk
Behind the Scenes: My TH-cam Toolkit 🛠
I use VidIQ to help create the best possible TH-cam videos for you! VidIQ helps me optimize my videos, research keywords, and understand what viewers are looking for. If you're a content creator, I highly recommend checking it out. You can sign up here (and I’ll get a small commission if you do): vidiq.com/numberninjadave
You need to explain where you get the sinh/h limit result to avoid circular reasoning because it is usually proven by L'hopital's which uses the derivative of sin.
If you want to use that limit result without L'hopital, the next best thing is using the geometric argument+squeeze theorem method, so you might wanna explain that upfront, otherwise this feels a bit shady since the limit is not "easy".
It is not proven by LHopitals. LH rule comes from it. I’m aware of the circular reason and cover it in another video. But we need to be mindful that it’s only circular if you are using LH rule on the limit, which I’m not here. I communicated it’s simply a known squeeze limit theorem.
Using the formal squeeze limit theorem to prove the limit is outside the scope of this video and would be a bit much to keep the video length like my typical lengths.
I appreciate the comments though! I’ll consider making a dedicated video on that squeeze limit theorem result
That's how you get F. If using out of scope with "trust me bro" is allowed for no reason, then it's not even a simple proof - just use Taylor series, where you need no limit, no sin sums, just polynomial derivative.
How about this way of doing it? This requires Euler's formula as a starting point. The standard way to prove Euler's formula, is with Taylor series, which would be circular reasoning. But there are alternatives to Taylor series to prove it, that don't require us to know in advance how it differentiate trig functions.
e^(i*x) = cos(x) + i*sin(x)
e^(i*x) - e^(-i*x) = [cos(x) + i*sin(x)] - [cos(-x) - i*sin(-x)]
By symmetry:
cos(-x) = cos(x)
sin(-x) = -sin(-x)
Thus:
e^(i*x) - e^(-i*x) = 2*i*sin(x)
Solve for sin(x):
sin(x) = -i/2 * [e^(i*x) - e^(-i*x)]
Take derivative, using the derivative of e^x, along with the constant coefficient and chain rules:
d/dx sin(x) = -i/2 * [i*e^(i*x) + i*e^(-i*x)]
d/dx sin(x) = 1/2 * [e^(i*x) + e^(-i*x)]
Looks familiar to what we had earlier for sine, except with a plus sign in the middle. Repeat the same process, adding exponential terms instead of subtracting them.
e^(i*x) + e^(-i*x) = [cos(x) + i*sin(x)] + [cos(-x) - i*sin(-x)]
e^(i*x) + e^(-i*x) = 2*cos(x)
Rearrange to make it look like what we have, for d/dx sin(x):
1/2*[e^(i*x) + e^(-i*x)] = cos(x)
Thus:
d/dx sin(x) = cos(x), QED
@@carultchnicely done!
@@AM-yk5yd My videos are broken into smaller tips and tricks and not full lectures. I’ll handle deciding what I want to keep in scope of a video.
Are you here to actually learn? Because if you’re only here to troll, you can go do that elsewhere and not on my channel.
I don’t think you read my responses in full, nor watched the video that answered your question on my channel (that’s your homework to find it).
Constructive conversation is great. Trolling gets you removed.
Math is very difficult 😮😢
It can be very tricky! But we can get through it!
photo on the cover is not the thing you solved. dy/dx(sinx) != dsinx/dx.
@@cankaankaya5001 y=sin x though, by substitution. But I appreciate the feedback. I’ll consider making the thumbnail even more clear.