Name: Classic solution to e^pi vs pi^e (no calculator) @blackpenredpen
Uploaded: 2026-09-18T08:17:00-05:00
Duration: 8 min 17 s
Description: Classic solution to e^pi vs pi^e (no calculator)

Classic solution to e^pi vs pi^e (no calculator)

blackpenredpen 2026-09-18 | Classic solution to e^pi vs pi^e (no calculator)

Hes 94 & still teaching calculus! @ycmathematicsphysicsandche5659

blackpenredpen 2024-05-10 | Thanks to the YouTube algorithm, I discovered Mr. Feng, a 94-year-old calculus teacher, @ycmathematicsphysicsandche5659 . He said in his note that teaching makes him happy and makes his life meaningful, which I resonate with so much. I want to make a video to introduce him to you! Please consider subscribing to his channel and supporting what he does. Thank you.
The solution to the derivative challenge problem: youtu.be/V9rrLp2CpPw?si=SadRcaYGZUVruIuj

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First time solving an A-Level maths exam! (90 minutes, uncut)

blackpenredpen 2024-04-29 | I will be doing a British A-Level further maths paper on the spot for the first time! This paper contains mainly algebra and calculus. Topics include complex numbers, hyperbolic equations, exponential equations, trigonometric identities and equations, first-order linear differential equations, determinant and the inverse of a 3x3 matrix, and more! Was I able to solve all the questions within 90 minutes? Was I able to get all the questions correct? Watch to find out.

The afterward youtu.be/9GFUyT-D6D0

Try the paper first: qualifications.pearson.com/content/dam/pdf/A-Level/Mathematics/2017/Exam-materials/9fm0-01-que-20220526.pdf

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Big thanks to my Patrons for the full-marathon support!
Ben D, Grant S, Erik S. Mark M, Phillippe S., Nicole D.
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Timestamps:
0:00 A-Level further math paper from June 2022
0:58 Q1 Solving a cubic equation with complex roots
6:23 Q2 Solving a 4-th degree hyperbolic equation
15:55 Q3 Solving a first-order linear differential equation with integrating factor
21:28 Q4 Solving a series problem with natural log
28:55 Q5 Finding the determinant and the inverse of a 3x3 matrix
40:50 Q6 Partial fraction decomposition & integral of a rational function
50:05 Q7 A complex number problem
1:00:11 Q8 A long trig identity
1:23:39 Q9 An improper integral with hyperbolic cosine

The notorious three-circle problem from the GCSE

blackpenredpen 2024-04-26 | Get started with a 30-day free trial on Brilliant: 👉brilliant.org/blackpenredpen ( 20% off with this link!)

Here's the last question from the 2022 GCSE maths paper that made the news. We have three circles as shown and each radius is 4 cm. We have to find the area of the shaded region in the middle. I made a horrible mistake last time when I said the area of a circular sector is r*theta. The correct formula should be A=1/2*r^2*theta and theta has to be in radians.

My first time doing a British GCSE maths paper: youtu.be/rJXfUiSwBzU?si=Ps79fuXWQ1xiQ_In
Reddit: reddit.com/r/theydidthemath/comments/12oholz/self_last_years_gcse_maths_question_that_made_the
The Sun News: thesun.co.uk/news/18647879/kids-left-baffled-impossible-gcse-question

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Big thanks to my Patrons for the full-marathon support!
Ben D, Grant S, Erik S. Mark M, Phillippe S.

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Solutions to the 2023 AP Calc AB FRQ

blackpenredpen 2024-04-16 | Get started with a 30-day free trial on Brilliant: 👉brilliant.org/blackpenredpen ( 20% off with this link!)

We will go over ALL the free response questions from the 2023 AP Calculus AB test to help you prepare for the upcoming AP exam. Topics included defined integral, average value of a function on an interval, change in position vs total distance traveled, implicit differentiation, related rates, derivative from a table, and more. Try all the questions first: apcentral.collegeboard.org/media/pdf/ap23-frq-calculus-ab.pdf and best of luck on your AP calculus test!

2 series problems from the AP Calc BC FRQ: youtu.be/pHSFB8_W3mc
Try this AP calculus BC FRQ from 2004: youtu.be/DHDC3oOy178?si=SpIiIByrPIJ6s615

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Big thanks to my Patrons for the full-marathon support!
Ben D, Grant S, Erik S. Mark M, Phillippe S.

💪 Get my math notes by becoming a patron: patreon.com/blackpenredpen
🛍 Shop my math t-shirt & hoodies: amzn.to/3qBeuw6
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#blackpenredpen #math #calculus #apcalculus

0:00 AP calculus exam is coming!
0:15 Thanks to Brilliant for sponsoring this video
1:13 Q1
15:41 Q2
27:39 Q3 differential equation
43:29 Q4
1:04:30 Q5
1:15:30 Q6

You see nonlinear equations, they see linear algebra! (Harvard-MIT math tournament)

blackpenredpen 2024-03-30 | Get started with a 30-day free trial on Brilliant: 👉brilliant.org/blackpenredpen ( 20% off with this link!)

This system of nonlinear equations is from the general round of the 2023 Harvard-MIT math tournament. hmmt.org/www/archive/271 I will present the linear algebra method I learned from their official solution to solve this system because I thought it was fascinating. It's from Harvard and MIT, of course, it is awesome!

3 ways of finding the determinant of a 3x3 matrix: youtube.com/watch?v=TkNeDxoRikY

Try "my first Harvard-MIT math tournament problem": youtu.be/jQuArBZO7PI?si=EvMgqNNDPHFZMppU
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Which is the worst math debate: 0^0, sqrt(1), 0.999...=1, or 12/3(4)?

blackpenredpen 2024-03-11 | These are the most debated math topics on the Internet but which one is the worst?
(A) 0 to the 0th power=1 or undefined. No calculus limit here.
(B) sqrt(1) = 1 or both +-1?)
(C) 0.999...=1 or not?
(D) order of operations 12/3(4)=1 or 16

More than 28,000 viewers voted in my recent poll and now let's discuss what each debate is all about.
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You wouldn’t expect this quadratic equation to have 6 solutions!

blackpenredpen 2024-03-03 | Surprisingly, the "quadratic" equation x^2+5abs(x)-6=0 has a total of 6 solutions (2 real and 4 complex solutions) which I did not expect. I came up with this equation purely by accident and I think it is super cool. It will feature complex numbers!

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#blackpenredpen #math #calculus #equation
Subscribe to @blackpenredpen for more fun math videos.

Using a logarithm property to make this equation easier!

blackpenredpen 2024-02-29 | Get started with a 30-day free trial on Brilliant: 👉brilliant.org/blackpenredpen ( 20% off with this link!)

We will solve this power equation x^ln(4)+x^ln(10)=x^ln(25). However, the process of solving this equation would require making one side equal to 0 but factoring every term by x^ln(4), and I don't think the expressions will look that nice. So instead, we can use the log property that a^logb(c)=c^logb(a) and change the equation to an exponential equation. The answer will involve the usage of the quadratic formula and we will also see the golden ratio.

Equation-of-the-year playlist: youtube.com/playlist?list=PLj7p5OoL6vGxHsQjgCw5Nfe8O3o8fu3LV

0:00 Math for fun, solving the power equation x^ln(4)+x^ln(10)=x^ln(25)
1:09 Proving a^logb(c)=c^logb(a)
3:18 Solving the equation in exponential form
8:35 Check out Brilliant

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#blackpenredpen #math #calculus #apcalculus

My first calculus 3 limit on YouTube

blackpenredpen 2024-02-20 | Get started with a 30-day free trial on Brilliant: 👉brilliant.org/blackpenredpen ( 20% off with this link!)

This is my first video on a multi-variable limit that you will see in your Calculus 3 class. We will evaluate the limit of y/x as (x,y) goes to (0,0) but how do we take care of this? We do get 0/0 indeterminate form but can we use L'Hopital's rule?

0:00 Limit of y/x as (x,y) goes to (0,0)
9:47 Check out today's sponsored, Brilliant!

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#blackpenredpen #math #calculus #apcalculus

Believe in geometry, not squaring both sides!

blackpenredpen 2023-12-27 | Get started with a 30-day free trial on Brilliant: 👉brilliant.org/blackpenredpen ( 20% off with this link!)

We will solve the equation that has two square roots sqrt(16-x^2)+sqrt(9-x^2)=5 with a cool geometric triangle trick. In the end, we will also see if ChatGPT can solve this equation by using geometry. Check out a similar video on the proof of the Pythagorean Theorem and its inverse: youtu.be/itfXCKpmQPs

0:00 Solving sqrt(16-x^2)+sqrt(9-x^2)=5 by using geometry
5:15 Check out Brilliant
6:13 Can ChatGPT solve this with geometry?

🛍 Shop my math t-shirt & hoodies: amzn.to/3qBeuw6
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AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
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Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
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Solution of the transcendental equation a^x+bx+c=0

blackpenredpen 2023-12-19 | Learn contest math on Brilliant: 👉brilliant.org/blackpenredpen (now with a 30-day free trial plus 20% off with this link!)

Today we will find a general solution to the transcendental equation a^x+bx+c=0. Here's an example 2^x-3x-1=0 youtu.be/GJbzsmccFtw
Check out a detailed the Lambert W function introduction: youtu.be/Qb7JITsbyKs?si=bKnZW_lqlDjiKQuf

0:00 solve a^x+bx+c=0
7:02 Check out Brilliant
8:06 bonus part (the conditions on this formula)

🛍 Shop the Lambert W function t-shirt: amazon.com/dp/B0C8FLMTFC
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💪 Support the channel and get featured in the video description by becoming a patron: patreon.com/blackpenredpen
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt Zahra Parhoun Hassan Alashoor Kaakaopuupod bbaa Joash Hall Andr3w11235 Cadentato Joe Wisniewski Eric Maximilian Mecke Jorge Casanova Alexis Villalobos Jm Law Siang Qi Tancredi Casoli Steven Sea Shanties Nick K Daniel Akheterov Roy Logan Abdennor Bel Tobias V Killbot2525 Stephen Ripple Spam Email YKTV # bruh Amanda Skelton
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My failed attempts to the integral of sqrt(sin^2(x))

blackpenredpen 2023-12-14 | I have been trying to integrate sqrt(sin^2(x)) and get the same answer as on WolframAlpha, which is -sqrt(sin^2(x))*cot(x)+C, but I just can't. So let me show you what I have tried (including how to integrate sqrt(x^2) with integration by parts) and maybe some of you will have an idea!

🛍 Shop Ultimate calc 2 integral sweatshirt: amazon.com/dp/B0C9L798F2
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💪 Support the channel and get featured in the video description by becoming a patron: patreon.com/blackpenredpen
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt Zahra Parhoun Hassan Alashoor Kaakaopuupod bbaa Joash Hall Andr3w11235 Cadentato Joe Wisniewski Eric Maximilian Mecke Jorge Casanova Alexis Villalobos Jm Law Siang Qi Tancredi Casoli Steven Sea Shanties Nick K Daniel Akheterov Roy Logan
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Thank you all!

if x+y=8, find the max of x^y

blackpenredpen 2023-12-09 | If x+y=8 for nonnegative x and y, then find the maximum value of x^y. This question seems like a typical calculus optimization problem, but to find the exact answer, we will need to use the Lambert W function. youtu.be/Qb7JITsbyKs?si=lc0IZ6aTrvewRBIh
Check out my 100 calculus 2 problems to help you with your calc 2 final: youtu.be/Kwyk_mteyNc?si=Dj_3rv2qeen7SiMi

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#blackpenredpen #math #calculus #apcalculus

Calculus teacher vs LHopitals rule students

blackpenredpen 2023-12-01 | If you are also a calculus teacher, then you know most students will just use L'Hospital's rule for every limit after they learn the rule. So let's investigate the limit as x goes to infinity of (the integral from 0 to x of sin(t)*e^cos(t))/x and see why we cannot use L'Hôpital's rule. This is a great challenging limit problem for your calculus students! #calculus #blackpenredpen #challenge

Related videos:
youtu.be/VAScsSpfKv8?si=UIO7P0ccOAUnxQCs
youtube.com/watch?v=57Eawy5IAaM

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💪 Support the channel and get featured in the video description by becoming a patron: patreon.com/blackpenredpen
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt Zahra Parhoun Hassan Alashoor Kaakaopuupod bbaa Joash Hall Andr3w11235 Cadentato Joe Wisniewski Eric Maximilian Mecke Jorge Casanova Alexis Villalobos Jm Law Siang Qi Tancredi Casoli Steven Sea Shanties Nick K Daniel Akheterov Roy Logan
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Thank you all!

combining rational exponents, but using calculus,

blackpenredpen 2023-11-22 | Learn more calculus on Brilliant: 👉 brilliant.org/blackpenredpen (now with a 30-day free trial plus 20% off with this link!)

The usual exponent rules are pretty easy to see when the exponents are whole numbers. For example, we know e^2 means e*e and e^3 means e*e*e, so e^2*e^3=e*e*e*e*e*e=e^5. But what if we have rational exponents such as e^(1/2)*e^(1/3)? How do we prove the rule of exponent in this case is equal to e^(5/6)?

Proving e^x*e^y=e^(x+y) by power series: youtu.be/r87AfxUwD60
Deriving the power series of e^x centered at 0: youtu.be/1OhMpXFb6yI?si=2u-5z1KINQPLz6sK
Check out the binomial theorem: youtu.be/cvhyJT9c0ac?si=2brCzT8yH5uAslkd

The Power Series For You merch
t-shirt 👉 amzn.to/47EXwNF
hoodie 👉 amzn.to/46oCpyj

0:00 We know e^2*e^3=e^(2+3), but what if we have e^(1/2)*e^(1/3)?
1:12 Proving e^(1/2)*e^(1/3) by using power series
3:14 Cauchy product of two infinite series
7:38 Back to the proof
14:42 Check out Brilliant

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💪 Support the channel and get featured in the video description by becoming a patron: patreon.com/blackpenredpen
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt Zahra Parhoun Hassan Alashoor Kaakaopuupod bbaa Joash Hall Andr3w11235 Cadentato Joe Wisniewski Eric Maximilian Mecke Jorge Casanova Alexis Villalobos Jm Law Siang Qi Tancredi Casoli Steven Sea Shanties Nick K Daniel Akheterov Roy Logan
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Thank you all!

easy derivative but it took me 32 minutes

blackpenredpen 2023-11-18 | We will find the second derivative of sin(x^2) but without the chain rule, power rule, and product rule. Instead, we will just need ONE limit! This is a hidden gem for limit enthusiasts! The key is to use the limit definition of the second symmetric derivative to find the second derivative of sin(x^2). Note, that if the function is second-differentiable in the usual sense, then the second symmetric derivative is equal to the regular second derivative. Since sin(x^2) is always differentiable, so let's find d^2/dx^2(sin(x^2)) with this rarely-taught approach. We will also need two famous limits, the limit as x goes to 0 of sin(x)/x and the limit as x goes to 0 of (1-cos(x))/x.

0:00 finding the second derivative of sin(x^2) but no differentiation shortcuts
0:54 Definition of derivative, symmetric derivative, and second symmetric derivative in terms of limits
7:46 The real action!!
9:08 The formula for sin(A)+sin(B)
12:01 Back to the limit
29:50 Check the answer!

Check out my 100-limit video for more calculus limit practice. youtu.be/TglD4Y6lmQk?si=lR_jfa-gI8FVO7VS
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#blackpenredpen #math #calculus #apcalculus
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💪 Support the channel and get featured in the video description by becoming a patron: patreon.com/blackpenredpen
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt Zahra Parhoun Hassan Alashoor Kaakaopuupod bbaa Joash Hall Andr3w11235 Cadentato Joe Wisniewski Eric Maximilian Mecke Jorge Casanova Alexis Villalobos Jm Law Siang Qi Tancredi Casoli Steven Sea Shanties Nick K Daniel Akheterov Roy Logan
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Thank you all!

My all-in-one calculus problem

blackpenredpen 2023-11-14 | Learn more calculus on Brilliant: 👉brilliant.org/blackpenredpen (now with a 30-day free trial plus 20% off with this link!)

I made this all-in-one style calculus problem for you as an early Christmas gift. We will find the derivative of sin^2(x^2), which requires the chain rule twice, then we need to find a closed form for the infinite power series 1+x^2+x^4/2+x^6/6+..., then we have the limit of sqrt(x)/ln(x) and the limit of ln(x)/sqrt(x) as x goes to infinity. Finally, we will put everyone together and integrate it!

#calculus #math #challenge #blackpenredpen
🛍 Shop my math t-shirt & hoodies: amzn.to/3qBeuw6

0:00 Christmas is coming, so I made an all-in-one calc 2 problem or you
0:20 limit of ln(x)/sqrt(x) as x goes to infinity
1:45 derivative of sin^2(x^2), chain rule twice!
2:57 Power series for 1+x^2+x^4/2+x^6/6+...
4:00 solving the integral

----------------------------------------
💪 Support the channel and get featured in the video description by becoming a patron: patreon.com/blackpenredpen
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt Zahra Parhoun Hassan Alashoor Kaakaopuupod bbaa Joash Hall Andr3w11235 Cadentato Joe Wisniewski Eric Maximilian Mecke Jorge Casanova Alexis Villalobos Jm Law Siang Qi Tancredi Casoli Steven Sea Shanties Nick K Daniel Akheterov Roy Logan
----------------------------------------
Thank you all!

solving x^x=1 but x is not real!

blackpenredpen 2023-11-09 | How to solve x^x=1? We know 1^1=1 and 0^0 has no agreement (some say 1, some say undefined) but you are about to learn that (2.213...+3.113...i)^(2.213...+3.113...i) will also give you 1. Today we will solve a basic tetration equation x^x=1 but we will use the Lambert W function and get the complex solutions! Check out another cool exponential equation e^e^x=1: youtu.be/ckc9F0VjZ3k?si=m91azalG4twF0nTo

🛍 Shop my math t-shirt & hoodies: amzn.to/3qBeuw6
----------------------------------------
💪 Support the channel and get featured in the video description by becoming a patron: patreon.com/blackpenredpen
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt Zahra Parhoun Hassan Alashoor Kaakaopuupod bbaa Joash Hall Andr3w11235 Cadentato Joe Wisniewski Eric Maximilian Mecke Jorge Casanova Alexis Villalobos Jm Law Siang Qi Tancredi Casoli Steven Sea Shanties Nick K Daniel Akheterov Roy Logan
----------------------------------------
Thank you all!

the sine triangle problem

blackpenredpen 2023-11-03 | Can we find x so that we have sin(x), sin(2x), and sin(3x) on a right triangle? Yes! I actually have done the log triangle, exp triangle, so today this is the sine triangle for you! This is a perfect problem to challenge any precalculus or even calculus student!

Subscribe to @blackpenredpen for more fun math lessons.
#calculus #trigonometry #math #mathforfun #blackpenredpen

Check out the related videos:
log triangle problem: youtu.be/CMdJPwEbE8A
exp triangle problem: youtu.be/qKytYveIOmA?si=tujFerP3CccFKJrV
triple angle identity for sine: youtu.be/aSq9vwGHLTg

🛍 Shop my math t-shirt & hoodies: amzn.to/3qBeuw6
----------------------------------------
💪 Support the channel and get featured in the video description by becoming a patron: patreon.com/blackpenredpen
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt Zahra Parhoun Hassan Alashoor Kaakaopuupod bbaa Joash Hall Andr3w11235 Cadentato Joe Wisniewski Eric Maximilian Mecke Jorge Casanova Alexis Villalobos Jm Law Siang Qi Tancredi Casoli Steven Sea Shanties Nick K Daniel Akheterov Roy Logan
----------------------------------------
Thank you all!

I couldnt solve x^x^x=2, so I solved x^x^(x+1)=2 instead

blackpenredpen 2023-10-29 | We will solve a tetration of a tetration equation x^x^(x+1)=2, which becomes (x^x)^(x^x)=2. This equation has one real solution and it's solvable by using the Lambert W function. Check out how I used Newton's method to get the approximated root to x^x^x=2 youtu.be/ef-TSTg-2sI?si=jNBw8TXYrftVIKb5

What’s the Lambert W function? youtu.be/Qb7JITsbyKs?si=05xAbHwlaj9ChD_W

🛍 Shop my math t-shirt & hoodies: amzn.to/3qBeuw6
#blackpenredpen #math #mathforfun
----------------------------------------
💪 Support the channel and get featured in the video description by becoming a patron: patreon.com/blackpenredpen
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt Zahra Parhoun Hassan Alashoor Kaakaopuupod bbaa Joash Hall Andr3w11235 Cadentato Joe Wisniewski Eric Maximilian Mecke Jorge Casanova Alexis Villalobos Jm Law Siang Qi Tancredi Casoli Steven Sea Shanties Nick K Daniel Akheterov Roy Logan
----------------------------------------
Thank you all!

Beyond the Gaussian integral

blackpenredpen 2023-10-27 | Learn more calculus on Brilliant: 👉brilliant.org/blackpenredpen (now with a 30-day free trial plus 20% off with this link!)

Here's a challenging integral for Calculus 2 students! We will evaluate the improper integral 1-e^(-1/x^2) from 0 to infinity. This integral actually converges thanks to the "1-". This integral is closely related to the most famous integral in the world, the Gaussian integral. In fact, we can use integration by parts, u-substitution, and L'Hosptial's rule to make the connection with the Gaussian integral. This can be given to any calculus 2 or calculus 3 students if they would like to have a challenging integral. #calculus #mathforfun #blackpenredpen

Check out the related videos:
How Laplace solved the Gaussian integral, 👉 youtu.be/tCPQSobqFh4?si=eCeLOIRLl9CrL7Lv
The original way with polar coordinate 👉 youtu.be/7kvxMrbPvhs?si=YCjkwB1g5iMPa7o1
The error function introduction 👉 youtu.be/jkytxdedxhU?si=cJJfu1ibrgrLjIlV

0:00 My pick for the most famous integral in the world
0:14 integral of 1-e^(-1/x^2) from 0 to inf
9:17 Check out Brilliant

🛍 Shop my math t-shirt & hoodies: amzn.to/3qBeuw6
----------------------------------------
💪 Support the channel and get featured in the video description by becoming a patron: patreon.com/blackpenredpen
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt Zahra Parhoun Hassan Alashoor Kaakaopuupod bbaa Joash Hall Andr3w11235 Cadentato Joe Wisniewski Eric Maximilian Mecke Jorge Casanova Alexis Villalobos Jm Law Siang Qi Tancredi Casoli Steven Sea Shanties Nick K Daniel Akheterov Roy Logan
----------------------------------------
Thank you all!

all solutions to 2^x-3x-1=0 (transcendental equation)

blackpenredpen 2023-10-23 | Learn how to solve this transcendental equation 2^x-3x-1=0 with the Lambert W function! This is definitely not a regular math equation that you see in school!
Subscribe to @blackpenredpen so you don't miss the "fixed point iteration" video!

Lambert W function introduction: youtu.be/Qb7JITsbyKs?si=5z1DnLDsqUgGsocc
For another challenge, check out this video: youtu.be/Rg3dBosfZ3Y?si=Zmdy1Y_YnJJOFCPW

🛍 Shop my math t-shirt & hoodies: amzn.to/3qBeuw6
----------------------------------------
💪 Support the channel and get featured in the video description by becoming a patron: patreon.com/blackpenredpen
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt Zahra Parhoun Hassan Alashoor Kaakaopuupod bbaa Joash Hall Andr3w11235 Cadentato Joe Wisniewski Eric Maximilian Mecke Jorge Casanova Alexis Villalobos Jm Law Siang Qi Tancredi Casoli Steven Sea Shanties Nick K Daniel Akheterov Roy Logan
----------------------------------------
Thank you all!
#algebra #math #blackpenredpen

finally 0^0 approaches 0 after 6 years!

blackpenredpen 2023-10-18 | I finally came up with an example of a limit with the indeterminate form 0^0 and it approaches 0. Here's the video from 2017 "Can 0^0 approach 0?" youtu.be/Gcl_9KIdpso?si=3dpKINUkFaxgXdka

Check out 0^0 approaches e: youtu.be/zyQtN7Zbi-c

🛍 Shop my math t-shirt & hoodies: amzn.to/3qBeuw6
----------------------------------------
💪 Support the channel and get featured in the video description by becoming a patron: patreon.com/blackpenredpen
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt Zahra Parhoun Hassan Alashoor Kaakaopuupod bbaa Joash Hall Andr3w11235 Cadentato Joe Wisniewski Eric Maximilian Mecke Jorge Casanova Alexis Villalobos Jm Law Siang Qi Tancredi Casoli Steven Sea Shanties Nick K Daniel Akheterov Roy Logan
----------------------------------------
Thank you all!
#calculus #math #mathforfun #blackpenredpen

I want all trig functions in one integral!

blackpenredpen 2023-10-14 | Learn more calculus on Brilliant: 👉brilliant.org/blackpenredpen (now with a 30-day free trial plus 20% off with this link!)

how calculus teachers create integrals for your test! I took me about an hour to craft this integral which features all the trigonometric functions, the natural logarithm function, the pi-th power, and a derivative for the denominator. This integral looks scary but the solution is actually very straightforward! Enjoy! #calculus #math #trigonometry #blackpenredpen

Check out my 100 integrals for more integration practice for your Calculus 1 or Calculus 2 class. youtu.be/dgm4-3-Iv3s?si=lTybJlpTMFdQINXr

0:00 how calculus teachers create integrals for your test
6:30 check out Brilliant:
7:25 BOOM!

🛍 Shop my math t-shirt & hoodies: amzn.to/3qBeuw6
----------------------------------------
💪 Support the channel and get featured in the video description by becoming a patron: patreon.com/blackpenredpen
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt Zahra Parhoun Hassan Alashoor Kaakaopuupod bbaa Joash Hall Andr3w11235 Cadentato Joe Wisniewski Eric Maximilian Mecke Jorge Casanova Alexis Villalobos Jm Law Siang Qi Tancredi Casoli Steven Sea Shanties Nick K Daniel Akheterov Roy Logan
----------------------------------------
Thank you all!

Berkeley Math Tournament calculus tiebreaker

blackpenredpen 2023-10-11 | To sign up for the Berkeley Math Tournament (BMT), please visit: https://berkeley.mt/
If you would also like to sponsor BMT, please visit: https://berkeley.mt/sponsors/

This calculus tutorial contains the solution to the 2022 Berkeley Math Tournament calculus tiebreaker! It contains a limit, derivative, and an integral at calculus 2 (i.e. AP calculus BC) level. The time limit is 15 minutes but you will also have to race with other times during the tournament. I love these math competition problems because they really force you to think and they really challenge your problem-solving skills. It's okay if you can't solve these competition problems. I, too, can't solve many of them either. It really takes lots of effort and practice in order to be good at these. Finally, I just want to tell you, to be sure to enjoy the learning process and have fun!

0:00 2022 Berkeley Math Tournament calculus tiebreaker
0:08 Q1, evaluate a limit with an integral
3:10 Q2, find the 2022nd derivative of f(x)=e^x*sin(x) at x=0
7:36 Q3, evaluate the integral of arctan(x)/x from 1/e to e
#calculus #tutorial #berkeley #blackpenredpen

🛍 Shop my math t-shirt & hoodies: amzn.to/3qBeuw6
----------------------------------------
💪 Support the channel and get featured in the video description by becoming a patron: patreon.com/blackpenredpen
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt Zahra Parhoun Hassan Alashoor Kaakaopuupod bbaa Joash Hall Andr3w11235 Cadentato Joe Wisniewski Eric Maximilian Mecke Jorge Casanova Alexis Villalobos Jm Law Siang Qi Tancredi Casoli Steven Sea Shanties Nick K Daniel Akheterov Roy Logan
----------------------------------------
Thank you all!

I integrated the quadratic formula

blackpenredpen 2023-10-07 | Ultimate math for fun! Last year we differentiated the quadratic formula, so today we will integrate the quadratic formula! This is my first time integrating the quadratic formula and I was surprised that the answer turned out to be pretty nice! This is a super fun problem to challenge any Calculus 2 student. In the end, we will check if ChatGPT can integrate the quadratic formula for us.
Check out how to differentiate the quadratic formula: youtu.be/JEcE-wDRMCk?si=Uq1p1KYNj-Kpu5g2

0:00 We will integrate the quadratic formula
7:48 ChatGPT vs integral of the quadratic formula

🛍 Shop the Gaussian Integral t-shirt: amzn.to/45jTJnt
💪 Get my math notes by becoming a patron: patreon.com/blackpenredpen
#blackpenredpen #math #calculus #apcalculus #tutorial

💪 Support the channel and get featured in the video description by becoming a patron: patreon.com/blackpenredpen
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt Zahra Parhoun Hassan Alashoor Kaakaopuupod bbaa Joash Hall Andr3w11235 Cadentato Joe Wisniewski Eric Maximilian Mecke Jorge Casanova Alexis Villalobos Jm Law Siang Qi Tancredi Casoli Steven Sea Shanties Nick K Daniel Akheterov Roy Logan
----------------------------------------
Thank you all!

how is e^e^x=1 solvable??

blackpenredpen 2023-09-25 | The exponential equation e^x=0 has no solutions, not even in the complex world, but e^e^x=1 does have solutions! I was surprised to see how WolframAlpha actually gave the solutions to this seemingly impossible equation and I would like to show you how to solve it! The trick is to write 1 in the complex polar form. Subscribe to @blackpenredpen for more math for fun videos. #math #complexnumbers #blackpenredpen #fun #tutorials

🛍 Euler's Identity e^(iπ)+1=0 t-shirt: amzn.to/427Seae
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AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt Zahra Parhoun Hassan Alashoor Kaakaopuupod bbaa Joash Hall Andr3w11235 Cadentato Joe Wisniewski Eric Maximilian Mecke Jorge Casanova Alexis Villalobos Jm Law Siang Qi Tancredi Casoli Steven Sea Shanties Nick K Daniel Akheterov
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Thank you all!

the first 5 days of calculus 2 but the integrals get harder!

blackpenredpen 2023-09-19 | Learn more calculus on Brilliant (now with a 30-day free trial): 👉brilliant.org/blackpenredpen (20% off with this link!)

Today we will see what your first 5 days in your calculus 2 or AP calculus BC class is like. Calculus 2 is known as integral calculus and we usually start with learning all the integration techniques. We will cover the integral of x^2 with the reverse power rule, the integral of x*e^(x^2) with u-substitution, the integral of x^2*e^x with integration by parts, the integration of sqrt(1+x^2) with trigonometric substitution, and the integration of (2x+1)/(x^2+3x+2) with partial fraction decomposition.

0:00 day 1, reverse power rule
0:20 day 2, u sub
1:27 day 3, integration by parts
2:29 day 4, trig sub
6:16 day 5, partial fraction decomposition
8:51 Check out Brilliant

🛍 Shop my math t-shirt & hoodies: amzn.to/3qBeuw6
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💪 Support the channel and get featured in the video description by becoming a patron: patreon.com/blackpenredpen
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt Zahra Parhoun Hassan Alashoor Kaakaopuupod bbaa Joash Hall Andr3w11235 Cadentato Joe Wisniewski Eric Maximilian Mecke
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Thank you all!

limit of 1/(1-cos(ln(sin(tan^-1(e^(x^2)))))) as x goes to infinity

blackpenredpen 2023-09-13 | As a calculus teacher, I love creating my own calculus problems and this crazy limit problem is my creation today. I wanted to put this on my Calculus 1 quiz this week but I decided not to because it's only week 3. So I am making a video on it! We have the limit of 1/(1-cos(ln(sin(tan^-1(e^(x^2)))))) as x goes to infinity. What's the answer?

#blackpenredpen #math #calculus #tutorial
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💪 Support the channel and get featured in the video description by becoming a patron: patreon.com/blackpenredpen
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt Zahra Parhoun Hassan Alashoor Kaakaopuupod bbaa Joash Hall Andr3w11235 Cadentato Joe Wisniewski Eric Maximilian Mecke
----------------------------------------
Thank you all!

how to simplify trigonometric identity problems

blackpenredpen 2023-09-05 | Master simplifying and proving trigonometric identities by working along with these 24 trig identity problems. I put the problems in the multiple-choice format so the problems are more challenging. Moreover, we will also derive the triple-angle identity, the quadruple-angle identity for cosine, i.e. cos(3x) and cos(4x), and the power reduction identity for sin^4(x). Mastering these problems will prepare you for your calculus class, especially when we do derivatives and integrals.
#trigonometry #math #precalculus

File 👉 patreon.com/posts/24-trig-identity-88803595
Solutions 👉 patreon.com/posts/my-hand-written-88804184
🛍 Shop my math t-shirt & hoodies: amzn.to/3qBeuw6

Thanks to @takureido3122 & @sohampinemath1086 for the time stamps!!!
Intro - 00:00
1 - 00:51 — sinx+cotxcosx
2 - 04:40 — (secx-cosx)/sinx
3 - 07:16 — cotx/(cscx-sinx)
4 - 09:00 — (1+2cosx)/(2+secx)
5 - 10:45 — 1/(1-sinx)+1/(1+sinx)
6 - 13:00 — (2+cot²x)/csc²x-1
7 - 15:36 — tan(x+π/4)
8 - 19:11 — cos(3x)
9 - 24:09 — cos(4x)
10 - 29:32 — sec(sin^-1x)
11 - 32:42 — cos(2tan^-1x)
12 - 36:56 — tan(2sin^-1x)
13 - 42:44 — csc²x+sec²x
14 - 45:41 — cosx cos(2x)
15 - 49:38 — sin⁴x
16 - 57:22 — sin⁴x-cos⁴x
17 - 1:02:04 — tanx+tany
18 - 1:04:47 — 2tanx/(1+tan²x)
19 - 1:07:25 — 1/(secx-1)+1/(secx+1)
20 - 1:12:08 — (sinx+tanx)/(1+cos(-x))
21 - 1:16:14 — cos²x-sin⁴x sec²x
22 - 1:14:50 — (sinx+cosx)²
23 - 1:20:18 — sinx/(1-cotx)+cosx/(1+tanx)
24 - 1:23:57 — sec²(x/2)
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💪 Support the channel and get featured in the video description by becoming a patron: patreon.com/blackpenredpen
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt Zahra Parhoun Hassan Alashoor Kaakaopuupod bbaa Joash Hall Andr3w11235 Cadentato Joe Wisniewski Eric Maximilian Mecke
----------------------------------------
Thank you all!

How to do epsilon-delta proofs (ultimate calculus guide)

blackpenredpen 2023-09-02 | This is the ultimate calculus study guide for your university-level calculus and real analysis class. We will do 24 rigorous proofs for limits, including the epsilon-delta proofs, epsilon-N proofs (when x approaches infinity), M-delta (when the limit is infinity), and M-N proofs (when x goes to infinity and f(x) also goes to infinity) for limits. It's definitely the GPA saver that you need for college. Make sure you know the 4 big rigorous definitions for limits here: 👉 youtu.be/onrBRLXefuQ
File (public): 👉 patreon.com/posts/24-rigorous-88483708
My hand-written notes for all these 24 proofs: (Patreons only) 👉patreon.com/posts/notes-for-24-88670205

Join this channel to get access to the perks:
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0:00 24 limit proofs with definitions
Q1. 0:29 epsilon-delta proof, linear case
Q2. 3:51 epsilon-delta proof, rational function
Q3. 12:04 epsilon-N proof, x approaching infinity, rational function
Q4. 17:18 M-delta proof, x approaching infinity, rational function
Q5 22:19 epsilon-N proof
Q6 28:34 epsilon-delta proof, quadratic case
Q7 32:26 M-N proof, square root function
Q8 36:06 epsilon-delta proof, square root function
Q9 42:47 εδ linear case
Q10 45:00 εδ limit proof, *hard*
Q11 52:21 limit of 1/x^2 as x goes to infinity
Q12 55:17 limit of sqrt(x-3) as x goes to infinity
Q13 58:24 limit of x^3 as x goes to 2
Q14 1:04:43 εδ linear case
Q15 1:06:40 εN proof
Q16 1:15:50
Q17 1:19:00
Q18 1:21:50
Q19: 1:26:46
Q20: 1:30:07
*start at 1:31:41, please change all the ε to M
Q21 1:32:24
Q22 1:36:42
Q23 1:41:21
*at 1:42:28, it should be 1/abs(x-3) is greater than 1/δ. And then say 1/(abs(x-3))^2=1/(x-3)^2 is greater than 1/δ^2
Q24 1:44:15 limit of x^2/(x+1) as x goes to infinity
*at 1:48:26, we need to add the condition that N is greater than 1. So please write N=max{1, 2ε}
----------------------------------------
💪 Support the channel and get featured in the video description by becoming a patron: patreon.com/blackpenredpen
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt Zahra Parhoun Hassan Alashoor Kaakaopuupod bbaa Joash Hall Andr3w11235 Cadentato Joe Wisniewski Eric Maximilian Mecke
----------------------------------------
Thank you all!

#calculus #GPAsaver #math #blackpenredpen

4 BIG limit definitions

blackpenredpen 2023-08-31 | Here are the 4 rigorous definitions for limits (epsilon-delta, epsilon-N, M-delta, M-N definitions) that you need to know if you are taking University-level calculus and real analysis. We will start with the infamous epsilon-delta definition of a limit, followed by the other variations for limits involving infinities. I will point out the difference between these definitions and also explain how to interpret the math symbols. Check out my playlist on how to write a rigorous proof for a limit: youtube.com/playlist?list=PLj7p5OoL6vGwzCW5N_hyGa88BzGQHv1ax

#calculus #math #blackpenredpen

🛍 Shop my math t-shirt & hoodies: amzn.to/3qBeuw6
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💪 Support the channel and get featured in the video description by becoming a patron: patreon.com/blackpenredpen
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt Zahra Parhoun Hassan Alashoor Kaakaopuupod bbaa Joash Hall Andr3w11235 Cadentato Joe Wisniewski Eric Maximilian Mecke
----------------------------------------
Thank you all!

National Taiwan University graduate school entrance exam problem

blackpenredpen 2023-08-26 | Learn calculus on Brilliant: 👉brilliant.org/blackpenredpen (20% off with this link!)

The integral of x*sin(x)/(1+cos^2(x)) from 0 to pi is actually a famous exam problem integral. I saw a post on Dcard about this integral and I just had to integrate it. I use the property that the integral of f(x) from a to b is the same as the integral of f(a+b-x) from a to b. This is commonly known as the king property and I have a proof here: youtu.be/9xaGYqiOkPM?si=eXQuXRoHdvUbI2CQ

#calculus #math #maths #exam
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🛍 Shop my math t-shirt & hoodies: amzn.to/3qBeuw6
----------------------------------------
💪 Support the channel and get featured in the video description by becoming a patron: patreon.com/blackpenredpen
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt Zahra Parhoun Hassan Alashoor Kaakaopuupod bbaa Joash Hall Andr3w11235 Cadentato Joe Wisniewski Eric Maximilian Mecke
----------------------------------------
Thank you all!

epsilon-N definition for a limit at infinity (introduction & proof writing tips)

blackpenredpen 2023-08-23 | I will introduce the epsilon-N definition of a limit and will also show you how to write a rigorous proof for a limit as x goes to infinity. I will help you understand the definition by using a specific epsilon value, graphs of the function, and the line y=L-epsilon, and solving for a corresponding N. And remember "given, choose, suppose, and check".
#calculus #math #maths #college #blackpenredpen

This is similar to the epsilon-delta proofs for limits and you can check out that introduction here: 👉 youtu.be/DdtEQk_DHQs

0:00 I will help you understand the εN definition for a finite limit at infinity
0:19 the 4 main cases of a rigorous definition of a limit
2:24 the εN definition
4:14 how to easily write a rigorous limit proof
11:32 an actual example with ε=0.02 and find a N

#calculus #blackpenredpen #realanalysis #math #college

🛍 Shop my math t-shirt & hoodies: amzn.to/3qBeuw6
----------------------------------------
💪 Support the channel and get featured in the video description by becoming a patron: patreon.com/blackpenredpen
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt Zahra Parhoun Hassan Alashoor Kaakaopuupod bbaa Joash Hall Andr3w11235 Cadentato Joe Wisniewski Eric Maximilian Mecke
----------------------------------------
Thank you all!

Regular math vs competition math

blackpenredpen 2023-07-20 | Let's try a regular math problem and a Princeton competition math problem.
Regular math: Compute the sum of all positive divisors of the number 186
Competition math: Compute the sum of all positive integers whose positive divisors sum to 186.
This competition problem is from the 2022 Princeton University Math Competition, the number theory section. As you will see, the competition problem requires way more math knowledge and thinking rather than just a standard routine process. This is also a reason why it is more fun to solve math competition problems.

Princeton University Math Competition site: jason-shi-f9dm.squarespace.com
The test: static1.squarespace.com/static/570450471d07c094a39efaed/t/642cf9f637301b7f0aa1d2e3/1680669174825/Number_Theory_B_2022_.pdf

🛍 Shop math t-shirt & hoodies: amazon.com/shop/blackpenredpen
----------------------------------------
💪 Support the channel and get featured in the video description by becoming a patron: patreon.com/blackpenredpen
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt Zahra Parhoun Hassan Alashoor Kaakaopuupod bbaa Joash Hall Andr3w11235 Cadentato Joe Wisniewski Eric Maximilian Mecke
----------------------------------------
Thank you all!

ultimate counting problem from Oxford MAT

blackpenredpen 2023-07-06 | Learn math and science from Brilliant: 👉brilliant.org/blackpenredpen (20% off with this link!)

I saw this beautiful question from the 2020 Oxford Math Admissions Test, which was for both math and computer science applicants. The question has multiple parts but the first part gives away a very big hint, so I decided to modify this question a bit and present it to you guys. I am sure this could easily be a math competition-level problem and I had tons of fun solving it! See the original Oxford math and computer science admissions test problem on page 8 here:
maths.ox.ac.uk/system/files/attachments/test20_0.pdf

0:00 I modified an Oxford math admission problem
24:07 Check out Brilliant
#math #blackpenredpen #oxford #numbertheory

🛍 Shop math t-shirt & hoodies: amazon.com/shop/blackpenredpen
----------------------------------------
💪 Support the channel and get featured in the video description by becoming a patron: patreon.com/blackpenredpen
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt Zahra Parhoun Hassan Alashoor Kaakaopuupod bbaa Joash Hall Andr3w11235 Cadentato Joe Wisniewski Eric Maximilian Mecke
----------------------------------------
Thank you all!

one super interesting cubic question from the Stanford math tournament

blackpenredpen 2023-06-22 | Enhance your math contest skills from Brilliant: 👉brilliant.org/blackpenredpen (20% off with this link!)

This question is from the Stanford Math Tournament (SMT) the algebra tiebreaker section stanfordmathtournament.com/archive/2023. We are given a cubic function f(x)=x^3-6x^2+25/2x-7 and we know there exists a closed interval [a,b] such that the sequence x, f(x), f(f(x)), ... is bounded for x on [a, b]. This question is fascinating because not all cubic functions have such properties. I will use the fixed-point approach for this problem, how would you solve this? #brilliant #math #stanford #blackpenredpen

0:00 Stanford Math Tournament algebra tiebreaker
1:00 Check out Brilliant
2:10 My solution to this problem
7:12 Why my solution works for this problem
Note: I avoided calculus derivative here because I wanted to keep it with just algebra. I will do another video on the fixed-point iteration later.
9:56 the official solution

💪 Support the channel and get featured in the video description by becoming a patron: patreon.com/blackpenredpen
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt Zahra Parhoun Hassan Alashoor Kaakaopuupod bbaa Joash Hall Andr3w11235 Cadentato Joe Wisniewski Eric Maximilian Mecke
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Thank you all!

How to solve quadratic equations (by factoring, completing the square, & quadratic formula)

blackpenredpen 2023-06-16 | Master all techniques for solving quadratic equations with this algebra tutorial! We'll cover the factoring method, square roots, completing the square, and the quadratic formula with 24 detailed worked-out examples. The last 4 equations also feature quadratic equations with complex solutions. Note, "i" is the imaginary unit where i^2=-1 and sqrt(-1)=i. This video is perfect for algebra and precalculus students. Teachers and tutors, feel free to use this video and worksheet with your students.

#algebra #math #blackpenredpen

worksheet 👉 patreon.com/posts/my-algebra-only-81417933
my written solution 👉 patreon.com/posts/my-algebra-81418158

0:00 24 quadratic equations with only "1, 2, 3, 4"
0:31 Q1, solving a quadratic equation by factoring
4:55 Q2, by taking the square roots on both sides
9:43 Q3, expand first, then solve by factoring
14:00 Q4, completing the square
20:14 Q5, the quadratic formula
26:40 Q6
30:03 Q7
33:52 Q8
36:11 Q9
37:55 Q10
43:10 Q11
45:28 Q12
48:10 Q13
50:37 Q14
52:07 Q15, a special way to complete the square
59:00 Q16
1:00:30 Q17
1:02:55 Q18
1:06:03 Q19
1:12:49 Q20,
1:17:02 Q21, quadratic equation with complex solutions
1:20:12 Q22,
1:23:49 Q23,
1:32 44 Q24,

💪 Support the channel and get featured in the video description by becoming a patron: patreon.com/blackpenredpen
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt Zahra Parhoun Hassan Alashoor Kaakaopuupod bbaa Joash Hall Andr3w11235 Cadentato Joe Wisniewski Eric
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Thank you all!

how is i^x=2 possible?

blackpenredpen 2023-05-30 | Learn more complex numbers from Brilliant: 👉brilliant.org/blackpenredpen (20% off with this link!)

Can the power of the imaginary unit i ever give us 2? We know i^1=1, i^2=-1, i^3=-i, i^4=1, and so on. So is it possible for us to find x so that i^x=2? Well, the answer is yes but it is not real. #ComplexNumbers #Exponentiation #ImaginaryNumbers #Mathematics #blackpenredpen

0:00 We know i^1=1, i^2=-1, i^3=-i, i^4=1, and so on. So is it possible for us to have i^x=2?
7:35 Check out Brilliant

Get an Euler's Identity Tee on my Amazon store: amzn.to/3OGnESd
Check out "Can 1^x=2?" youtu.be/9wJ9YBwHXGI
----------------------------------------
💪 Support the channel and get featured in the video description by becoming a patron: patreon.com/blackpenredpen
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt Zahra Parhoun Hassan Alashoor Kaakaopuupod bbaa Joash Hall Andr3w11235 Cadentato Joe Wisniewski Eric
----------------------------------------
Thank you all!

exact value of sin(10 degrees)

blackpenredpen 2023-05-13 | We will use the cubic formula to find a formula for sin(x/3) and we will do the classic trig problem of finding the exact value of sin(10 degrees). By the way, proving sin(10 degrees) is irrational is a classic math problem called the coffin problem. Check it out here: youtu.be/cN0vDlWjFiY

Notes on Patreon: patreon.com/posts/notes-on-sin-x-3-82966449
Cubic formula: youtu.be/ULsPhWmqhyc
Check out my unit circle tee on Amazon: amzn.to/3O9jge7
Why sin(50 degrees) and sin(70 degrees) showed up: youtu.be/JqagcCrdpQQ

0:00 what's a formula for sin(x/3), i.e. 1/3 angle formula for sine
0:23 deriving sin(3x) by using double-angle formula
4:34 using the cubic formula (the depressed version)
11:11 attempting to get sin(10 degrees) but we ran into some issues
19:11 finally got sin(10 degrees)
but I wonder if we can write sin(10 deg) as a radical expression without using the imaginary unit i
----------------------------------------
💪 Support the channel and get featured in the video description by becoming a patron: patreon.com/blackpenredpen
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt Zahra Parhoun Hassan Alashoor Kaakaopuupod bbaa Joash Hall Andr3w11235 Cadentato Joe Wisniewski Eric
----------------------------------------
Thank you all!

proving the limit of a product is the product of the limits, epsilon-delta definition

blackpenredpen 2023-05-02 | We will prove the limit of a product is the product of the limits (assuming the limits exist) by using the epsilon-delta definition of a limit. This is a classic proof that you will see in your advanced calculus class or real analysis class. This is hard proof especially when you first see it. Be sure you put in the effort and study it well. #calculus #realanalysis #blackpenredpen

Notes for this video are available on my Patreon: 👉 patreon.com/posts/notes-proof-of-82385985

Introduction and more examples of the epsilon-delate definition of a limit: 👉youtube.com/playlist?list=PLj7p5OoL6vGwzCW5N_hyGa88BzGQHv1ax

0:00 Review some limits and their epsilon-delta definition
2:48 The proof!

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10% off with the code "WELCOME10"
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💪 Support the channel and get featured in the video description by becoming a patron: patreon.com/blackpenredpen
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt Zahra Parhoun Hassan Alashoor Kaakaopuupod bbaa Joash Hall Eric
----------------------------------------
Thank you all!

precalculus law of sine, but be careful

blackpenredpen 2023-04-28 | Learn more about the law of sine and cosine on Brilliant: brilliant.org/blackpenredpen (now with a 30-day free trial)

I came up with this tricky test question challenge for my precalculus students. Start with an obtuse triangle, give students the length of all three sides and one of the acute angles. Ask the students to find the measure of the obtuse angle. Now many students will proceed by using the Law of Sine because the formula is shorter compared to the Law of Cosine. But because the range of the inverse sine function is _________, we will have to be extra careful with we use the Law of Sine!

Written solutions for this problem: patreon.com/posts/81996867
My trig worksheets (for anyone): patreon.com/posts/81996734

Here's another precalculus challenge: youtu.be/-aedH9Pusr4
Proofs of the Law of Sine & Cosine: youtu.be/843Uw-Hj-OQ
#precalculus #math #blackpenredpen #mathteacher

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10% off with the code "WELCOME10"
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💪 Support the channel and get featured in the video description by becoming a patron: patreon.com/blackpenredpen
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt Zahra Parhoun Hassan Alashoor Kaakaopuupod bbaa Joash Hall
----------------------------------------
Thank you all!

0:00 my favorite precalculus test question
5:02 a small mistake. c should be 5.8
8:24 Check out Brilliant

so you want to see the cubic formula

blackpenredpen 2023-04-18 | Here's the complete derivation of the cubic formula for ax^3+bx^2+cx+d=0. We will first review the formula for the quadratic equation x^2+px+q=0. Then we will take three main steps to solve the cubic equation. This ultra-algebra proof is a must-see for any math enthusiast looking to expand their knowledge and deepen their understanding of the roots of cubic equations. See my Patreon for my hand-written notes: patreon.com/posts/notes-on-cubic-81794694

de Moivre quintic formula 👉 youtu.be/L8W684pCHyc

0:00 we will derive the cubic formula
0:07 review on solving the quadratic equation x^2+px+q=0 and the pq-formula
6:17 part1, get rid of the x^2 term in the y-world (let x=y-b/(3a))
17:23 part 2, solving y^3+py+q=0 (let y=z-p/(3z))
39:27 part 3, put everything together
#maths #imaginarynumber #blackpenredpen

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10% off with the code "WELCOME10"
----------------------------------------
💪 Support the channel and get featured in the video description by becoming a patron: patreon.com/blackpenredpen
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt bbaa Joash Hall
----------------------------------------
Thank you all!

calculus teacher vs I know it already student

blackpenredpen 2023-03-28 | Learn more calculus on Brilliant: brilliant.org/blackpenredpen (now with a 30-day free trial)

We will tackle a famous calculus problem using the definition of the derivative, forcing even the most confident students to dig deep and truly understand the concept. Watch as we derive the derivative of f(x)=x^2*sin(1/x), including the challenging part of handling the derivative at x=0, where the function is not differentiable. But we don't stop there - we also introduce the squeeze theorem, a powerful tool to evaluate limits, to further deepen our understanding of calculus. This video is perfect for anyone looking to enhance their calculus knowledge and master the fundamental concepts of derivatives.

Related video: Calculus Teacher vs Power Rule Students: youtu.be/57Eawy5IAaM

0:00 If you are also a calculus teacher
0:39 f(x)=x^2*sin(1/x) when x is not 0 and f(0)=0
7:13 check out Brilliant

🛍 Shop math t-shirts & hoodies: blackpenredpen.creator-spring.com
10% off with the code "WELCOME10"
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💪 Support the channel and get featured in the video description by becoming a patron: patreon.com/blackpenredpen
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt
----------------------------------------
Thank you all!

the last question on my precalculus test

blackpenredpen 2023-03-17 | Here's the last question on a precalculus test I gave out this week. I asked the students to find the inverse of a rational function. In order to make this more fun, I gave them an involutory function. That is, the inverse function is actually the same as the original, f^-1(x)=f(x) or f(f(x))=x. Some students noticed the answer and told me it was a tricky question. #math #precalculus #blackpenredpen
Read more on involutory functions: en.wikipedia.org/wiki/Involution_(mathematics)

0:00 how I have fun when I teach math classes
0:33 the precalculus way to find the inverse of a function
3:28 finding the involution for rational functions
13:21 a much better way!

🛍 Shop math t-shirts & hoodies: blackpenredpen.creator-spring.com
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***Thanks to ALL my lovely patrons for supporting my channel and believing in what I do***
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
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Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt
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if your calculus teacher still doesnt believe the DI method...

blackpenredpen 2023-02-28 | Try Brilliant with 30 days free: 👉 brilliant.org/blackpenredpen (20% off with this link!) #calculus #brilliant #maths #math

In this video, we will show you why your calculus students deserve to learn and use the DI method for integration by parts. While it's true that the DI and u-dv methods are equivalent, the DI method has many advantages. It's simpler, more intuitive, and it reduces the risk of mistakes. By using the DI method, your students will be able to tackle more complex integration problems with ease. We'll go over examples of integrating x^2ln(x) and x^2cos(x) using both methods, and you'll see for yourself how much easier and faster the DI method is. Don't let your students struggle with the traditional u-dv method any longer! #calculus #integrationbyparts #DIMethod #blackpenredpen #matheducation

Here are the 3 stops for the DI method: youtube.com/watch?v=2I-_SV8cwsw
Why I don't teach the LIATE method: youtu.be/z33rYYUM4d8

0:00 integral shortcut? answer in 5 seconds : )
0:07 why i am making this video
0:43 proving the integration by parts formula
2:02 example 1, integral of x^2*ln(x)
5:30 why the DI method is really the same as integration by parts
8:34 example 2, the integral that got 6M+ views on Instagram
12:22 using DI method for the integral of x^2*cos(x)
15:56 check out Brilliant

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***Thanks to ALL my lovely patrons for supporting my channel and believing in what I do***
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt
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i-th root of -1

blackpenredpen 2023-01-28 | Learn more complex numbers from Brilliant: 👉 brilliant.org/blackpenredpen (20% off with this link!)

We know the principal square root of -1 is the imaginary unit i but what if we have the i-th root of -1? We will do a quick review on converting a complex number from the standard form a+bi to its polar form re^(i*theta). Then we will see the surprising answer that the i-th root of -1 will actually give us.

Related videos:
the proof of the Euler's identity (with the Taylor series): youtu.be/Jqt67tJCiwQ

i-th root of i: youtu.be/hy-ZSvjpqlc

🛍 Shop Euler's Identity t-shirt (as seen in the video): blackpenredpen.creator-spring.com/listing/etotheipiplus1is0?product=2
10% off with the code "WELCOME10"

0:00 what's the i-th root of -1
5:30 check out Brilliant
6:08 challenging question (-1)^pi
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***Thanks to ALL my lovely patrons for supporting my channel and believing in what I do***
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald
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💪 If you would also like to support this channel and have your name in the video description, then you could become my patron here patreon.com/blackpenredpen

Using Feynmans technique TWICE! (the integral of sin^3(x)/x^3 from 0 to inf)

blackpenredpen 2023-01-12 | We will evaluate the improper integral of sin^3(x)/x^3 from 0 to infinity by using Feynman's technique of integration (aka differentiation under the integral sign, Feynman's integration trick, or Leibniz's Rule). This is definitely one of the coolest integration techniques (but unfortunately it is not often taught in a calculus class).

See the integral of sin(x)/x from 0 to infinity: youtu.be/s1zhYD4x6mY
integral of sin^2(x)/x^2 from 0 to infinity: youtu.be/jQz1gQ24OHc?t=22092

🛍 Shop math t-shirts & hoodies: http://bit.ly/bprpmerch
10% off with the code "WELCOME10"
----------------------------------------
***Thanks to ALL my lovely patrons for supporting my channel and believing in what I do***
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N
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💪 If you would also like to support this channel and have your name in the video description, then you could become my patron here patreon.com/blackpenredpen

the hardest integral from the BMT integration bee

blackpenredpen 2023-01-08 | The Fall 2023 Berkeley Math Tournament will be held on the UC Berkeley campus on Saturday, Nov. 4th and I will be there! For more info and registration, check out https://berkeley.mt/events/bmt-2023/.

I solved this monster improper integral during the Extreme Calculus Calculus (another 100 integrals). Full video youtu.be/jQz1gQ24OHc

Subscribe to @bprpcalculusbasics
Follow my IG for pictures & short videos instagram.com/blackpenredpen
Shop math t-shirts & hoodies: http://bit.ly/bprpmerch 10% off with the code "WELCOME10"
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***Thanks to ALL my lovely patrons for supporting my channel and believing in what I do***
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N
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💪 If you would also like to support this channel and have your name in the video description, then you could become my patron here patreon.com/blackpenredpen

100 integrals (almost 9 hours nonstop)

blackpenredpen 2022-12-25 | 100 integrals, part 2! This video features calculus 3 integrals and non-elementary integrals with special functions. We use a variety of integration techniques, including integration by parts, u-substitution, trig substitution, partial fraction decomposition, reverse power rule, integral reduction formulas, Weierstrass substitution, improper integrals, Feynman's technique, power series, the Gaussian Integral, double integrals, Jacobian, polar coordinates, and more! Join us in this nonstop calculus adventure and boost your integration skills. #calculus #integration #mathchallenge #nonstop #specialfunctions #techniques #improperintegrals #polarcoordinates #powerseries #feynmantechnique

Get the file & answer key on my Patreon 👉 patreon.com/posts/102752235
100 integrals (original) 👉 youtu.be/dgm4-3-Iv3s
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Big thanks to my Patrons for the full-marathon support!
Ben D, Grant S, Mark M, Phillippe S.
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Thanks to Nathaniel Barnett for the time stamps!
0:00 100 integrals part 2!
Q1, 0:00:46 integral of 2022
Q2, 0:03:03 integral by trig identities and u-sub
Q3, 0:08:23 integral by trig sub
Q4, 0:12:34 integral by u-sub and integration by parts
Q5, 0:17:02
Q6, 0:19:42
Q7, 0:30:07
Q8, 0:33:45
Q9, 0:37:58
Q10, 0:45:16
Q11, 0:48:30
Q12, 0:53:16
Q13, 0:59:06
Q14, 0:58:34
Q15, 1:00:41
Q16, 1:04:56
Q17, 1:07:27
Q18, 1:12:27
Q19, 1:14:42
Q20, 1:18:50
Q21, 1:22:30 integral of x^n*e^x
Q22, 1:25:14 integral of (ln(x))^n
Q23, 1:27:40 integral of sin^n(x)
Q24, 1:37:44 integral of tan^n(x)
Q25, 1:42:58 integral of sec^n(x)
Q26, 1:52:14
Q27, 1:54:45
Q28, 1:57:45
Q29, 2:01:49
Q30, 2:04:49
Q31, 2:24:38 integral of sec(x) by Weierstrass Substitution
Q32, 2:27:17
Q33, 2:31:25
Q34, 2:35:00
Q35, 2:37:20
Q36, 2:42:02
Q37, 2:44:24
Q38, 2:50:18
Q39, 2:51:53
Q40, 2:56:12
Q41, 3:00:21
Q42, 3:04:07
Q43, 3:06:45
Q44, 3:11:58
Q45, 3:19:02
Q46, 3:24:19
Q47, 3:28:36
Q48, 3:31:54
Q49, 3:33:44
Q50, 3:43:40
Q51, 3:49:16
Q52, 3:55:44
Q53, 3:59:50
Q54, 4:03:18 integral of x/(1+e^x)
Q55, 4:08:21 integral of e^x/(1+x)
Q56, 4:09:21 integral of e^e^x
Q57, 4:10:27 integral of ln(ln(x))
Q58, 4:12:40
Q59, 4:14:28
Q60, 4:19:48 integral of the Lambert W function
Q61, 4:25:39 improper integral of 1/x^p from 1 to inf
Q62, 4:28:52
Q63, 4:29:32
Q64, 4:32:55
Q65, 4:35:29 the Gaussian Integral
Q66, 4:50:04
Q67, 4:55:42
Q68, 4:58:36 integral of ln(x) from 0 to 1
Q69, 5:02:56
Q70, 5:10:32
Q71, 5:13:24 integral of ln(x)/(x-1) from 0 to 1
Q72, 5:21:08 integral of (x-1)/ln(x) from 0 to 1
Q73, 5:28:08 integral by symmetry
Q74, 5:34:14 integral by even and odd function properties
Q75, 5:43:16
Q76, 5:51:17
Q77, 5:52:58
Q78, 6:00:28 the Dirichlet integral, the integral of sin(x)/x from 0 to inf
Q79, 6:08:10
Q80, 6:13:28
Q81, 6:37:48 double integral & change the order of integration
Q82, 6:45:45
Q83, 6:49:24
Q84, 6:54:00
Q85, 6:57:25
Q86, 7:03:50
Q87, 7:07:33
Q88, 7:11:11
Q89, 7:17:27
Q90, 7:21:37
Q91, 7:28:52
Q92, 7:37:47
Q93, 7:55:15
Q94, 8:07:00
Q95, 8:10:24
Q96, 8:21:05
Q97, 8:22:55
Q98, 8:26: 30
Q99, 8:29: 59
Q100, 8:33:33
Q101, 8:38:32 the BIG integral from the Berkeley Math Tournament

*Thanks to ALL my lovely patrons for supporting my channel and believing in what I do*
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N