Anti Pythagoras 2: the Line, Ellipse, and Infinite Triplets Dr Peyam 2022-04-24 | Anti Pythagoras 2: the Line, the Ellipse, and the Infinity. How to generate infinitely many anti Pythagorean triplets of the form b^2 = a^2 + c^2 - ac The triplets relate the sides of a triangle where one angle is 60 degrees. It is a variation of the Pythagorean theorem from 8th grade math. It has to do with intersecting an ellipse with a line of rational slope and performing some clever algebra like expanding quadratic polynomials and quadratic forms and comparing numerator and denominators of fractions with integers. Then we give an example of a triangle with such a triplet. Special thanks goes to Ian Fowler who has introduced me to this problem. Ian would like to give a big shout out to Emidio Iacobucci and Frank Cirone for their part in the adventure.0:00 Introduction1:40 Rational Slopes2:40 Finding Nemo6:00 The triplets are bornAnti Pythagorean theorem: youtu.be/TMR1oYkCKEsYT channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
Gauss would be proud Dr Peyam 2022-12-10 | Gauss would be proud! I calculate the integral of x^2n e^-x^2 from -infinity to infinity, using Feynman's technique, as well as the Gaussian integral and differentiation. This integral appears over and over again in quantum mechanics and is useful for calculus and physics students. It's a very nice and challenging exercise in integration, especially for preparing for finals, enjoy! This idea has been recommended by my friend Walden, thank you again! You should check out his YouTube channel @chinesegaymoviereviews9468 and his TikTok channel learnnorthernchineseGaussian Integral: youtu.be/kpmRS4s6ZR4Gaussian Playlist: youtube.com/playlist?list=PLJb1qAQIrmmCgLyHWMXGZnioRHLqOk2bWSubscribe to my channel: youtube.com/drpeyamTikTok: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
Google and eigenvalues Dr Peyam 2022-12-07 | Google and eigenvalues. We describe the Pagerank algorithm, which was one of the algorithms used by Google for their search engine. For this, we rank the websites using an importance vector vector and write the system as a Markov chain, using matrices. Then we diagonalize the matrix using linear algebra, more precisely eigenvalues and eigenvectors. Finally, by taking limits of the matrix, we can determine which website reigns supreme. This probably has applications in data science and neural networks as well.Matrix Limits and Markov Chains: youtu.be/G0V3UzifI_0Diagonalization: youtu.be/EblrLncv7a0Subscribe to my channel: youtube.com/drpeyamTikTok: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
Which two implies the third? Dr Peyam 2022-12-03 | Which two implies the third? Here is a fun algebra problem! Which two of the following three identities implies the other one? We have a+b=1 and a^2 + b^2 = 1 and 1/a+1/b=1. Can you guess what the answer is? This is a must-see for any arithmetic and geometry lovers, and especially for high school and college mathematics students. I even relate it to lines and circles! Another fun problem: youtube.com/watch?v=QICEl0QNuTwSubscribe to my channel: youtube.com/drpeyamTikTok: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
Solving exact differential equations Dr Peyam 2022-11-30 | Solving exact differential equations. I solve exact differential equations, which are of the form Pdx+Qdy = 0. For this, we use multivariable calculus, more specifically conservative vector fields and line integrals. The solutions can be written in explicit form or implicit form, in terms of level sets and level curves with gradients. At the end I show how to derive the formula, which just relies on the fundamental theorem of line integrals. A must-see for every calculus and ode student, enjoySubscribe to my channel: youtube.com/drpeyamTikTok: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
cos = 2 (in Spanish) Dr Peyam 2022-11-26 | cos = 2 en espanol. Voy a resolver la ecuacion cos = 2 utilizando los nombres complejos y la formula de Euler, que dice que exp ix = cos x + i sen x. Es muy apropriado por los estudiantes de matematicas, es un verdadero placer de algebra, buen aprevecho! Subscribe to my channel: youtube.com/drpeyamTikTok: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
Solving an ODE using separation of variables Dr Peyam 2022-11-23 | Solving an ODE using separation of variables. I find the solution of a differential equation by splitting it, putting all the x terms on one side and all the t terms on the other side, and integrating. This gives an answer in implicit form, which sometimes we can write in explicit form. This is a very common technique taught in calculus classes.Subscribe to my channel: youtube.com/drpeyamTikTok: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
playing with triangles: a quasi-Pythagorean identity Dr Peyam 2022-11-19 | Playing with triangles: a quasi-Pythagorean identity. I highlight a beautiful identity coming from geometry, which has to do with equilateral triangles and complex numbers. This has been inspired by a Tweet by Steven Strogatz from Cornell University. For this, we use Euler's formula and rotations, and some very simple algebra. This is a must see for anyone who likes math and education and hard geometry problems with elegant solutions. It is reminiscent of the Pythagorean theorem a^2 + b^2 + c^2 and the binomial formulaanti-Pythagorean theorem: youtube.com/watch?v=TMR1oYkCKEsYT channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
Solving an ode using integrating factors Dr Peyam 2022-11-16 | Solving an ode using integrating factors. We solve a differential equations using a method called integrating factor, where you write the equation in standard form and then multiply by exp integral P. This converts it into product rule form, and after integrating we can easily solve for y. This is a very common technique for solving first order scalar equations and is extremely useful in calculus. Subsribe to my channel: youtube.com/drpeyamTikTok: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
a crazy tangent no oneβs even heard of Dr Peyam 2022-11-12 | a crazy tangent no oneβs even heard of. Here's an amazing calculus fact you didn't know: If you have a cubic polynomial with three roots, then the tangent line at the midpoint of the first two roots has to go through the third root. This is useful in the design of projectiles and missiles, as well as for fox hunting. It uses the tangent line definition using derivatives and some simple algebra. A must-see for college calculus students and geometry and graph loversYT channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
my math YouTube advice Dr Peyam 2022-06-30 | advice from a math YouTuber. Here are some tips on how to use YouTube in your teaching and how you can start your own YouTube channel. This is the talk I was supposed to give at the 2022 Joint Mathematics Meetings before it got cancelled. Spreading the math: using YouTube to communicate with the rest of the world.YT channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
not your AVERAGE Putnam limit (2021 Putnam A2) Dr Peyam 2022-06-26 | not your AVERAGE Putnam limit. We evaluate a limit with exponential functions and averages, taken from the William Lowell Putnam Math Competition. This is problem A2 from the 2021 Putnam exam. We do this by taking logarithms, hospitalβs rule, derivatives of exponential functions, and the definition of e as a limit. This is a must see for calculus 1 students and anyone interested in math Olympiad problems, enjoy! YT channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
what is the area of this Neumann Oval ? Dr Peyam 2022-06-23 | what is the area of this Neumann Oval ? We calculate the area of the Neumann oval, which is the image of the unit disk in R2 under the mapping z + z^2/2. For this we need a new way of changing variables which involves the derivative squared. This method uses Jacobians, double integrals, polar coordinates. At the end we explain why there is a |fβz|^2 term, but it basically follows from the Cauchy Riemann equations. This is a must see for complex analysis and multivariable calculus lovers!
Feynman Integral from Reddit Dr Peyam 2022-06-19 | Reddit Feynman Integral. We calculate the integral from 0 to 1 of x^2 - 1 / lnx using two methods: the Feynman technique and classical u substitution. Both methods are useful in physics and a must learn for beginning calculus students and anyone interested in integrals and integration.0:00 Feynman way3:49 u sub wayFeynman integration: youtu.be/sdWpmwutSHEοΏΌοΏΌYT channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
number comparison: 57^8 vs 62^7 Dr Peyam 2022-06-16 | you wonβt believe which number is bigger!! We compare 57^8 and 62^7 by taking ratios and comparing exponents. Moreover we use the binomial theorem which is used to expand out quadratic formulas and higher power of polynomials. This is a must see for any algebra or math student! Which number is larger? Watch this video to find out YT channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
How many revolutions on this Ellipse? Dr Peyam 2022-06-12 | How many revolutions on this Pythagorean ellipse? Suppose an ellipse rolls over a sine curve in such a way that one revolution of the ellipse equals to one period of the curve, how are the axes of the ellipse related to the amplitude. The answer is surprisingly related to the Pythagorean theorem. This is a really cool Putnam problem that we solve using calculus and multivariable calculus, more precisely by calculating arclength two ways with integrals and using a u substitution. Itβs a must see for calculus and geometry lovers.YT channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
a neat integral equation π Dr Peyam 2022-06-09 | a neat integral equation. I solve an equation with only integrals, without using any calculus or differentiation whatsoever! Which function f satisfies integral of f times integral of 1/f is x? Instead, I use the famous Cauchy Schwarz Bunyakovsky inequality, which relates dot products and norms and is used widely in linear algebra and math, enjoy!how to solve integral equations: youtu.be/LljEAnjyS9UYT channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
GIGANTIC number comparison πͺ Dr Peyam 2022-06-05 | Gigantic number comparison. We compare 22^33 and 33^22 using two methods. One is by taking ratios and comparing exponents. The other one is by using calculus, more precisely differentiations and critical points and maximum/minimum, which can be generalized to other exponents like pi^e and e^pi and also x^y vs y^x. This is a must see for any calculus student, enjoy! YT channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
an integro differential equation? Dr Peyam 2022-06-02 | an integro differential equation. How to solve a an equation that has derivatives and integrals. The key for this ode is the Laplace transform, which turns differentiation into multiplication and convolution into products as well. This has lots of applications to predator prey models and the lotka volterra equation. We also give another method using the chain rule for integrals.Laplace integral gone BANANAS πyoutu.be/a5l4owYxjRwYT channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
Impossible fraction question? Dr Peyam 2022-05-29 | impossible fraction equation? Is it true that 5/3 + 3/5 is equal to 1? A tricky math algebra question! We solve the equation x/y + y/x = 1 by putting under the common denominator, using the quadratic formula, and complex exponentials. The solution can be elegantly interpreted in terms of rotations and Euler's formula. We solve two variations: x/y + y/x = 0 and x/y + y/x = 2 which are much simpler and can be written in terms of lines and linear functions. 0:00 Introduction 3:00 Variation4:40 Variation 05:23 Verification (finally)YT channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
where did the pi go? area of a superellipse Dr Peyam 2022-05-26 | calculating the area of a super ellipse. Here we find the area enclosed by a pseudo ellipse, which is a generalization of an ellipse but with n. Special cases include squares, circles, and ellipses. After a change of variables, we transform this geometric figure into an integral, which involves the beta function and which has been calculated previously using the Laplace transform. The answer includes the gamma function and factorials, but where did the pi go? This is a must see for calculus and math students, enjoy!Laplace integral: youtu.be/a5l4owYxjRwYT channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
an outrageous equation: cos + sin + tan = 2 Dr Peyam 2022-05-22 | solving an outrageous equation: cos(x) + sin(x) + tan(x) = 2. Another beautiful equation with complex numbers, which features complex exponentials, the quadratic formula, polar coordinates, dividing by imaginary numbers, complex conjugates, complex roots, quartic equations, intermediate value theorem, modulus, absolute values, and periodicity. A must-see for students taking calculus or algebra or analysis, or who enjoy math in general.0:00 complex exponentials2:00 a quartic equation7:30 a mathematical exploration9:10 the answerYT channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
quaternion square root of -1 Dr Peyam 2022-05-19 | quaternion square root of -1. We calculate the square root of -1 using the quaternions, which involves knowing how to multiply quaternion numbers. The answer will surprise you, because it involves spheres and it will make you see complex numbers in a new way, as north and south poles of balls. Quaternions are generalizations of imaginary numbers with i j k, and are useful in physics and quantum mechanics and roboticsYT channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
an unnatural logarithm? Dr Peyam 2022-05-15 | an unnatural logarithm? imaginary logarithm vs imaginary base. We calculate log base i of 2 and log base 2 of i, using complex exponentials, Eulerβs formula, and natural logs. We also prove the neat result that they are reciprocals of each other, using properties of logarithms. This is a must see for college algebra, calculus, and complex analysis students and undergraduates, enjoy! YT channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
the outstanding Laplace method for solving systems of ode Dr Peyam 2022-05-12 | the extraordinary Laplace method for solving systems of ode. We solve a system of differential equations in a direct and easy way, without the use of linear algebra or eigenvalues. This method uses the laplace transform, very widely used in physics and engineering, which turns differentiation into multiplication. It is defined with an integral with exponential functions, and is very suited for convolutions and dirac delta distributions. To solve this, we use L on both sides and solve for the solutions using Cramer's rule. This is good when the two solutions or particles are coupled, and can be used for example to describe planetary motion in physics.0:00 Introduction1:15 Laplace Transforms3:36 Cramer's rule7:00 SolutionLaplace integral: youtu.be/a5l4owYxjRwYT channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
which logarithm is bigger? Dr Peyam 2022-05-08 | The Battle of the Logs. Which is bigger, log base 3 of 5 or log base 5 of 9? The answer may surprise you. We solve this equation by taking the difference, putting on a common denominator, writing it as a single base, and then comparing 3 to the square root of 2 with closer and closer fractions. This logarithmic inequality is applied math and analysis at its finest, and a treat for algebra 2 students and mathematics undergraduates, especially those who like problem solving and logarithmic and exponential equations.YT channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
Can you simplify this cubic cosh formula? Dr Peyam 2022-05-05 | Can you simplify this cubic cosh formula? How to evaluate expressions with hyperbolic trig functions. We write cosh^3 in terms of cosh3x, similar to double angle formulas in trig. We do this by writing cosh in terms of exponential functions and using the binomial theorem. This has a very interesting application to plastic ratios and allows us to write it in a more simplified form. This video will make you go Oh My Cosh!!!Plastic Ratio: youtu.be/CpBeRv-P5RMYT channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
Peyam goes PUTNAM: Destroying a Monster Polar Integral (2021 Putnam A4) Dr Peyam 2022-05-01 | Peyam goes Putnam: Destroying a monster polar integral. Calculating a hard math competition integral that appeared on the 2021 William Lowell Putnam Exam A4, using calculus techniques such as polar coordinates, averages, change of variables and jacobians, rotations, weierstrass substitution, and inverse trig substitution. In simplest terms, the question is: What is the limit of the integral as the radius of the domain goes to 0. Watch a math professor get destroyed by this seemingly innocent integral. Can you handle the math heat?0:00 The Integral3:30 Rotation6:00 Cleaning this up9:50 Polar polar polar18:50 Weierstrasssssssssssss22:50 The answerBig thanks goes to Kiran Kedlaya, for maintaining his Putnam exam webpage: kskedlaya.org/putnam-archiveYT channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
Can you solve this HARD math PhD complex number problem? Dr Peyam 2022-04-28 | Can you solve this HARD complex number PhD problem? Suppose a complex number z satisfies z^n + (1-iz)^n = 0, then what is the imaginary part of z? This problem is from a complex analysis qualifying exam question given to entering PhD students at UC Irvine, and was suggested to me by a current postdoc at my university. It sounds very complicated but can be solved using first year complex analysis student techniques. It involves modulus of complex numbers, but not arguments or logarithms or cauchy's integral formula, as I originally thought.Complex Analysis Overview: youtu.be/66AlliKQc-gComplex Analysis Playlist: youtube.com/playlist?list=PLJb1qAQIrmmCRmgJJJjCmBpJRhIleVNV1YT channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
Laplace integral gone BANANAS π Dr Peyam 2022-04-21 | Calculating integral using Laplace transforms. Evaluating integral of x^m (1-x)^n from 0 to 1 using convolution and Laplace transforms of functions, which is useful for engineering systems. I use a u sub and then apply the transform to power functions, which then turns it into a product of transforms. This is much faster and elegant than expanding and using the binomial theorem. There will be factorials involved and I also mention an extension to gamma functions. This is a must-see for calculus students, engineers, mathematicians, and differential equations lovers.0:00 Introduction0:50 Convolution4:00 Laplace transform6:07 The answerConvolution Intuition: youtu.be/_8LwWEGNyyMYT channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
the anti Pythagorean theorem? Dr Peyam 2022-04-17 | an anti Pythagorean theorem. Here is a geometry identity related to Pythagoras from 8th grade math which you may not know about. It relates the sides of a triangle where one angle is 60 degrees. It all starts with a high school problem about triangles and circles and uses the law of cosine from precalculus and leads to a special 8 7 5 triangle with integer sides. Then we'll see if there are other triangles of this form, which involves the quadratic formula. In a surprising turn of events, this is related to ellipses and lines. Special thanks goes to Ian Fowler who has introduced me to this problem. He is a retired high school math teacher and graded math competition problems at the University of Waterloo in CanadaPart 2: youtu.be/Sxq2hA7qZYM0:00 Introduction0:30 Problem3:10 Special Triangles4:50 Thirty Years Later6:50 MiracleLaw of Cosines: youtu.be/yJZJURqKYUcYT channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
what is the plastic ratio? Dr Peyam 2022-04-14 | what is the plastic ratio? I define the plastic number and explain its properties, first as a cubic equation and a Fibonacci sequence to count bunnies, then as a fractal cube root with self similarities, and finally a cool formula with hyperbolic cosine function substitution. This is a cool journey through algebra, polynomials, inverse functions, and calculus. Proof that I did it before mathologer: youtu.be/SMeNVyJjd2o0:00 Introduction0:36 Fibonacci1:45 Fractal3:10 Hyperbolic trigCalculating hyperbolic trig functions: youtu.be/My2S-xbLPG0YT channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
are imaginary numbers real? (feat ZPhysics) Dr Peyam 2022-04-10 | are imaginary numbers a part of nature? The wave function is used in quantum mechanics to predict the values of position, energy, and momentum of particles. Is the wave function real or complex? We will try and guess the equation that the wave function solves, by starting with a solution and using Euler's formula, and ultimately derive Schrodinger's partial differential equation, which is the basis of all quantum mechanics. It will include Einstein's postulate that the energy of a photon is given by Planck's constant times the angular frequency. ZPhysics Channel: youtube.com/channel/UC3L5MO3gJTlB09wmmHGW5QgSubscribe to my channel: youtube.com/drpeyamTikTok: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
solving a fractional differential equation Dr Peyam 2022-04-07 | solving a fractional differential equation. I solve an equation with half derivatives, by using techniques from calculus like differentiation. The process is clever yet elementary, and suitable for high school and college students. The solution involves gamma function and factorials and gaussian functions, even has a nice trick at the end with polynomials and exponents. There even is square root of pi involved. Applications include quantum mechanics and broken viscous physical processes0:00 the differential equation0:55 calculating the terms4:30 putting it together6:48 the solution and applicationsHalf derivative video: youtu.be/gaAhCTDc6oACredit to the problem goes to vibingmath on InstagramYT channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
Can 2^x be negative? Dr Peyam 2022-04-03 | Can 2^x ever be negative? We solve the equation 2^x is less than 0 by using complex exponentials and Euler's formula with i and pi and logarithms. The solutions in the complex plane are straight lines that are parallel and nicely laid out. So yes, this exponential equation can be negative, unlike what you may have learned in algebra and precalculus. YT channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
a unifying product quotient rule proof Dr Peyam 2022-03-31 | Deriving the product and quotient rules using the chain rule. I prove the formulas for the product and quotient of derivatives from calculus by applying the chain rule to a clever function of several variables. This is like a unified theory of everything, but for the chain rule and shows how powerful it is. I'm hoping this is useful for any undergraduate taking calculus and analysis and it gives a useful way of remembering it for math exams.0:00 Introduction0:28 Product Rule2:07 Quotient RuleClassical Quotient rule proof: youtu.be/xCcXgAdud5ICool Quotient rule proof: youtu.be/HBRzARgo27wSubscribe to my channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
equilateral triangle on a parabola Dr Peyam 2022-03-27 | equilateral triangle on a parabola. Find two points on the parabola y = 1-x^2 such that the triangle formed by the x axis and going through those points is equilateral. It is a nice problem that combines calculus and geometry, more specifically tangent lines, intercepts, distances, and inscribed triangles. This was inspired by a instagram post by UC Irvine math.Subscribe to my channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
Solving x^2022 = 1 by the LONGEST factoring I have ever done Dr Peyam 2022-03-24 | Solve all 2022 roots of unity. I solve the root equation x^2022 = 1, to start off the new year. I do this the algebraic way by using difference of squares and factoring and long division. Then I find the complex solutions using Euler's formula and De Moivre's formula and plot them on the unit circlex^2022 = 1 (old version): youtu.be/2C8F_JtpsEYTwo complex solutions in x^2021 = x^2020 + 1: youtu.be/IOkyoXSBllEYT channel: youtube.com/drpeyamTikTok: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
Calculating sin of square root of i Dr Peyam 2022-03-06 | Calculating sin of square root i. Using complex exponentials and Euler's formula, we calculate the values of trigonometric functions of complex numbers, more specifically sine of the radical of the imaginary number. We also compare it to sqrt sin i which is much easier to compute with cosh and sinh. Subscribe to my channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
matrix choose a matrix Dr Peyam 2022-02-17 | matrix choose a matrix. Calculating the number of matrix combinations of a matrix, using techniques from linear algebra like diagonalization, eigenvalues, eigenvectors. Special appearance by simultaneous diagonalizability and commuting matrices. In the end, I mention the general case using the gamma function transformation, a generalization of the factorial. Enjoy this linear algebra extravaganza.Matrix root of matrix: youtu.be/4vPrf-k0_bMMatrix Factorial: youtu.be/aVOc0xLITtcEuler's Secret: youtu.be/I2GjpW0mOj8Subscribe to my channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
β€οΈ area of an actual heart β€οΈ Dr Peyam 2022-02-14 | Calculating the area of an actual heart. In honor of valentine's day, I evaluate the area of a heart by using integrals under parametric curves. This much fancier than the cardioid. It involves the chain rule and some trigonometric integrals. It also explores symmetry and even and odd features of the graph, and at the end there is a nice simplification into linear terms. No polar coordinates or double integrals involved, just some nice multivariable calculus. Area of cardioid: youtube.com/watch?v=YYpZHnrtR9gYT channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
Half integral of x Dr Peyam 2022-02-11 | Defining the half integral, which is halfway between a function and its integral. Done by analogy with half derivatives, by finding formulas for the derivatives of power functions. It uses the factorial which is then generalized to the gamma function with Gaussian integrals. I also mention trig functions and applications to broken processes. This is a delight for any calculus student or anyone who is interested in fractional derivatives.0:00 Introduction0:30 Analogy2:58 Gamma Function5:00 Half integral of x9:37 Properties and ApplicationsHalf Derivative: youtu.be/gaAhCTDc6oAYT channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
Zeros of polynomial vs derivative Dr Peyam 2022-02-07 | We will see how the zeros of the derivative of a polynomial are related to the zeros of the original polynomial. This is called the Gauss Lucas Theorem for zeros of derivatives, which is an elegant result from complex analysis that relates it with the convex hull of the original roots. I also mention Marden's theorem, which is a special case of a triangle, and which has to do with the Steiner inellipse and foci. Enjoy this geometry and calculus adventure.0:00 Introduction0:09 Roots of P3:00 Plotting the roots geometrically5:15 Gauss Lucas Theorem6:52 Marden's TheoremGauss Lucas (full version): youtu.be/5DwKA9MVt-Q (members only)Gauss Lucas Proof: youtu.be/phthkaWOx6o (members only)YT channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
How many tangents to this ellipse? (no calculus) Dr Peyam 2022-01-12 | How many tangent lines are there to two intersecting ellipses? Intuitively there should be four, but why? Here a solution that uses no calculus whatsoever, just pure algebra, suitable for high school and algebra 2 students. It's a very nice application of the quadratic equation, followed by systems of equations. This video was inspired by a TikTok post.YT channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
all solutions to x^2022=1 Dr Peyam 2022-01-07 | Solve all 2022 roots of unity. I solve the root equation x^2022 = 1, to start off the new year. I do this the algebraic way by using difference of squares and nth roots, and the complex numbers way by using complex exponentials and euler identity and de moivre's formula. Then I plot the solutions on the unit circleSupport me on Patreon: patreon.com/drpeyamYT channel: youtube.com/drpeyamTikTok: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
Two complex solutions in x^2021=x^2020+1 Dr Peyam 2021-12-31 | Finding roots of an equation with 2021 and 2020. Solve some roots of x^2021 = x^2020 + 1 using modular arithmetic, difference of cubes, and some complex numbers. See how this equation is related to the unit circle in the complex plane and to Euler's formula. It even involves quadratic equations from 2021 and radicals and might be related to an important math paper. Happy new year everyone, hope this year is off to a good startOriginal Stackexchange Problem: math.stackexchange.com/questions/3609765/how-to-find-a-complex-root-of-x2021-x20201-while-this-complex-root-alsoYT channel: youtube.com/drpeyamTikTok: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
π cos + sin = 2 π Dr Peyam 2021-12-17 | Solving cos(x) + sin(x) = 2. Another beautiful equation with complex numbers, which features complex exponentials, the quadratic formula, polar coordinates, dividing by imaginary numbers, complex conjugates, and complex roots! A must-see for students taking calculus or algebra or analysis YT channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
life changing integration by parts trick Dr Peyam 2021-12-12 | Letβs learn a life-changing integration by parts trick. Once you learn this integration technique for you calculus 2 class, many integrals will be much easier. The trick here is to choose a clever antiderivative with a smart constant. 0:00 Intro 0:15 Integral x arctan x2:20 Integral ln x+23:30 Integral arctan square root x+1YT channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
Good luck on finals!!! Dr Peyam 2021-12-09 | Good luck on finals, may you ace them!!!Subscribe to my channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
calc 1 final be like (integral version) Dr Peyam 2021-12-08 | This is how a calculus 1 final exam problem might look like in college. We calculate the price Integral of x^2 from 0 to 1 in four different ways:0:00 The Problem0:29 Reverse Power Rule 4:20 Analysis 10:10 Lebesgue Integral This is in response to blackpenredpen's video, where he does the same thing with the derivative of x^2Calc 1 final be like: youtube.com/watch?v=wUD7n_ohMQk Subscribe to my channel: youtube.com/drpeyamTikTok channel: tiktok.com/@drpeyamInstagram: instagram.com/peyamstagramTwitter: twitter.com/drpeyamTeespring merch: teespring.com/stores/dr-peyam
