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Check yourself. Physics.Mathematics
United States
เข้าร่วมเมื่อ 19 เม.ย. 2023
Math Olympiad Toolkit | For what a, equation has a single solution? | No Lambert W-function used.
Learn how graphical analysis can help to solve a complicated problem related to equation a^x = x.
0:00 Problem formulation
00:13 Number of roots for various a
03:11 One-root case analysis
07:11 Conclusion
The graphical analysis is used to find parameter a of equation a^x = x at which this equation has a single solution. It helps to solve even more general problem, namely, to classify the number of roots of that equation using the value of a.
For more rigorous way to find number of roots as function of a watch video at th-cam.com/video/80_-3O0TO0M/w-d-xo.html
#math #Olympiad #algebra #exponentialequation #multiplesolutions #graphicalanalysis
0:00 Problem formulation
00:13 Number of roots for various a
03:11 One-root case analysis
07:11 Conclusion
The graphical analysis is used to find parameter a of equation a^x = x at which this equation has a single solution. It helps to solve even more general problem, namely, to classify the number of roots of that equation using the value of a.
For more rigorous way to find number of roots as function of a watch video at th-cam.com/video/80_-3O0TO0M/w-d-xo.html
#math #Olympiad #algebra #exponentialequation #multiplesolutions #graphicalanalysis
มุมมอง: 172
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Math Olympiad Toolkit | Find number of roots of equation aTOx=x (a positive) as function of a
มุมมอง 4012 ชั่วโมงที่ผ่านมา
Learn how graphical analysis can help to solve a complicated problem. No Lambert W-function used. 00:00 Problem formulation 00:11 Transformation of equation 00:47 Analysis and plotting of function ln(x)/x 03:45 Determining maximum of ln(x)/x 04:48 Determining number of roots of a^x=x 07:27 Conclusion #math #Olympiad #algebra #equation #multiple-solutions
Math Olympiad Problem | Knowing arithmetic and geometric mean of two numbers, find their difference
มุมมอง 51วันที่ผ่านมา
What is the difference of two positive numbers if their arithmetic mean is 2√3 and their geometric mean is 3? A simple trick will help you to solve this problem easily. The problem is designed for high school and undergraduate students. #matholympiad #algebra #highschool #arithmeticmean #geometricmean #trick
Math Olympiad Toolkit | Equation 2^(x+a) + x^2 = 3 has a single solution | Find a
มุมมอง 5321 วันที่ผ่านมา
Learn how graphical analysis can help to solve a complicated problem. 0:00 Problem formulation 00:16 Case a = 0 01:47 Case a greater than 0 02:55 Case a smaller than 0 04:22 Finding a for single solution 10:47 Conclusion The graphical analysis to find parameter a of equation 2^(x a) x^2 = 3 at which this equation has a single solution is used. It helps to solve even more general problem, namely...
Math Olympiad Problem | Can the side of a given square have an integer length? | Tricky solution!
มุมมอง 401หลายเดือนก่อน
Can the square with area A = 69303990366014 have the side of integer length? The answer is NO, The proof is based on the fact that the remainder after dividing of the square of any integer by 3 is either 0 or 1. 00:00 Problem formulation 00:39 Important property of integers 01:00 Analysis of the area 1:55 Proof of the property of integers 4:02 End screen The problem can be solved by high school...
Logical puzzle | Where will the egg roll? | Only 15% answer correctly
มุมมอง 16หลายเดือนก่อน
Funny logical problem which can be solved even by kids. It is taken from the book of Raymond Smullyan “What is the name of this book?” #logics #egg #roof #equilibrium
Math Olympiad Toolkit |Rule of thumb to solve cubic equation| You must know this!
มุมมอง 40หลายเดือนก่อน
Learn simple rule of thumb to solve the cubic equations which often suggested at Math Olympiads. May be very useful! 00:00 Problem formulation 00:48 Determining the first root 01:57 Factorization of the polynomial of third degree 05:20 Determining two other roots 06:20 Conclusion Even though this rule of thumb is not universal it can often help to solve cubic equations or even equations of high...
Logical Problem | Sum of two coins equal 30 c, yet one of them is not a nickel | Is it possible?
มุมมอง 20หลายเดือนก่อน
Funny logical problem of the famous American mathematician Raymond Smullyan. It can be solved by everybody who can think logically. Much more brilliant logical problems one can find in the book of Raymond Smullyan “What is the name of this book?” #logics #twocoins #smullyan
Complex numbers | Imaginary unit "i" | How to pronounce it: "i" or "iota"? | You must know this!
มุมมอง 34หลายเดือนก่อน
Why do different people pronounce the imaginary unit symbol differently? You must know this! #complexnumbers #imaginaryunit #pronunciation
Tricky Math | Find all roots of equation x^Sqrt[x] = 1/2 | Help to the students
มุมมอง 131หลายเดือนก่อน
First, the number of roots of equation x^Sqrt[x] =1/2 is determined using graphical method. Then all roots are found using representation of ½ by two tricky ways. 00:00 Problem formulation 00:07 Plotting graph of y = x^Sqrt[x] 02:11 Determining the number of roots 03:19 Determining first root 04:22 Determining second root 05:17 Conclusion The proof that lim x^Sqrt[x] = 1 (x to 0) is here: th-ca...
Tricky Math | Find extrema of unusual function | Help to the students
มุมมอง 28หลายเดือนก่อน
The extrema of the function y = x^Sqrt[x] are found. The solution involves logarithmic differentiation, logarithmic equations, and analysis of points of extremum. In addition, the plot of the function is sketched. 00:00 Problem formulation 00:21 Logarithmic differentiation 01:58 Finding the point of extremum 02:31 Analysis of extremum 03:33 Plotting the graph of the function 04:46 End screen Th...
Tricky Math| Resolving nontypical indeterminate form | Help to the students
มุมมอง 1112 หลายเดือนก่อน
An example of the limit containing nontypical indeterminate form (0 to 0) is considered. The main idea is to transform this indeterminate form to more typical one (0/0) and apply the l’Hopital’s rule. 00:00 Problem formulation 00:37 Rewriting the limit 2:00 Solution of the proxy problem 3:45 Solution of the main problem #limit #tricky #indeterminate_forms #zerotopowerzero
Math Olympiad Toolkit | Multiple solutions of equation 2^x + x^2 = 3 | Rule of thumb
มุมมอง 1572 หลายเดือนก่อน
Learn the rule of thumb to solve the equations like this: 2^x x^2 = 3, which are often suggested at Math Olympiads. The rule is not universal but can be very useful! 00:00 Problem formulation 00:17 Integer root 01:05 Number of solutions 02:45 Estimating second root 04:47 End screen The rule of thumb is formulated for solution of equation 2^x x^2 = 3. This equation has more than one root the exi...
Complex numbers |Exponential and logarithmic functions | Complete solution of equations like 1^x = 3
มุมมอง 1322 หลายเดือนก่อน
Believe or not but all solutions of equations like 1^x = 3 which are popular on the Internet are incomplete!!! 00:00 Problem formulation 00:20 Usual (Incomplete) solution 02:31 Complete solution 08:10 Discussion #mathematics #complexnumbers #exponentialequations
Math Olympiad Toolkit | Rule of thumb for equations not considered at your school | 2^x + x^3 = 3
มุมมอง 522 หลายเดือนก่อน
Math Olympiad Toolkit | Rule of thumb for equations not considered at your school | 2^x x^3 = 3
Conservation of angular momentum | Rotating disc with moving cockroach
มุมมอง 1173 หลายเดือนก่อน
Conservation of angular momentum | Rotating disc with moving cockroach
Static equilibrium. A wheel raising over a step obstacle of given height.
มุมมอง 1223 หลายเดือนก่อน
Static equilibrium. A wheel raising over a step obstacle of given height.
Inelastic collision | What is conserved? | Linear momentum? Kinetic energy? | Analysis of mistakes.
มุมมอง 723 หลายเดือนก่อน
Inelastic collision | What is conserved? | Linear momentum? Kinetic energy? | Analysis of mistakes.
Energy conservation | Frisbee flight | Change of mechanical energy
มุมมอง 604 หลายเดือนก่อน
Energy conservation | Frisbee flight | Change of mechanical energy
Vertical free fall motion |Final velocity and time of flight | Simple problem #2
มุมมอง 204 หลายเดือนก่อน
Vertical free fall motion |Final velocity and time of flight | Simple problem #2
Vertical free fall motion | Find final velocity and time of flight | Simple problem #1
มุมมอง 104 หลายเดือนก่อน
Vertical free fall motion | Find final velocity and time of flight | Simple problem #1
Speed of a satellite orbiting a planet. Mistakes analysis
มุมมอง 714 หลายเดือนก่อน
Speed of a satellite orbiting a planet. Mistakes analysis
Motion of a block along horizontal surface in the presence of friction and vertical force.
มุมมอง 714 หลายเดือนก่อน
Motion of a block along horizontal surface in the presence of friction and vertical force.
Calculating average velocity | Two simple problems | Mistakes analysis.
มุมมอง 904 หลายเดือนก่อน
Calculating average velocity | Two simple problems | Mistakes analysis.
Math Olympiad problem | Straightforward solution vs. tricky one
มุมมอง 97 หลายเดือนก่อน
Math Olympiad problem | Straightforward solution vs. tricky one
Projectile motion. A dart hits a target. Simple example of free fall kinematics.
มุมมอง 837 หลายเดือนก่อน
Projectile motion. A dart hits a target. Simple example of free fall kinematics.
Polarization of light | Two polarizers | Intensity of the transmitted light
มุมมอง 1209 หลายเดือนก่อน
Polarization of light | Two polarizers | Intensity of the transmitted light
Physics Olympiad problem | Ray optics | Illumination of bottom of a tunnel using plane mirror.
มุมมอง 11110 หลายเดือนก่อน
Physics Olympiad problem | Ray optics | Illumination of bottom of a tunnel using plane mirror.
Electromagnetic waves. Electric and magnetic components, Intensity, and power. Typical problem.
มุมมอง 7610 หลายเดือนก่อน
Electromagnetic waves. Electric and magnetic components, Intensity, and power. Typical problem.
Electromagnetic Induction | Solenoid and Coil | Faraday’s Law | Lenz’s Law | Crazy Mistake
มุมมอง 10010 หลายเดือนก่อน
Electromagnetic Induction | Solenoid and Coil | Faraday’s Law | Lenz’s Law | Crazy Mistake
Wrong solution. 1,4447^2=2,09 1,4447^3=3,015 1,4447^4=4,36...
4 is not a solution of a^x = x at a = a0 and cannot be used for verification. Solution is x = e. If you check x = e you will see that everything is OK.
An even quicker solution 2|A, as 2|4 4∤A as 4∤14 Thus A can't be a perfect square
Thanks for alternative solution!
Careful application of fixed-point or Newton-Raphson iteration will find all real roots. A complex-arithmetic version of fixed-point or Newton-Raphson iteration will find all roots.
Thak you for your comment. At this point, my goal was to show that after finding a solution by simple method (e.g. by trial and error) one must think about existence of other solutions. This step is absent in many videos on the Internet.
x≈-1.63658
Thank you for doing! next iterations!
Watch related problem at th-cam.com/video/IBYmy8fDInc/w-d-xo.html
Is it possible to use Lumbert W- funcrion?
I am not sure.
Very simple and understandable solution! Thank you.
It is an old problem but not always can solve it. Thank you.
I think you cant say that n=3m, you can only say that N=3m
3 is a prime number. For this reason, the only choice for N to be multiple of 3 is n=3m
Good work! Keep it up!
Will try:)
beautiful
Thank you! 😊
Amaizing! Solution is so simple!
100% agree
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You have ignored the resting time of 20 minutes, which will reduce the average velocity.
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