With Calculus, it easier for x^n graphs.
For example, take the derivative of a positive quadratic function and you get a linear function, and of it, where the line touches the x axis, that is the x-value of the minimum point or 0-slope point of the quadratic.
Another example, for the two points on a cubic equation, you can simple take its deravite and it becomes a quadratic. You can take its roots and these x values are the x values of the two points on a cubic graph. If there arent any roots, then the cubic equation simply doesnt have two "sharp" points.
Ok thats pretty basic stuff, but without calculus:
Quadratics: a) Factorising method which can reveal the coordinates of this 0-slope point. b) Use a pretty basic formula for x and y value. These two methods are actually quite similar in terms of their logic.
Now, cubics. I think by factorising a,b,c and d terms, one can figure out a formula. If it is, what is it? And a bit of a harder question, can it be done "logically" using factorising or simpler algebra?
A cubic equation f(x)=a(x-b)(x-c)(x-d) which has three real roots. The question is to prove that the tangent of the average of any two roots passes through the third root. I was stuck on this for a long time, and the hint says that show f’( (b+c) /2 ) = -a/4(b-c)^2 . I was been able to show this but couldn’t understand why this is the proof.
Hi, so I'm doing a problem and the question asks:
Solve for x in the equation: 12x^3+8x^2-x-1=0
I did some guess and check and found one of the factors, so I have the answers x=-1/2 and 1/3. But I'm not sure how else I can solve it without guess and check? I tried factoring by pulling out but that didn't work, so I'm kind of stuck. Anyone willing to explain to me? Thank you!
http://twitter.com/canoo/status/1413215890200428544
Hi guys, could someone please explain to me why x=2 is an oblique point of inflection. Thanks :)
Hey guys, thanks so much to u/deschan2021 for adding me, ur a legend.
Could someone pls explain to me why @ x=4 is a horizontal point of inflection. Thanks!
Could someone also pls explain to me why at x=2 is an oblique point of inflection, thanks!
https://preview.redd.it/9lt92ibjkpi71.png?width=654&format=png&auto=webp&s=0e6ea853f960836fc3bf03a5597698857d1690fc
So, I do some data science work that’s normally in python and for which I have to fit a cubic spline to about 70 data points. I need the spline function since I also have to solve for additional x-axis values.
In python it’s fairly easy with Scipy and Interp1d libs, but... I need to develop it to circulate to other users. I’ve made a front end with Qt but it’s a bit janky.
Now, I could make a 10 times better front end in Flutter in about a tenth of the time but I can’t find any way to create a cubic spline function. Things are made a bit more complex because if you search for ‘flutter’ and ‘spline’ there are a thousand hits to do with animation curves since it’s one of the options. So TLDR - does anyone know of a library that will create cubic spline function (not just plot the spline) in dart / flutter?
Anyone come across websites using the cubic bezier function the wild? Learning front end I'd like to see what's possible with this particular function of CSS.
https://www.youtube.com/watch?v=VvF6XCGFaUQ
-Can a cubic polynomial P have more than three (real) zeroes?
-How many local extreme values can a cubic function have?
-Do cubics have inflection points?
-If a cubic has three zeroes, what is the x-coordinate of the inflection point?
Last minute review of how to apply ideas relating to derivatives to analyzing cubic functions. Added in lots of pictures so you can see how derivatives affect things. (Also, a slight review of quadratics since those are a cubic's derivative)
All the vids so far:
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Series: https://www.youtube.com/watch?v=nqqPuT7PmEM
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Flow Chart for series: https://www.youtube.com/watch?v=e19RmVGUGb0
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FTC (accumulation: https://www.youtube.com/watch?v=u8DGiK11gVU
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FTC (derivative): https://www.youtube.com/watch?v=mgeMhi8pdrg
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Particle: https://www.youtube.com/watch?v=M-T1Q5o4nMM
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Volume: https://www.youtube.com/watch?v=FZtsJoHhRGk
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Polar: https://www.youtube.com/watch?v=PPOEj7piH_E
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Parametric: https://www.youtube.com/watch?v=kmG9ee2BXKM
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Logistic differential equation: https://www.youtube.com/watch?v=q8Rr8m2TAo8
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Numerical integration: https://www.youtube.com/watch?v=7vRSYZwSXrk
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Maclaurin series: https://www.youtube.com/watch?v=SHu_z4YpBmw
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Cubics: https://www.youtube.com/watch?v=VvF6XCGFaUQ
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Torus (solid of revolution): https://www.youtube.com/watch?v=TweCrW1ENhk
My professor says I need three roots for this function: x^3 - x^2 - 2x - 1=0.
I only get one using Newton's method, I don't know how to get three.
Am I applying the correct formula or is there something else I should be using?
If I have x(x^2-2x-2)=3 Am I allowed to say x=3, together with the other non solutions (i)? The basics got me hard rn
I really need help with this problem. With the information given I found 4 equations, two f’(x) and two f(x). I tried solving them but I’m not getting anywhere. Any help is appreciated
Hey all,
I'm working on finding the inverse of the equation y = 3x^2 - 2x^3, aka the "Ease Function of Order 3". I only care about inverting the part of the original function where 0 < x < 1. However, When I plug into Wolfram Alpha, I only get the outer branches of the inverse and not the middle, so basically every part of the original function that is not between 0 and 1.
My thinking is that WA is ignoring the "plus-or-minus" cases and just spitting out the first solution, which is the uppermost branch. So I tried flipping signs and only got the lowermost branch.
All I want is the middle branch. Anyone better at inverting cubics than I?
It's all here, the question and my work.
You can probably tell I started phoning it in towards the end because I have no clue what is going on with this problem! lol. I've been coming back to it for the last couple of days and I give up. I hate this solution and know it sucks!
DO NOT GIVE ME THE ANSWER just yet, but I would appreciate a helpful nudge and information about whether I am warm or cold so far.
Like how do I find formula for an ax^3 + bx^2 + cx + d with Just points I am given?
Hey everyone,
I'm new here and I have just started looking at cubic functions in algebra and I'm trying to gain insight into something but I'm not even sure what question to ask, but I'll try anyway.
Please take a look at the graph.
https://preview.redd.it/8k5pcb4dplz41.png?width=1920&format=png&auto=webp&s=0ab430d01b9075d53f9d026e34ce0ce609d45222
As you can see I've graphed x^(2) , x^(3) and x^(3)-x^(2). What I'm trying to understand is the curve of x^(3)-x^(2) when x<1. I understand the inflection point of x^(3)-x^(2) is where the difference between x^(2) and x^(3) starts decreasing rather than increasing, which also happens to be 2/3rds of the way through. Why does the difference between them start getting smaller at 2/3rds which also happens to be the exponents in the equation. Also if I do x^(4) - x^(3) then the inflection point happens to be at exactly 3/4's and so on. Can anyone shed some light on this for me? Is there even any insight to be gained here or am I just wasting my time? I'm trying to get an intuitive understanding of why this is the case.
I've seen multiple people make a quadratic that goes through three points, but I've never seen a cubic that goes through four points. I'm guessing some of the functions could be greatly simplified, but I do all my graphs without looking anything up, and this is as simplified as I could get it:
https://www.desmos.com/calculator/r0slnoidco
If any of you are curious how it works I'd be happy to explain (be prepared - its significantly more complicated than quadratics because quadratics have a useful property that cubics don't)
