So, if you're interested in cube roots of a number (as you are in this example), you will always have three roots. When you take the nth root of a number, there are always n different roots In essence, the takeaway from it in the context of this problem is as follows: The reason comes from the Fundamental Theorem of Algebra. The 1 calculator that I found that showed a real solution was Symbolab. I know this is also the case for -5/3, try that on paper too. If you have any insight on the weirdness, please give me some insight below! Thanks for your time! :D I asked my math teacher and he said it should be real. But when you type in (-7/3)^(-7/3) onto most calculators, google included: I was finding y when x = -7/3 when I noticed something strange. I was using pencil and paper for most of my problems to determine whether the y values (given x) were real or not. Then I started to find as many points I could in the negative section to see if there was a pattern. This lead me to wonder whether there are points in between the integers, and there are! Such as when x = -1/3, yet not all are real numbers. I recognized that some values are complex and cannot be graphed, yet some can, such as (-1, -1) or (-2, 0.25). I noticed the function abruptly stopped at the y-axis and did not extend into the second or third quatrains at all. I'm not exactly good at math so forgive me if this question seems really silly.Ībout a week ago, I was wondering in my math class what the graph x^x looks like, and since we were using the desmos calculator in class, I tried it out. But remember to add C.Don't have a question about this video, but I don't really have a good category to put it. If we are lucky enough to find the function on the result side of a derivative, then (knowing that derivatives and integrals are opposites) we have an answer. Which teaches us to always remember "+C". And the increase in volume can give us back the flow rate.The flow still increases the volume by the same amount.The derivative of the volume x 2+C gives us back the flow rate:Īnd hey, we even get a nice explanation of that "C" value. The integral of the flow rate 2x tells us the volume of water: Derivative: If the tank volume increases by x 2, then the flow rate must be 2x.Integration: With a flow rate of 2x, the tank volume increases by x 2.Imagine the flow starts at 0 and gradually increases (maybe a motor is slowly opening the tap):Īs the flow rate increases, the tank fills up faster and faster: This shows that integrals and derivatives are opposites! We can integrate that flow (add up all the little bits of water) to give us the volume of water in the tank. The input (before integration) is the flow rate from the tap. So we wrap up the idea by just writing + C at the end. So when we reverse the operation (to find the integral) we only know 2x, but there could have been a constant of any value. and the derivative of x 2+99 is also 2x,īecause the derivative of a constant is zero.and the derivative of x 2+4 is also 2x,.It is there because of all the functions whose derivative is 2x: The symbol for "Integral" is a stylish "S"Īfter the Integral Symbol we put the function we want to find the integral of (called the Integrand),Īnd then finish with dx to mean the slices go in the x direction (and approach zero in width). Integration can sometimes be that easy! Notation That simple example can be confirmed by calculating the area:Īrea of triangle = 1 2(base)(height) = 1 2(x)(2x) = x 2
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