![]() ![]() Negative two x squared times, we have here, one minus four x squared. We're going to have, this is equal to e to the Negative two x squared, I'll do that in green. ![]() I'm going to try toįigure out where this is either undefined or where We have that x over there and let's see, can we simplify it at all? Well obviously both of these terms have an e to the negative two x squared. Times negative four x, and of course we have this x over here. That's going to be what, negative four x. We're going to multiply that times the derivative of negative two X squared, well that's just going to be e to the Negative two x squared with respect to negative two That is going to be equal to- We'll just apply the chain rule. Now the derivative of e to the negative two x squared over here. This first part is going to be equal to e to the negative two x squared. What is this going to be? Well all of this stuff in magenta, the derivative of x with respect to x, that's just going to be equal to one. The derivative of e to the negative two x squared Derivative of the x times e to the negative of two x squared plus X squared plus the derivative with respect to x of e to With respect to x of x, so it's going to be that, times e to the negative two We're going to have toĪpply some combination of the product rule and the chain rule. Let's think about how we canįind the derivative of this. The derivative of this with respect to x is either equal Numbers for f we want to figure out all the places where Short for if and only if, f prime of c is equal to zero We would say c is a critical number of f, if and only if. This video and think about, can you find any critical numbers of f. To x times e to the negative two x squared, and we want toįind any critical numbers for f. So it is not considered as a critical point in this case. Apparently x is not a critical point in this case, because when x=3 not only it's derivative h'(3) is undefined but also main function h(3) is also undefined. P.S.: I got it a minute after I submitted the question. It's not just isolated quiz, I did a bunch of them and every time you try to present critical x when f'(x) is undefined it renders my answer incorrect. Yet if you present that both x=3.5 and x=3 are critical number quiz returning answer than 3 is an incorrect answer. Now clearly h'(x) is 0_ when x = 3.5, but also _h'(x) is undefined when x = 3 If you take derivative of that you would get: H(x) = e^2x/(x-3) and a question asking to find critical points. When I'm doing a quiz on Khan Academy I see this: Application to reality is not necessary.0:54 Sal mentioned that 'c' is a critical number 'iff' f'(c) = 0 or f'(c) undefined.īut I did few exercises on Khan Academy that ignores points when f'(x) is undefined and states that answer as incorrect. Now we have this concept of "the complex numbers" that we can further explore. ![]() For any two complex numbers a and b, a^b is complex. The product of two complex numbers is complex.ĥ. The sum of two complex numbers is complex.Ĥ. There is a complex number i such that i²= -1.ģ. To get the complex numbers, we do a similar thing. Any relation to real life is just the result of people applying these abstractions to real-world problems. The word "field" just means that they follow 9 certain rules, like "for every real number x, x+0=x" Likewise, "ordered" just adds about 3 more rules, and "complete" adds one more. The misleadingly-named real numbers are defined as a complete ordered field. Numbers are just concepts that follow certain rules. Yet a vast majority of the real numbers are irrational. That would require splitting atoms and quarks in impossible ways. You can't have exactly √2 apples, or any irrational number of apples. You mention counting, but most numbers aren't used for counting either. Can you show me a 3? Not a drawing or a representation of a 3, but the actual number 3? Of course not. None of the numbers you use in life are real.
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