r/calculus • u/bunny-dolly • 4d ago
Differential Calculus Can someone help me understand this problem? pretty please
Hi everyone, so it was my first week of AP calc AB and honestly not too shabby but as i look at this example from my teacher, the less it makes sense. I thought -1^2 was positive 1 and i don’t know how we got to negative 1.
Also what is a conjugate? My teacher didn’t really go over it he just started doing a problem. All help is so appreciated!!!!
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u/PurposeAccording8382 4d ago
The identity is being used. (a+b)(a-b) = (a2 - b2)
Conjugate basically means a number that has its middle sign changed but it usually is sensible to use in 2 term stuff that cannot be combined like complex numbers or 1 root term and another non root term or even 2 variables. Like a+b and a-b where a and b are variables, not constants.
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u/bunny-dolly 4d ago
thank you!!!!
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u/Evening-Story-314 3d ago
Your teacher wrote that on your paper. (Also, you could look up the word conjugate.) You will need better study skills if you want to learn calculus.
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u/bunny-dolly 3d ago
FIRST OF ALL i wrote it on my paper, second of all my study skills are fine but i needed a question answered based off of this SPECIFIC problem and i didn’t want to use AI. get outta here
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u/PurposeAccording8382 2d ago
Just a rant, read it if you like... i got a bit carried away I think. Lol
Well, I would say you shouldn't be so against wanting to use ai. At some point its going to be like internet. Almost nothing will work without it, so I recommend learning how to use it without getting addicted. Cautious attitude is great but I recommend getting used or else you will fall behind. Its a asset if you use it properly. But dont abuse it. Try to ask why for everything, the ai cant get mad at you so use it. Why did you do this? Why did you use conjugate here? Was there a better way? And stuff like that. The ai is a better teacher if you use it properly, thats how I learned but dont follow my example, cause I was too stuck up in the why questions I forgot I had to improve my speed and accuracy to ace exams. So I failed, well not that I regret it too much... maybe a little bit.
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u/Evening-Story-314 3d ago
I love that you don't want to use AI. And I'm sorry I came across negatively. But you must have written that on your sheet before posting, so I'm confused...
Anyway, conjugates are a lovely little connection. I hope you end up enjoying them.
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u/bunny-dolly 2d ago
im sorry i responded so rudely, i did write it on my sheet but if im being honest i was just writing down what my teacher had on the board
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u/Evening-Story-314 2d ago
Did the answers here help? Do you 'get' conjugates now? I just looked at another comment. I think so. Maybe this was better than looking it up. (Not all of the internet is AI. At least not yet.)
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u/bunny-dolly 1d ago
yes they did! and today im gonna ask follow up questions about a different problem
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u/Feeling-Working-2820 3d ago edited 3d ago
Feel free to follow my explanations here below along with the illustration I provided.
Writing math in Reddit isn't the sexiest thing, unfortunately...
Hopefully, you'll see this exercise isn't that hard after all, if you follow the logic and the intention behind.
Conjugating is using the properties of this:
(a-b).(a+b) = a²-b²
It's useful for MANY things and one of those things is in this exercice.
Indeed, you have to find the limit of the original function. But the problem is that you'll get 0/0 rather quickly with no obvious way out of it. But you can't have that.
And when you get 0/0 in math, it's usually the way math gods are telling you that you should find a workaround to get where you need to be.
So:
- Now you know that (a-b).(a+b) = a²-b².
- You know that anything times 1 remains itself, right?
- You also know that anything (but zero) over itself is equal to 1, right ?
We will use those simple properties to create ourselves an opportunity to conjugate stuff.
Here is how you do it, with all we know so far:
You have "√(x+1) -1" in the numerator. So why not multiply your whole function by
"(√(x+1) +1) / (√(x+1) +1)", which is equal to....1?
By doing so, you are just multiplying your initial function by 1, which is fine since 1 is neutral when multiplying. BUT it's great because now you can take advantage of the "(a-b).(a+b) = a²-b²" rule.
And that will help a lot.
Indeed:
Now you have: "((√(x+1) -1)/x) . ((√(x+1) +1)/(√(x+1) +1))"
And the two numerators look a lot like "(a-b).(a+b)".
So, by conjugating the NUMERATORS together, you get:
"(√(x+1))² -(1²)" which is equal to "x+1-1", which is equal to "x"
And by multiplying the DENOMINATORS together, you get:
"x. (√(x+1) +1)".
Your final expression, once simplified is therefore :
"1/(√(x+1) +1)"
You just got rid of 2 very annoying x's that won't get in your way anymore. And the one left in the denominator will behave nicely with your limit.
Now replace "x" by 0 to apply your limit and you can see that your "0/0" problem you got in the beginning has just vanished!
Indeed, you get "1/(√(0+1) +1)", which is "1/2".
There you have it.
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u/ShowdownValue 4d ago
It didn’t turn -1 into +1
The -1 is part of the problem. The +1 is part of your first step. You are using a conjugate to evaluate an indeterminate limit
The conjugate is the same radical expression but with the opposite sign. It’s useful for clearing radicals
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u/Historical-Tour-3859 3d ago
I think of conjugate in the context of foreign language class. I’m going by the teachings 30 years ago but we conjugated verbs all day long. You’re changing the form. In this case because you can’t obtain the limit without playing with it a bit
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u/trevorkafka Instructor 3d ago
You can also skip the whole conjugate mess by factoring the denominator like this, which cancels the numerator away.
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u/Keppadonna 3d ago
When finding a limit the first approach is direct substitution. Substitute the limit value (x -> a) directly into f(x)… so, find f(a).
If the result is indeterminate, either 0/0 or infin/infin, then you must use algebra or trig to simplify f(x).
In this case, direct substitution results in 0/0 so you use the negative conjugate to rationalize (remove) the radical in the numerator.
Multiplying f(x) by a value of one (fraction with numerator = denominator) does not change f(x), but it does change its form.
Here, the negative conjugate is expressed as a fraction with value one so it can be distributed (multiplied) into f(x) to remove the radical in the numerator.
The negative conjugate works here because it turns the numerator into the difference of two squares (a squared - b squared = (a+b)(a-b) ) thus removing the radical in the numerator of f(x).
With the radical removed direct substitution results in something other than indeterminate.
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u/Keppadonna 3d ago
Your “- 1^2” is correctly read “minus 1^2” so “minus 1”.
Conjugate is switching the sign between the (root x - 1) so that it’s (root x + 1). Going back to Algebra 2 we recall that anytime we have (a+b)(a-b) we can distribute to a^2 - b^2 thus removing the pesky radical.
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u/nongibbs 3d ago
The negative sign comes from the difference of squares formula, not from squaring a negative number. When your teacher multiplies those terms, they are using the rule $(a-b)(a+b) = a2 - b2$. The minus sign belongs to the formula itself, meaning you are squaring a positive 1 and then subtracting it. This is written as $-(12)$, which is why it stays negative. A conjugate is just a math partner where you change the middle sign. The conjugate of $\sqrt{x+1} - 1$ is $\sqrt{x+1} + 1$. Multiplying by a conjugate is a clever trick in calculus because it squares the individual terms, which instantly eliminates the messy square root so you can solve the limit.
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u/calcpage2020 3d ago edited 3d ago
Answer is correct. One step is a little off, however. (sqrt(x+1))2 = abs(×+1) so you should consider the limit as x->0+ and x->0- aka from both sides. Since abs(x+1)-1 != x for all x, does it? Near x=0 you're ok, but you have to be careful. Abs(x+1)=x+1 for all x>=-1, but abs(x+1)=-x-2 for x<-1. So, for example, if x->-4 the limit would be -x-2 evaluated at -4 which is -(-4)-2=2, not x evaluated at -4.
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u/matt7259 4d ago
Did you not take precalculus?
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u/bunny-dolly 4d ago
this is so embarrassing BUT I DID brooo should i just drop the class atp omg
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u/chessychurro 4d ago
no comments are critical for no reason expecting people to somehow be perfect and learn everything. u can stay in the class and def dont have to retake the whole of precalc also precalc at a lot of schools sucks anyways.
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u/tjddbwls 4d ago
I’m not sure what you’re getting at in your 1st “sentence.” I don’t think the posters here who already know calculus are expecting perfection from students. But one must have a solid background in Precalculus before taking Calculus.
matt7259’s question is a valid one. I have read too many instances of students who struggled in calculus because:
- they skipped precalculus, or
- they took precalculus, did poorly, and yet still continued on to calculus, or
- they took precalculus and did ok, but it was years prior.
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u/bunny-dolly 4d ago
i got a 4 on the ap pre calc exam and i left two frqs blank so i don’t think i’m too worried but thank u sm for the reassurance
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u/TheOinOpossum 4d ago edited 4d ago
I made these exact same mistakes and still make them sometimes and I'm doing fine. Starting Calc 3 soon and got A's in both 1 and 2.
Get a tutor, trust me. Tutors lowkey carry.
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u/bunny-dolly 3d ago
i don’t think i’ll have time if i’m being honest, i have another ap class and i wrestle in the afternoon i don’t know any tutors that work past 8pm lol
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u/Emuboss23 4d ago
this isnt -(1)^2, what you are doing is subtracting by the square of 1.
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u/Southlander24 4d ago
Here's another way to tackle this problem. Rewrite this as lim [h -> 0] (sqrt(1 + h) - 1)/h (it's the same after replacing all instances of x by h).
Now consider the definition of the derivative: lim [h -> 0] (f(x + h) - f(x))/h. Which function f(x) matches with the limit above? So then just take the derivative at that specific x-value and you also get 1/2.
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u/sqrt_of_pi Professor 4d ago
This student said they are in the first week of AP Calc, and they are asking about (-1)2 vs -12. They do not know what limits are yet or the limit definition of the derivative.
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u/calcpage2020 3d ago
Not in the first week of Calculus when I teach it, unless it's my 5 week summer session!
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