use calculus to approximate 1.999^4 (no calculators)
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- čas přidán 28. 08. 2024
- Use calculus to approximate 1.999^4 (no calculators).
We will use local linear approximation to approximate 1.999^4 and we will also use differential to do the same. This is a calculus 1 homework problem from James Stewart's Single Variable Calculus textbook.
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I actually got the same answer by finding the first 2 terms in the binomial expansion of (2+x)^4, with x = -0.001. Neat!
The Beautiful Mind, Dear Dr Barker is here ... Cool.
Binomial? You made a video about it?
I should check your channel, yes.
I think this should always hold because the linearization method in calculus is using Taylor expansion and ignoring all terms after 2nd derivative, where the term with the 1st derivative is exactly the same expression as the 2nd term in the binomial expansion of the shifted function.
How to approximate 1.999^4 whilst using a calculator:
1. type in 1.999^4
2. add 0.0000000001 to the answer, because we want an approximation, not the actual answer
lol
BPRP: use calculus to approximate 1.999^4 (no calculators)
Also BPRP: *uses calculator*
forest trees
Easier to use the formula directly with x = 1.999 and a= 2, that gives L(1.999) = 16 + 32*(-0.0001) = 16 - 0.032 = 15.968.
Trying to simplify the general expression of L(x) leads to a more complicated calculation in this specific case.
another way of calculating 32*1.999 at the last step of the linearization, could be writing 1.999=2-(1/1000) and then proceed with applying distributive property:
32*1.999-48 = 32* (2-1/1000) - 48 = 64 - 0.032- 48 = 16 - 0.032 = 15.968
Yup
Yes good, the same as i thought that make it so easy without calculator
Best approximation of (1.999)^4 is 2^4 I think
not accurate enough without calculus
@@nick46285 As an engineer, this is good enough for me
For me I just did 2 to the power of 4 then minus 0.032
The first was easier to compute before distributing the 32, I even think that the 2 methods look the same in that step, but as always maybe there is a detail that dont make them the same and that I am missing because I am not a mathematician
dear Just Calculus,
can you please add main channel (RedPenBlackPen) into "Channels" submenu of this channel?
This would allow for better navigation on mobile than "main channel" button in "About" submenu
You should have left off at L(x) = 16 + 32 (x - 2), or better, since wanting to use a number less than 2, L(x) = 16 - 32 (2 - x), then it is basically the same as the differential method...
Ah yes. That would have been much better
great explanation
16 i guess lol
I watched this before on your main channel and I liked it very much lol
Using binomial theorem to approximate>>>
Him: no calculators
Also him: uses calculator for 32(1.999)
Very interesting!
2
*Calculus not calculator* ... *I love it*
Thank you Teacher 💖
2^4=16, or even 0, is okay, he said approximate, but didn’t say how accurate
For April fools you should have made a video titled “use a calculator not calculus”
I tried this method using a random number like (2.3)^4 but i didint get the same answer as the calculator. Is it because that this method Is used for certain numbers?
If the x (2.3) is further from your first guess (2), you need more terms in the linearization process. Or you use iterative techniques, like Newton-Raphson method.
2.3 if too far away from 2 for this method to work well.
Useless
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lim _{x -> 2} x⁴= 16
**fast approximation**
I just found 2⁴ which is 16
and since (1.999)⁴