Tangent Approximation

[masterpizza]

Can someone please explain to me how to do this/what this accomplishes/shows us so I can have a working understanding of it. My instructor went over it but I am a bit confused.

EX ( ?101
EX) [sup:1kk8yvht]3[/sup:1kk8yvht] ?7.9

 

[BigGlenntheHeavy]

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[soroban]

Hello, masterpizza!

These are problems in differential approximations.

This is the basic idea . . .

We have a function: .\(\displaystyle f(x)\:=\:y\)

If we change \(\displaystyle x\) by some small amount, \(\displaystyle \Delta x\)
. . the value of \(\displaystyle y\) will also change:. . \(\displaystyle f(x + \Delta x) \:=\:y + \Delta y\)

For small values of \(\displaystyle \Delta x\), the differential \(\displaystyle dy\) is approximately \(\displaystyle \Delta y.\)
. . That is:. . \(\displaystyle f(x+\Delta x) \;\approx\;y + dy\)

\(\displaystyle (1)\;\sqrt{101}\)

The function is the square-root function: .\(\displaystyle f(x) \:=\:\sqrt{x} \:=\:x^{\frac{1}{2}}\)

Note that 101 is "close" to 100, and we know that: .\(\displaystyle \sqrt{100} = 10\)

So, \(\displaystyle \sqrt{100+1}\) is equal to 10 "plus a little more".


We have: .\(\displaystyle x = 100,\;dx = 1,\:\text{ and }\,y \:=\:x^{\frac{1}{2}}\)

\(\displaystyle \text{The differential is: }\;dy \:=\:\tfrac{1}{2}x^{-\frac{1}{2}}dx \:=\:\frac{dx}{2\sqrt{x}}\)

\(\displaystyle \text{Substitute: }\:dy \:=\:\frac{1}{2\sqrt{100}} \:=\:\frac{1}{20} \:=\:0.05\quad\text{(the "little more")}\)


\(\displaystyle \text{Therefore: }\:\sqrt{101} \;\approx\;10 + 0.05 \:=\:\boxed{10.05}\)


Check

The actual value of \(\displaystyle \sqrt{101}\) is: .\(\displaystyle 10.04987562...\)

So our approximation is very good!

\(\displaystyle (2)\;\sqrt[3]{7.9}\)

\(\displaystyle \text{We have: }\:f(x) \:=\:\sqrt[3]{x} \:=\:x^{\frac{1}{3}}\)

We note that 7.9 is close to 8, and we know that: .\(\displaystyle \sqrt[3]{8} = 2\)

So that \(\displaystyle \sqrt[3]{8 - 0.1}\) is equal to 2 "minus a little less".


\(\displaystyle \text{We have: }\:x = 8,\;dx = -0.1,\;y \,=\,x^{\frac{1}{3}}\)

\(\displaystyle \text{The differential is: }\:dy \:=\:\tfrac{1}{3}x^{-\frac{2}{3}}dx \:=\:\frac{dx}{3\sqrt[3]{x^2}}\)

\(\displaystyle \text{Substitute: }\:dy \;=\;\frac{-0.1}{3\sqrt[3]{8^2}} \:=\:-\frac{0.1}{12}\quad\text{(the "little less")}\)


\(\displaystyle \text{Therefore: }\:\sqrt[3]{7.9} \;\approx\;2 - \frac{0.1}{12} \;=\;\boxed{1.9916666...}\)


Check

\(\displaystyle \sqrt[3]{7.9} \;=\;1.991631701...\)


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Question: Why do we have to know this?

Suppose you are on Survivor (without a calculator).

And your very lives depend on knowing \(\displaystyle \sqrt{101}\).

You can work out 10.05, writing in the dirt with a stick . . .


Trust me on this, guys.
Chicks really dig this stuff!


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Before you start composing your hate mail . . .

Yes, I know that "chicks" is a sexist term.

But saying "chicks" is not a felony
. . . . . although it is a Ms-demeaner.

 

[Deleted member 4993]

Baby chickens always like short-cuts - what is this noise about felony and what-not????