Question

Linear approximation question

Answer 1

Use the linear approximation of:

\[f(x)=\sin ^2(x)\]

at x=\[\pi/6\]

to estimate the value of \[\sin ^2(31) \]

Answer 2

I got:

\[\frac{ 1+4\sqrt{3} }{ 4 }\]

Is that right?

Answer 3

@satellite73

Answer 4

That's 31 degrees btw.

Answer 5

god another stupid math teacher mistake, sorry this problem is not correct

Answer 6

i can tell you how they want you to do it, but it is not right

Answer 7

Tell me anyways.

Answer 8

My answer is nowhere close to the actually value of 0.265 btw.

Answer 9

find the equation of the line tangent to the graph of \(y=\sin^2(x)\) at \((\frac{\pi}{6},\frac{1}{2})\)

Answer 10

sorry that would be \((\frac{\pi}{6},\frac{1}{4})\)

Answer 11

I was gonna say :P .

Answer 12

derivative is \(2\sin(x)\cos(x)\)

Answer 13

Yep...

Answer 14

evaluate at \(\frac{\pi}{6}\) get \(\frac{\sqrt{3}}{2}\)

Answer 15

We could change that to 2sin(2x) ...

Answer 16

linear approximation is therefore
\[y-\frac{1}{4}=\frac{\sqrt{3}}{2}(x-\frac{\pi}{6})\]

Answer 17

But putting 31 degrees gets you nowhere close to the actual value at all.

Answer 18

or if you prefer
\[y=\frac{\sqrt{3}}{2}\left(x-\frac{\pi}{6}\right)+\frac{1}{4}\]

Answer 19

Yes I converted the pi/6 .

Answer 20

well you have to convert 31 degrees to radians, not the other way around

Answer 21

Why? O_o .

Answer 22

\[\sin^2(x)=\frac{1-\cos(2x)}{2}\]

Answer 23

Coudn't I do it both ways?

Answer 24

sine and cosine are functions

Answer 25

that is, they are functions of number, regular old numbers. they only correspond to functions of angles if the angles are measured in radians, not in degrees

Answer 26

so for example, if you are thinking of sine as a function of angles measure in radians, then the derivative of that version of sine is NOT cosine

Answer 27

Hmm... Okay.

Answer 28

that is why this is a stupid question
your input needs to be numbers, not degrees

Answer 29

Darn... I have to use a calculator then :/ .

Answer 30

yeah your job is to convert 31 degrees to radians, then replace \(x\) by that number and see what you get
it is a stupid question, and a stupid calculator exercise, stupider still because if you need a calculator you might as well put it in degree mode and find \(\sin^2(31)\)

Answer 31

Well I got it by using radians. Thanks anyways :) .

Answer 32

yw

Answer 33

hope it is clear that the linear approximation would not be
\[y=\frac{\sqrt{3}}{2}(x-30)+\frac{1}{4}\] because the derivative of sine is not cosine if you are working in degrees, so everything from there on is wrong

Answer 34

further discussion you can read here
http://answers.yahoo.com/question/index?qid=20080129223349AAi6win
or maybe find a better explanation on line

Answer 35

I just left it as:

\[\frac{ 45+\pi \sqrt{3} }{ 180 }\]

Answer 36

You're being too harsh @satellite. You only need to convert 1 degrees (x-pi/6) into radians, which is approximately 0.02.

So \[\Delta y = \sqrt{3}/100 \approx 0.017\]

This gives the answer 0.267 and can be done without paper.

Answer 37

@beginnersmind : I just needed to know that I had to use radians. I did the rest of it my own way.

Answer 38

I know I meant satellite was harsh in saying it's a stupid question by your teacher. You can actually get a reasonable estimate without a calculator.