Question

Find the linear approximation to f(x) at x = x o. Graph the function and its linear approximation. f(x) = sin 3x, x 0, = 0

Answer 1

@electrokid can u help?

Answer 2

@paddyfitz can u help?

Answer 3

so, a linear approximation at \(x=x_0\) means, that
if \(x\) changes by \(\Delta x\), estimate linearly, what \(f(x+\Delta x)\) would be.

Answer 4

ok so i tried can u check it?

Answer 5

nvm i think i did it wrong :/

Answer 6

no, if x changes by dx, what is the change-> dy?
\[f'(x)=3\cos(3x)\\\implies \Delta f(x_0)=3\cos(3x_0)\times\Delta x\\
f(x_0+\Delta x)=f(x_0)+\Delta f(x_0)\\
\qquad=\sin(3x_0)+3\cos(3x)\Delta x\]

Answer 7

ohh ok what does the triangle represent?

Answer 8

delta-> a small change in..

Answer 9

ohh ok so

Answer 10

following Leibnitz's notation
\[\lim_{\Delta x\to0}\frac{\Delta y}{\Delta x}=\frac{dy}{dx}\]

Answer 11

ohh okay so what would be the linear approx for this problem

Answer 12

yep

Answer 13

infact, for any problem...
try this out
set \(f(x)=\ln(x)\),
follow the same steps,
to find \(\ln(0.1)\), use \(x_0=0, \Delta x=0.1\)
and see how close you are

Answer 14

or may be x=1.1

Answer 15

okay

Answer 16

okay i'm stuck

Answer 17

haha. lets get you un-entangled.
where you at?

Answer 18

so will i start like this f(0) + f'(0) ( x - 0)

Answer 19

?

Answer 20

yes. that is it

Answer 21

but since \(\ln(0)\) is undefined, lets take "x=1" and "dx = 0.1"

Answer 22

okay

Answer 23

so can u help?