# Thread: linear approximation

1. ## linear approximation

1.Find the linear approximation Lo(x) to the function f(x)=sin(x) at x=0
2.use linear approximation to estimate sin(0.03)

2. Originally Posted by bobby77
1.Find the linear approximation Lo(x) to the function f(x)=sin(x) at x=0
I expect you are expecting an answer of the form:

$\displaystyle Lo(x)=f(0)+x\,f'(0)=0 + x\, \cos(0)=x$

2.use linear approximation to estimate sin(0.03)
Using the above:

$\displaystyle \sin(0.03)\approx Lo(0.03)=0.03$

RonL

3. Originally Posted by bobby77
1.Find the linear approximation Lo(x) to the function f(x)=sin(x) at x=0
2.use linear approximation to estimate sin(0.03)
The equation of a non-vertical line is passing through point $\displaystyle (x_0,y_0)$ and slope $\displaystyle m$ is,
$\displaystyle \boxed{y-y_0=m(x-x_0)}$

You need to find the equation of tangent line at point $\displaystyle (x_0,y_0)=(0,\sin 0)=(0,0)$.

And the slope is the derivative at the point,
$\displaystyle \cos x$ at $\displaystyle x=1$ yields $\displaystyle x=0$.

Thus,
$\displaystyle y-y_0=1(x-0)$
Thus,
$\displaystyle y=x$
Is the linearazation line.

Thus,
$\displaystyle \sin (.03)\approx .03$ (in radians ).