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Thread: linear approximation

  1. #1
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    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)
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by bobby77 View Post
    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
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  3. #3
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    Quote Originally Posted by bobby77 View Post
    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 ).
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