Using the Tangent Line to Approximate Function Values

Key Questions

  • How do you use the tangent line approximation to approximate the value of ln(1004) ?

    A formula for a tangent line approximation of a function f, also called linear approximation , is given by

    f(x)~~f(a)+f'(a)(x-a),
    which is a good approximation for x when it is close enough to a.

    I'm not sure, but I think the question is about approximate the value ln(1.004). Could you verify it please? Otherwise we will need to know an approximation to ln(10).

    In this case, we have f(x)=ln(x), x=1.004 and a=1.

    Since,
    f'(x)=1/x=>f'(1)=1 and ln(1)=0, we get

    ln(1.004)~~ln(1)+1*(1.004-1)=0.004.

    Reginaldo D. · 1 · Oct 28 2014
  • How do you find the tangent line approximation for f(x)=sqrt(1+x) near x=0 ?

    We need to find the derivative of f(x). We need to use the Chain Rule to find the derivative of f(x).

    f(x)=sqrt(1+x)=(1+x)^(1/2)

    f'(x)=(1/2)(1+x)^((1/2-1))*1

    f'(x)=(1/2)(1+x)^((1/2-2/2))

    f'(x)=(1/2)(1+x)^((-1/2))

    f'(x)=1/(2sqrt(1+x))

    f'(0)=1/(2sqrt(1+0))=1/(2sqrt(1))=1/2=0.5

    AJ Speller · 3 · Sep 24 2014
  • How do you find the linear approximation of a function?

    The linear approximation L(x) of a function is done using the tangent line to the graph of the function. The equation of the tangent line at x=a is given by

    y=f'(a)(x-a)+f(a),

    the linear approximation is

    L(x)=f'(a)(x-a)+f(a).

    Wataru · · Sep 21 2014

Questions

Applications of Derivatives