Math (linear approximation)

posted by .

Find a linear approximation of the function f(x)=(1+x)^(1/4) at a=1, and use it to approximate the numbers (.95)^(1/4) and (1.1)^(1/4).
Round your answers to the nearest thousandth

Cheers in advance!

  • Math (linear approximation) -

    Any curve can be approximated by a straight line, in a small enough interval. So, we want the tangent line at x=0, which will be very close to the curve, if we stay close enough to x=0.

    y=(1+x)^(1/4)
    y(0) = 1

    y' = 1/4 * (1+x)^(-3/4)
    = 1/[4(1+x)^3]

    y'(0) = 1/4

    So, the line y = x/4 + 1 is tangent to f(x) at x=0

    let x = -0.05
    y = -.0125 + 1 = 0.9875

    If x = .1
    y = .025 + 1 = 1.025

    Just to check,

    .95^(1/4) = 0.9873
    1.1^(1/4) = 1.0241

    Since |.95 - 1| < |1.1 - 1| the approximation is better

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. Calculus

    Suppose that you can calculate the derivative of a function using the formula f(x) = 2f(x) + 4x. If the output value of the function at x=4 is 3 estimate the value of the function at 4.012. Linear Approximation right?
  2. Math

    Use differential, i.e., linear approximation, to approximate (125.4^(1/3)) as follows: Let f(x)=x^(1/3) . The linear approximation to f(x) at x=125 can be written in the form y=mx+b where m is: m=1/75 b=?
  3. calculus

    se differential, i.e., linear approximation, to approximate (8.4)^(1/3) as follows: Let f(x)=(x )^(1/3). The linear approximation to f(x) at x=8 can be written in the form y=mx+b where m is: and where b is: Using this, we find our …
  4. Calc....

    Verify the given linear approximation at a = 0.Then determine the values of x for which the linear approximation is accurate to within 0.1. (Enter your answer using interval notation. Round your answers to three decimal places.) ln(1 …
  5. Help please ???

    Verify the given linear approximation at a = 0.Then determine the values of x for which the linear approximation is accurate to within 0.1. (Enter your answer using interval notation. Round your answers to three decimal places.) (1+2x)^(1/4)¡Ö …
  6. Calculus(Urgent help)

    f(x)=x^3-3x^2+3x+1 near 2 (at a=2) A. Use Linear approximation to approximate. b.is this underestimate or overestimate?
  7. math

    Verify the given linear approximation at a = 0. Then determine the values of x for which the linear approximation is accurate to within 0.1. (Enter your answer using interval notation. Round your answers to three decimal places.)
  8. math

    Verify the given linear approximation at a = 0. Then determine the values of x for which the linear approximation is accurate to within 0.1. (Enter your answer using interval notation. Round your answers to three decimal places.) 1/(1 …
  9. linear approximation

    a) Find the linear approximation of the function f(x)=4cos(3x^2) near x = 0 b) Use the approximation found in part (a) to estimate the value of f(pi/20)
  10. Math

    Use differential, (i.e. linear approximation), to approximate cube root of 64.1 as follows: Let f(x) = cube root of x. The linear approximation to f(x) at x = 64 can be written in the form y = mx+b. Compute m and b and find cube root …

More Similar Questions