Math

posted by .

Use linear approximation to show that each function below can be approximated by the given expression when |x| is small

i.) sinx = x

ii.) e^x = 1+x

  • Math -

    using Taylor series,

    sinx = x - x^3/3! + x^5/5! - ...
    e^x = 1 + x + x^2/2! + ...

    when |x| is small, the higher powers vanish.

    Or, just using calculus,

    y=sinx
    y' = cosx
    at x=0, y=0, y'=1
    so y can be approximated by the line through (0,0) with slope 1, or y=x

    similarly for e^x
    The line through (0,1) with slope 1 is y = 1+x

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. Calculus

    Find the linear approximation L(x) of the given function at a, and use it to approximate the function value at the given x-value. f(x)=x^(1/3), a=-27, x=-28 I found the derivative to be f'(x)= 1/[3(-27)^(2/3)] but it does not exist …
  3. Math (linear approximation)

    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!
  4. Math

    a.) Given that f(3)=5 and f'(x)=x/((x^3)+3), find the linear approximation of f(x) at x=3. b.)If the linear approximation is used to estimate the value of f(2.9), will it be an overestimation or underestimation?
  5. Calculus

    a.) Given that f(3)=5 and f'(x)=x/((x^3)+3), find the linear approximation of f(x) at x=3. b.)If the linear approximation is used to estimate the value of f(2.9), will it be an overestimation or underestimation?
  6. Math

    Use linear approximation to show that each function below can be approximated by the given expression when |x| is small i.) sinx ~ x ii.) e^x ~ 1+x iii.) ln(1+x) ~ x
  7. Math

    Use the linear approximation (1+x)^k=1+kx to find an approximation for the function (3+3x)^(1/3) for values of x near zero. Help would be greatly appreciated!
  8. 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)
  9. calc

    For each function below find the best linear approximation (linearization) at the given value. f(x) = x^-1 at x = 2 y = 1/2 f ' (x) = -1x^-2 at x = 2 : -1/4 y - 1/2 = -1/4 (x-2) y - 1/2 = -1/4x + 1/2 y = -1/4x + 1 Use the appropriate …
  10. Math - Linear Approximation

    a) Find a linear approximation of y=sinx at x=pi/6 b) use part (a) to approximate sin(61pi/360) and sin(59pi/360) I just really have no idea how to approach this problem. I know the formula is y=f(a)+f'(a)(x-a). Does that mean it would …

More Similar Questions