10.10 Lagrange Error Bound

Cards (56)

  • What does the Lagrange Error Bound provide an upper limit for?
    Error in Taylor approximation
  • Rn(x)R_{n}(x) represents the remainder or error
  • The value M in the Lagrange Error Bound formula is the maximum value of the (n + 1)th derivative of the function on the interval containing x and a.
  • Steps to calculate the Lagrange Error Bound
    1️⃣ Find the (n + 1)th derivative of the function
    2️⃣ Determine the maximum value M of the derivative on the interval
    3️⃣ Plug the values into the Lagrange Error Bound formula
  • Match the variables in the Lagrange Error Bound formula with their descriptions:
    R_{n}(x) ↔️ Remainder or error
    M ↔️ Maximum value of (n + 1)th derivative
    n ↔️ Degree of the Taylor polynomial
    a ↔️ Center of the Taylor series
  • The Lagrange Error Bound helps ensure that the Taylor polynomial provides a reliable approximation of the function within a specified interval.
  • The remainder term, Rn(x)R_{n}(x), quantifies the difference between the true function value and the approximation provided by the Taylor
  • What formula is used to calculate the remainder term based on the Lagrange Error Bound?
    Rn(x)M(n+1)!xan+1|R_{n}(x)| \leq \frac{M}{(n + 1)!} |x - a|^{n + 1}
  • The value M in the remainder term formula is the maximum value of the (n + 1)th derivative of the function on the interval containing x and a.
  • The remainder term, R_{n}(x)</latex>, represents the error in approximating a function using its Taylor or Maclaurin
  • What does the Lagrange Error Bound estimate the maximum error in?
    Approximating with Taylor polynomial
  • The remainder term, Rn(x)R_{n}(x), quantifies the difference between the true function value and the approximation provided by the nth degree Taylor polynomial
  • What is the formula for the Lagrange Error Bound?
    Rn(x)M(n+1)!xan+1|R_{n}(x)| \leq \frac{M}{(n + 1)!} |x - a|^{n + 1}
  • The value M in the Lagrange Error Bound formula is the maximum value of the (n + 1)th derivative of the function on the interval containing x and a.
  • Rn(x)R_{n}(x) in the Lagrange Error Bound represents the remainder
  • What is the second-degree Taylor polynomial P2(x)P_{2}(x) of f(x)=f(x) =ex e^{x} around a=a =0 0?

    1+1 +x+ x +x22 \frac{x^{2}}{2}
  • The error bound for approximating e^{0.5}</latex> using P2(0.5)P_{2}(0.5) is approximately 0.0343.
  • The maximum error in approximating e0.5e^{0.5} using P2(0.5)P_{2}(0.5) is approximately 0.0343
  • What does the Lagrange Error Bound formula provide an upper limit for?
    The approximation error
  • The Lagrange Error Bound formula is |R_{n}(x)| \leq \frac{M}{(n + 1)!} |x - a|^{n + 1}</latex>.
  • In the Lagrange Error Bound formula, MM is the maximum value of the (n+1)th(n + 1)^{th} derivative on the given interval
  • Match the condition for using the Lagrange Error Bound with its description:
    Function is Differentiable ↔️ f(x)f(x) must be differentiable up to at least n+n +1 1 derivatives
    Maximum Derivative Value Known ↔️ MM, the maximum value of the (n+1)th(n + 1)^{th} derivative, must be known or estimated
    Taylor Polynomial Exists ↔️ f(x)f(x) must have a Taylor polynomial Pn(x)P_{n}(x) centered at aa
  • Steps to calculate the Lagrange Error Bound
    1️⃣ Determine the Taylor polynomial Pn(x)P_{n}(x)
    2️⃣ Find the (n+1)th(n + 1)^{th} derivative of f(x)f(x)
    3️⃣ Calculate or estimate MM
    4️⃣ Apply the Lagrange Error Bound formula
  • The third derivative of f(x)=f(x) =ex e^{x} is exe^{x}.
  • In the example of approximating exe^{x}, the maximum value of the third derivative MM at x=x =0.5 0.5 is approximately 1.649
  • The maximum error in approximating e^{0.5}</latex> using the second-degree Taylor polynomial is approximately 0.0343.
  • What does the Lagrange Error Bound estimate the maximum error for?
    Taylor polynomial approximation
  • Match the component of the Lagrange Error Bound with its description:
    Rn(x)R_{n}(x) ↔️ Remainder (error)
    MM ↔️ Maximum value of f(n+1)(x)f^{(n + 1)}(x)
    nn ↔️ Degree of Taylor polynomial
    aa ↔️ Center of Taylor series
  • The Lagrange Error Bound formula is Rn(x)M(n+1)!xan+1|R_{n}(x)| \leq \frac{M}{(n + 1)!} |x - a|^{n + 1}, where nn represents the degree of the Taylor polynomial
  • What does the remainder term R_{n}(x)</latex> represent in Taylor polynomial approximation?
    The approximation error
  • The maximum value of the (n+1)th(n + 1)^{th} derivative in the Lagrange Error Bound is denoted by MM.
  • What is the fourth derivative of exe^{x}?

    exe^{x}
  • The remainder term for approximating exe^{x} with a third-degree Maclaurin polynomial atx = 0.1</latex> is approximately 0.0000046
  • The third-degree Maclaurin polynomial provides a highly accurate approximation of e0.1e^{0.1}.
  • Match the component of the Lagrange Error Bound with its description:
    Rn(x)R_{n}(x) ↔️ Remainder (error)
    MM ↔️ Maximum value of f(n+1)(x)f^{(n + 1)}(x)
    nn ↔️ Degree of Taylor polynomial
    aa ↔️ Center of Taylor series
  • What is the maximum value of the (n+1)th(n + 1)^{th} derivative in the Lagrange Error Bound formula?

    MM
  • The Lagrange Error Bound formula is Rn(x)M(n+1)!xan+1|R_{n}(x)| \leq \frac{M}{(n + 1)!} |x - a|^{n + 1}, where aa represents the center of the Taylor series
  • What is the formula for the Lagrange Error Bound?
    Rn(x)M(n+1)!xan+1|R_{n}(x)| \leq \frac{M}{(n + 1)!} |x - a|^{n + 1}
  • The remainder in the Lagrange Error Bound formula is denoted by R_{n}(x)
  • In the Lagrange Error Bound, M represents the maximum value of the (n+1)th(n + 1)^{th} derivative on the interval containing xx and a</latex>.