10.10 Lagrange Error Bound

• What does the Lagrange Error Bound provide an upper limit for?

  Error in Taylor approximation
• $R_{n}(x)$ 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, $R_{n}(x)$, 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?

  $|R_{n}(x)| \leq \frac{M}{(n + 1)!} |x - a|^{n + 1}$
• 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, $R_{n}(x)$, 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?

  $|R_{n}(x)| \leq \frac{M}{(n + 1)!} |x - a|^{n + 1}$
• 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.
• $R_{n}(x)$ in the Lagrange Error Bound represents the remainder
• What is the second-degree Taylor polynomial $P_{2}(x)$ of $f(x) =$$e^{x}$ around $a =$$0$?


  $1 +$$x +$$\frac{x^{2}}{2}$
• The error bound for approximating e^{0.5}</latex> using $P_{2}(0.5)$ is approximately 0.0343.
• The maximum error in approximating $e^{0.5}$ using $P_{2}(0.5)$ 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, $M$ is the maximum value of the $(n + 1)^{th}$ derivative on the given interval
• Match the condition for using the Lagrange Error Bound with its description:

  Function is Differentiable ↔️ $f(x)$ must be differentiable up to at least $n +$$1$ derivatives

  Maximum Derivative Value Known ↔️ $M$, the maximum value of the $(n + 1)^{th}$ derivative, must be known or estimated

  Taylor Polynomial Exists ↔️ $f(x)$ must have a Taylor polynomial $P_{n}(x)$ centered at $a$
• Steps to calculate the Lagrange Error Bound

  1️⃣ Determine the Taylor polynomial $P_{n}(x)$

  2️⃣ Find the $(n + 1)^{th}$ derivative of $f(x)$

  3️⃣ Calculate or estimate $M$

  4️⃣ Apply the Lagrange Error Bound formula
• The third derivative of $f(x) =$$e^{x}$ is $e^{x}$.
• In the example of approximating $e^{x}$, the maximum value of the third derivative $M$ at $x =$$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:

  $R_{n}(x)$ ↔️ Remainder (error)

  $M$ ↔️ Maximum value of $f^{(n + 1)}(x)$

  $n$ ↔️ Degree of Taylor polynomial

  $a$ ↔️ Center of Taylor series
• The Lagrange Error Bound formula is $|R_{n}(x)| \leq \frac{M}{(n + 1)!} |x - a|^{n + 1}$, where $n$ 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 $(n + 1)^{th}$ derivative in the Lagrange Error Bound is denoted by $M$.
• What is the fourth derivative of $e^{x}$?


  $e^{x}$
• The remainder term for approximating $e^{x}$ 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 $e^{0.1}$.
• Match the component of the Lagrange Error Bound with its description:

  $R_{n}(x)$ ↔️ Remainder (error)

  $M$ ↔️ Maximum value of $f^{(n + 1)}(x)$

  $n$ ↔️ Degree of Taylor polynomial

  $a$ ↔️ Center of Taylor series
• What is the maximum value of the $(n + 1)^{th}$ derivative in the Lagrange Error Bound formula?


  $M$
• The Lagrange Error Bound formula is $|R_{n}(x)| \leq \frac{M}{(n + 1)!} |x - a|^{n + 1}$, where $a$ represents the center of the Taylor series
• What is the formula for the Lagrange Error Bound?

  $|R_{n}(x)| \leq \frac{M}{(n + 1)!} |x - a|^{n + 1}$
• 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 $(n + 1)^{th}$ derivative on the interval containing $x$ and a</latex>.