10.10 Lagrange Error Bound

• What are Taylor polynomials used for?

  Approximating function values
• What is the general formula for the Taylor polynomial of degree $n$ centered at $a$?


  T_{n}(x) = \sum_{k = 0}^{n} \frac{f^{(k)}(a)}{k!}(x - a)^{k}</latex>
• The Lagrange Error Bound provides an upper bound for the error in a Taylor polynomial approximation.


  True
• The Lagrange Error Bound estimates the maximum error when using a Taylor polynomial to approximate a function.

  True
• To estimate the error in approximating $e^{0.2}$ with T_{2}(x) = 1 + x + \frac{x^{2}}{2}</latex>, we use $M =$$1.22$ as the upper bound for $|f'''(x)|$ on the interval [0, 0.2].
• The variable $n$ in the Lagrange Error Bound represents the degree of the Taylor polynomial.
• The Lagrange Error Bound estimates the maximum error when using a Taylor polynomial to approximate a function.


  True
• What is the function being approximated in the example given in the material?

  $e^{x}$
• What is the interval of approximation expressed as?

  $[a, x]$
• The maximum value of $f'''(x)$ for $f(x) =$$e^{x}$ on $[0, 0.2]$ is 1.22
• Match the term with its description:

  Center of the Taylor series ↔️ $a$

  Point of approximation ↔️ $x$
• What is the formula for the Taylor polynomial $T_{n}(x)$ of degree $n$ centered at $a$?


  $\sum_{k = 0}^{n} \frac{f^{(k)}(a)}{k!}(x - a)^{k}$
• The second-degree Taylor polynomial for $f(x) =$$e^{x}$ centered at a = 0</latex> is 1 + x + x^2/2
• What is the formula for the Lagrange Error Bound?

  |E_{n}(x)| \leq \frac{M}{(n + 1)!} |x - a|^{n + 1}</latex>
• The Lagrange Error Bound estimates the maximum error when using a Taylor polynomial.


  True
• Steps to identify the maximum value of the $(n + 1)$th derivative on a given interval:


  1️⃣ Calculate the $(n + 1)$th derivative of $f(x)$

  2️⃣ Identify the interval $[a, x]$

  3️⃣ Find critical points within the interval

  4️⃣ Evaluate the $(n + 1)$th derivative at endpoints and critical points

  5️⃣ Determine the maximum value
• What is the maximum value of $f'''(x)$ for $f(x) =$$e^{x}$ on $[0, 0.2]$?


  1.22
• The $(n + 1)$th derivative of $f(x) =$$e^{x}$ is $e^{x}$


  True
• The Lagrange Error Bound formula includes the term $|x - a|^{n + 1}$, which measures the distance between the point of approximation and the center
• What is the first step in applying the Lagrange Error Bound formula?

  Identify the degree $n$
• Match the variable in the Lagrange Error Bound formula with its meaning:

  $n$ ↔️ Degree of the Taylor polynomial

  $M$ ↔️ Upper bound for $f^{(n + 1)}(x)$

  $a$ ↔️ Center of the Taylor series
• A Taylor polynomial approximates the value of a function at a given point
• The approximation error in a Taylor polynomial is the absolute difference between the actual function value and the Taylor polynomial value.
• What is the formula for the Lagrange Error Bound?

  |E_{n}(x)| \leq \frac{M}{(n + 1)!} |x - a|^{n + 1}</latex>
• What is the Lagrange Error Bound formula?

  $|E_{n}(x)| \leq \frac{M}{(n + 1)!} |x - a|^{n + 1}$
• What does the variable $a$ represent in the Lagrange Error Bound formula?


  Center of Taylor series
• What does the variable $M$ represent in the Lagrange Error Bound formula?


  Upper bound of derivative
• Match the variables in the Lagrange Error Bound formula with their descriptions:

  $|E_{n}(x)|$ ↔️ Maximum error in approximation

  $n$ ↔️ Degree of Taylor polynomial

  $M$ ↔️ Upper bound of derivative
• Steps to find the maximum value of the $(n + 1)^{th}$ derivative:


  1️⃣ Calculate the $(n + 1)^{th}$ derivative

  2️⃣ Identify the interval

  3️⃣ Check for critical points

  4️⃣ Evaluate the derivative at critical points and endpoints

  5️⃣ Determine the maximum value
• What is the value of $f'''(0)$ for $f(x) =$$e^{x}$?


  1
• The maximum value of the third derivative of f(x) = e^{x}</latex> on $[0, 0.2]$ is $e^{0.2}$.


  True
• The interval of approximation for f(x) = e^{x}</latex> centered at $a =$$0$ and approximating at $x =$$0.2$ is [0, 0.2]
• The approximation error is the difference between the actual function value and the Taylor polynomial value.


  True
• Match the concept with its formula:

  Taylor Polynomial ↔️ $T_{n}(x) =$$\sum_{k = 0}^{n} \frac{f^{(k)}(a)}{k!}(x - a)^{k}$

  Error ↔️ $E_{n}(x) =$$|f(x) - T_{n}(x)|$
• In the Lagrange Error Bound formula, $M$ represents an upper bound for the absolute value of the $(n + 1)$th derivative
• What is the maximum value of $f'''(x)$ for $f(x) =$$e^{x}$ on $[0, 0.2]$?


  1.22
• The interval $[a, x]$ represents the range over which the function is approximated
• What does the term $a$ represent in the context of a Taylor series approximation?


  Center of the series
• The value $M$ in the Lagrange Error Bound formula is an upper bound for the $(n + 1)$th derivative on the interval $[a, x]$


  True
• What does a smaller Lagrange Error Bound indicate about the Taylor polynomial approximation?

  More accurate approximation