10.9 Finding Taylor Polynomial Approximations of Functions

• What is a Taylor polynomial used for?

  Function approximation
• What is the general form of the n-th degree Taylor polynomial?

  $P_{n}(x) =$$\sum_{k = 0}^{n} \frac{f^{(k)}(a)}{k!}(x - a)^{k}$
• The term $f^{(k)}(a)$ represents the k-th derivative of $f$ evaluated at $x =$$a$.
• What does $k!$ represent in the Taylor polynomial formula?


  Factorial of k
• The accuracy of a Taylor polynomial approximation increases as $x$ gets closer to a</latex>
• What does Taylor's Theorem allow us to do?

  Approximate functions and quantify error
• Taylor's Theorem requires the function f(x)</latex> to be $(n + 1)$ times differentiable on an open interval containing $a$.
• In Taylor's Theorem, $R_{n}(x)$ represents the remainder term.
• What is the formula for the remainder term $R_{n}(x)$ in Taylor's Theorem?


  $R_{n}(x) =$$\frac{f^{(n + 1)}(c)}{(n + 1)!}(x - a)^{n + 1}$
• Match the term with its definition:

  Taylor Polynomial ↔️ Approximates $f(x)$ near $x =$$a$

  Remainder Term ↔️ Represents the error between $f(x)$ and $P_{n}(x)$
• What is the formula for the Taylor polynomial of degree $n$ centered at $a$?


  $P_{n}(x) =$$\sum_{k = 0}^{n} \frac{f^{(k)}(a)}{k!}(x - a)^{k}$
• The remainder term provides a bound on the approximation error in Taylor's Theorem.
• What is the Taylor polynomial of degree 3 for f(x) = \sin(x)</latex> centered at $a =$$0$?


  $P_{3}(x) =$$x - \frac{x^{3}}{6}$
• What is the remainder term for the Taylor polynomial of degree 3 for $\sin(x)$ centered at $a =$$0$?


  $\frac{\sin(c)}{4!}x^{4}$
• Taylor's Theorem provides a way to approximate a function using a Taylor polynomial while also quantifying the error
• The Taylor polynomial of degree $n$ centered at $a$ for $f(x)$ requires $f(x)$ to be $(n + 1)$ times differentiable on an open interval containing $a$.
• What does the remainder term $R_{n}(x)$ in Taylor's Theorem represent?


  The error of approximation
• The Taylor polynomial $P_{n}(x)$ of degree $n$ is centered at a
• What is the formula for the remainder term R_{n}(x)</latex> in Taylor's Theorem?

  $\frac{f^{(n + 1)}(c)}{(n + 1)!}(x - a)^{n + 1}$
• The remainder term provides a bound on the difference between the actual function value and its Taylor approximation.
• Match the term with its definition:

  Taylor Polynomial ↔️ Approximates f(x)</latex> near $x =$$a$ using its derivatives

  Remainder Term ↔️ Represents the error between $f(x)$ and $P_{n}(x)$
• To find the Taylor polynomial of degree 3 for $\sin(x)$ centered at $a =$$0$, we need to evaluate the derivatives of $\sin(x)$ at 0
• What is the Taylor polynomial of degree 3 for $\sin(x)$ centered at $a =$$0$?


  $x - \frac{x^{3}}{6}$
• The center of a Taylor polynomial is the point about which the polynomial is expanded to approximate a function.a
• What is the general form of the Taylor polynomial of degree $n$ centered at $a$?


  $\sum_{k = 0}^{n} \frac{f^{(k)}(a)}{k!}(x - a)^{k}$
• The Taylor polynomial of degree 2 for $\cos(x)$ centered at $a =$$0$ is 1 - \frac{x^{2}}{2}</latex>.
• Which region is best approximated by the Taylor polynomial of degree 2 for $\cos(x)$ centered at $a =$$0$?


  Near $x =$$0$
• A Taylor polynomial is a polynomial approximation of a function $f(x)$ near a specific point called its center
• Match the component of the Taylor polynomial with its description:

  Derivatives of $f(x)$ ↔️ Evaluated at $a$

  Powers of $(x - a)$ ↔️ Used to approximate $f(x)$

  Factorials $k!$ ↔️ Divide the derivatives to simplify the polynomial
• What is the Taylor polynomial of degree 2 centered at $a =$$0$ for f(x) = e^{x}</latex>?


  $1 +$$x +$$\frac{x^{2}}{2}$
• What does $R_{n}(x)$ represent in Taylor's Theorem?


  The error term
• The remainder term in Taylor's Theorem is given by R_{n}(x) = \frac{f^{(n + 1)}(c)}{(n + 1)!}(x - a)^{n + 1}</latex>, where $c$ is a value between a and $x$
• Taylor's Theorem allows us to approximate $\sin(x)$ using a polynomial while quantifying the error.
• What is the center of a Taylor polynomial?

  The point a
• The accuracy of a Taylor polynomial approximation improves as $x$ gets closer to its center
• What is the general formula for the Taylor polynomial of degree $n$ centered at $a$?


  P_{n}(x) = f(a) + f'(a)(x - a) + \cdots + \frac{f^{(n)}(a)}{n!}(x - a)^{n}</latex>
• Steps to calculate the Taylor polynomial of $f(x)$ at $a$


  1️⃣ Find the first derivative $f'(x)$

  2️⃣ Find the second derivative $f''(x)$

  3️⃣ Calculate higher derivatives up to $f^{(n)}(x)$

  4️⃣ Evaluate derivatives at $x =$$a$
• To evaluate derivatives at the center $a$, replace $x$ with a
• The nth derivative of $e^{x}$ is always $e^{x}$.
• What is the first step in constructing a Taylor polynomial?

  Evaluate derivatives at $a$