5.2 Maclaurin and Taylor Series

• What is the general formula for the Maclaurin series of a function $f(x)$?


  $f(0) +$$f'(0)x +$$\frac{f''(0)}{2!}x^{2} +$$\cdots$
• The Maclaurin series expands a function around $x =$$0$.
• What is the Maclaurin series for $e^{x}$?


  $1 +$$x +$$\frac{x^{2}}{2!} +$$\frac{x^{3}}{3!} +$$\cdots$
• Match the centering point with the correct series:

  Maclaurin Series ↔️ $a =$$0$

  Taylor Series ↔️ Any $a$
• What is the formula for the Maclaurin series in terms of $f(0)$, $f'(0)$, and $x$?


  $f(0) +$$f'(0)x +$$\frac{f''(0)}{2!}x^{2} +$$\cdots$
• The Maclaurin series is used to expand a function around $x =$$0$.
• What is the Maclaurin series for $e^{x}$?


  $1 +$$x +$$\frac{x^{2}}{2!} +$$\frac{x^{3}}{3!} +$$\cdots$
• What is the general formula for the Taylor series of a function $f(x)$ centered at $a$?


  $f(a) +$$f'(a)(x - a) +$$\frac{f''(a)}{2!}(x - a)^{2} +$$\cdots$
• The Taylor series approximates a function near the point $a$.
• What is the Taylor series for $sin(x)$ about $a =$$\frac{\pi}{2}$?


  $1 - \frac{(x - \frac{\pi}{2})^{2}}{2!} +$$\frac{(x - \frac{\pi}{2})^{4}}{4!} - \cdots$
• Match the series with its centering point and use case:

  Taylor Series ↔️ $a$ ||| Approximates near any $a$

  Maclaurin Series ↔️ $a =$$0$ ||| Approximates near zero
• The Taylor series is a representation of a function $f(x)$ as an infinite sum of terms based on its derivatives evaluated at a specific point a
• The Taylor series is used to approximate $f(x)$ near the point $a$
• What is the Taylor series for $sin(x)$ about $a =$$\frac{\pi}{2}$?


  sin(x) = 1 - \frac{(x - \frac{\pi}{2})^{2}}{2!} + \frac{(x - \frac{\pi}{2})^{4}}{4!} - \cdots</latex>
• The Maclaurin series is a special case of the Taylor series centered at a = 0
• The Taylor series is centered at any point $a$, while the Maclaurin series is centered at $a =$$0$.
• Match the type of series with its centering point:

  Maclaurin Series ↔️ $a =$$0$

  Taylor Series ↔️ Any $a$
• The Maclaurin series is a special case of the Taylor series centered at a = 0
• The Taylor series can be centered around any value $a$.
• What is the Maclaurin series for $e^{x}$?


  $e^{x} =$$1 +$$x +$$\frac{x^{2}}{2!} +$$\frac{x^{3}}{3!} +$$\cdots$
• Match the series with its centering point:

  Maclaurin Series ↔️ $a =$$0$

  Taylor Series ↔️ Any $a$
• The Maclaurin series is a special case of the Taylor series centered at $a =$$0$.
• What is the Taylor series for $sin(x)$ about $a =$$\frac{\pi}{2}$?


  $sin(x) =$$1 - \frac{(x - \frac{\pi}{2})^{2}}{2!} +$$\frac{(x - \frac{\pi}{2})^{4}}{4!} - \cdots$
• The Maclaurin series is a special case of the Taylor series centered at $a =$$0$.
• For f(x) = e^{x}</latex>, the Maclaurin series requires evaluating derivatives at x = 0
• The Taylor series for $sin(x)$ about $a =$$\frac{\pi}{2}$ requires evaluating derivatives at $x =$$\frac{\pi}{2}$.
• What is the general formula for the Maclaurin series of a function $f(x)$?


  $f(0) +$$f'(0)x +$$\cdots$
• For the function $f(x) =$$e^{x}$, the Maclaurin series is 1
• What is the Maclaurin series for $e^{x}$?


  1 + x + \frac{x^{2}}{2!} + \cdots</latex>
• The Taylor series evaluates derivatives at $x =$$a$.
• The general formula for the Taylor series of $f(x)$ centered at $x =$$a$ is f(a)
• What is the Taylor series for $sin(x)$ centered at a = \frac{\pi}{2}</latex>?


  $1 - \frac{(x - \frac{\pi}{2})^{2}}{2!} +$$\cdots$
• Steps to apply the Maclaurin and Taylor series formulas

  1️⃣ Identify the function $f(x)$

  2️⃣ Choose the centering point $a$

  3️⃣ Calculate derivatives of $f(x)$

  4️⃣ Evaluate derivatives at $a$

  5️⃣ Apply the series formula

  6️⃣ Simplify the expansion
• For the Maclaurin series, the centering point $a$ is always 0.
• What value of $a$ is used for the Taylor series to approximate $f(x)$ near x = a</latex>?


  $A suitable value near x$
• The Maclaurin series formula for $f(x)$ is f(0)
• Match the function with its derivatives at $x =$$0$ for Maclaurin series:


  $cos(x)$ ↔️ $f(0) =$$1, f'(0) =$$0$

  $\sin(x)$ ↔️ $f(0) =$$0, f'(0) =$$1$

  $e^{x}$ ↔️ $f(0) =$$1, f'(0) =$$1$
• What is the Maclaurin series for $cos(x)$?


  $1 - \frac{x^{2}}{2!} +$$\cdots$
• The first derivative of $cos(x)$ is -sin(x)
• The Taylor series for ln(x)</latex> is centered at $a =$$1$.