Taylor Polynomial Approximation

A Taylor polynomial provides an approximation to a function $f(x)$ near a point $a$. It uses derivatives of the function at $a$ and is given by the formula:

$P_n(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x - a)^n$

Where:

• $P_n(x)$: The Taylor polynomial of degree $n$.
• $f^{(n)}(a)$: The $n$-th derivative of $f$ evaluated at $a$.
• $(x - a)^k / k!$: The term involving $x$ shifted around $a$ with a factorial normalization.

Steps for Taylor Approximation

1. Choose the Function and Point: Decide on the function $f(x)$ and the center point $a$ around which the approximation is made.
2. Compute Derivatives: Find $f(a), f'(a), f''(a), \ldots$ up to the desired degree $n$.
3. Build the Polynomial: Plug the values into the Taylor formula.
4. Approximation Range: Use the polynomial to approximate $f(x)$ for values of $x$ near $a$.

Common Uses

• Approximating non-polynomial functions like $e^x$, $\sin(x)$, $\cos(x)$, etc.
• Solving differential equations numerically.
• Analysis of functions near a specific point.

Example

Approximate $f(x) = e^x$ near $x = 0$ (Maclaurin Series)

1. $f(x) = e^x$, $f'(x) = e^x$, $f''(x) = e^x$, etc. (all derivatives are $e^x$).
2. At $x = 0$, $f(0) = 1$, $f'(0) = 1$, $f''(0) = 1$, etc.
3. Using the formula: $P_n(x) = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots + \frac{x^n}{n!}$ For $n = 3$, $P_3(x) = 1 + x + \frac{x^2}{2} + \frac{x^3}{6}$.

The higher the degree $n$, the better the approximation for $x$ near 0.

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Do you want detailed steps for a specific function, or have any further questions?

Follow-Up Questions

1. How does the Taylor polynomial compare with the actual function graphically?
2. What is the error term in Taylor polynomial approximations?
3. Can you explain the difference between Taylor and Maclaurin series?
4. How is Taylor series used in practical applications like physics or engineering?
5. How does the choice of $n$ impact the accuracy of the approximation?

Tip: The Taylor approximation works best near $x = a$; the further $x$ is from $a$, the larger the error.