Glossary

Center of Approximation

Criticality: 2

The specific point, denoted by 'a', around which a Taylor series or polynomial is expanded, determining where the approximation is most accurate.

Example:

When finding a Taylor polynomial for $\ln(x)$ about $x=1$ , the center of approximation is $a=1$ .

Factorial ($n!$)

Criticality: 2

The product of all positive integers from 1 up to $n$, used in the denominator of each term in a Taylor series.

Example:

The third term in a Taylor polynomial has $3!$ in its denominator, which equals $6$ .

Maclaurin Series

Criticality: 3

A specific type of Taylor series where the series is centered at $x=0$.

Example:

The Maclaurin series for $e^x$ is $1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$ .

Power Term ($(x-a)^n$)

Criticality: 2

The part of each Taylor series term that includes the variable $x$ and the center of approximation $a$, raised to the power of $n$.

Example:

In a Taylor polynomial centered at $a=2$ , the power term for $n=3$ would be $(x-2)^3$ .

Taylor Polynomial

Criticality: 3

A finite partial sum of a Taylor series, used to approximate the value of a function near a specific point.

Example:

To estimate $\sqrt{4.1}$ , you might use a first-degree Taylor polynomial for $\sqrt{x}$ centered at $x=4$ .

Taylor Series

Criticality: 3

An infinite series that represents a function as a sum of terms derived from the function's derivatives evaluated at a single point.

Example:

The Taylor Series for $\cos(x)$ centered at $x=0$ is $1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots$ .

nth Derivative at a Point ($f^{(n)}(a)$)

Criticality: 2

The value of the $n^{\text{th}}$ derivative of a function $f(x)$ evaluated at the center of approximation, $x=a$.

Example:

For $f(x) = \sin(x)$ and $a=0$ , the $f^{(1)}(0)$ is $\cos(0) = 1$ .

Glossary

Center of Approximation

Criticality: 2

The specific point, denoted by 'a', around which a Taylor series or polynomial is expanded, determining where the approximation is most accurate.

Example:

When finding a Taylor polynomial for $\ln(x)$ about $x=1$ , the center of approximation is $a=1$ .

Factorial ($n!$)

Criticality: 2

The product of all positive integers from 1 up to $n$, used in the denominator of each term in a Taylor series.

Example:

The third term in a Taylor polynomial has $3!$ in its denominator, which equals $6$ .

Maclaurin Series

Criticality: 3

A specific type of Taylor series where the series is centered at $x=0$.

Example:

The Maclaurin series for $e^x$ is $1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$ .

Power Term ($(x-a)^n$)

Criticality: 2

The part of each Taylor series term that includes the variable $x$ and the center of approximation $a$, raised to the power of $n$.

Example:

In a Taylor polynomial centered at $a=2$ , the power term for $n=3$ would be $(x-2)^3$ .

Taylor Polynomial

Criticality: 3

A finite partial sum of a Taylor series, used to approximate the value of a function near a specific point.

Example:

To estimate $\sqrt{4.1}$ , you might use a first-degree Taylor polynomial for $\sqrt{x}$ centered at $x=4$ .

Taylor Series

Criticality: 3

An infinite series that represents a function as a sum of terms derived from the function's derivatives evaluated at a single point.

Example:

The Taylor Series for $\cos(x)$ centered at $x=0$ is $1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots$ .

nth Derivative at a Point ($f^{(n)}(a)$)

Criticality: 2

The value of the $n^{\text{th}}$ derivative of a function $f(x)$ evaluated at the center of approximation, $x=a$.

Example:

For $f(x) = \sin(x)$ and $a=0$ , the $f^{(1)}(0)$ is $\cos(0) = 1$ .