AP Calculus BC Flashcards: Finding Taylor Polynomial Approximations of Functions
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Review key ideas with interactive flashcards. This set includes 10 cards to help you master important concepts.
What is the formula for the coefficient of the $n^{th}$ degree term in a Taylor polynomial for a function $f$ centered at $x=a$?
The coefficient of the $n^{th}$ degree term is given by the formula $\frac{f^{(n)}(a)}{n!}$, where $f^{(n)}(a)$ is the $n^{th}$ derivative of $f$ evaluated at $a$.
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What is the formula for the coefficient of the $n^{th}$ degree term in a Taylor polynomial for a function $f$ centered at $x=a$?
The coefficient of the $n^{th}$ degree term is given by the formula $\frac{f^{(n)}(a)}{n!}$, where $f^{(n)}(a)$ is the $n^{th}$ derivative of $f$ evaluated at $a$.
Why is a Taylor polynomial centered at $x=a$ most useful for approximating values of $f(x)$ when $x$ is near $a$?
The polynomial is constructed using the function's behavior (derivatives) exactly at $x=a$, so the approximation is most accurate near this center point.
What generally happens to the accuracy of a Taylor polynomial as its degree increases?
In many cases, as the degree of a Taylor polynomial increases, it will more closely approach the original function over some interval.
The 3rd-degree term of a Taylor polynomial for $f(x)$ centered at $x=2$ is $7(x-2)^3$. What is the value of $f'''(2)$?
Since the coefficient is $\frac{f'''(2)}{3!} = 7$, we can find $f'''(2)$ by calculating $7 \times 3! = 7 \times 6 = 42$.
What is the relationship between a Taylor polynomial's degree and the number of derivatives used to construct it?
An $n^{th}$ degree Taylor polynomial requires the function's value and the values of all its derivatives up to the $n^{th}$ derivative, all evaluated at the center point.
How can a function be represented at a specific point using a Taylor polynomial?
A function can be represented at a point by a Taylor polynomial, which is a polynomial whose coefficients are determined by the function's derivatives at that point.
What does the term "centered at $x=a$" signify in the context of a Taylor polynomial?
It signifies that the polynomial is constructed using the derivatives of the function evaluated at the point $x=a$ and is written in powers of $(x-a)$.
How would you use a 4th-degree Taylor polynomial for $f(x)$, centered at $x=1$, to approximate the value of $f(1.1)$?
First, represent the function as a 4th-degree Taylor polynomial centered at $x=1$. Then, substitute $x=1.1$ into this polynomial to find the approximate value.
What is the primary purpose of using a Taylor polynomial for a function $f$ centered at $x=a$?
Taylor polynomials for a function $f$ centered at $x=a$ are used to approximate the function's values for inputs near $x=a$.
To find the coefficient of the $(x-3)^5$ term in a Taylor polynomial for a function $f(x)$, what specific information do you need?
You need the value of the fifth derivative of $f$ evaluated at $x=3$, which is $f^{(5)}(3)$, and the value of 5 factorial (5!).