AP Calculus BC Flashcards: Finding Taylor Polynomial Approximations Of Functions

What this deck covers

This deck focuses on Finding Taylor Polynomial Approximations Of Functions, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus BC.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is the $n^{th}$ degree Taylor polynomial for $f(x) = (1+x)^k$ centered at $0$?

Answer: $1 + kx + \frac{k(k-1)x^2}{2!} + \cdots + \frac{k(k-1)...(k-n+1)x^n}{n!}$. Binomial series with binomial coefficients.

Flashcard 2: Determine the Taylor polynomial of degree 1 for $f(x) = \ln(x)$ at $x = 1$.

Answer: $0 + (x-1)$. $f(1)=0$, $f'(1)=1$ gives linear approximation.

Flashcard 3: Identify the convergence condition for a Taylor series.

Answer: Converges if $|x-a| < R$ (radius of convergence $R$). Series converges within its radius of convergence $R$.

Flashcard 4: State the Taylor series for $\text{cos}(x)$ centered at $0$.

Answer: $\text{cos}(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$. Alternating signs with even powers only.

Flashcard 5: What is the $1^{st}$ degree Taylor polynomial for $f(x) = \text{e}^{2x}$ at $x = 0$?

Answer: $1 + 2x$. Replace $x$ with $2x$ in exponential series.

Flashcard 6: What is the $n^{th}$ degree Taylor polynomial of $f(x) = \text{ln}(1+x)$ centered at $0$?

Answer: $x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots + (-1)^{n-1}\frac{x^n}{n}$. Alternating signs with terms $\frac{(-1)^{n-1}x^n}{n}$.

Flashcard 7: Find the $3^{rd}$ degree Taylor polynomial for $f(x) = \text{ln}(1+x)$ centered at $0$.

Answer: $x - \frac{x^2}{2} + \frac{x^3}{3}$. First three terms of natural log series.

Flashcard 8: Calculate the $2^{nd}$ degree Taylor polynomial for $f(x) = \text{e}^x$ centered at $0$.

Answer: $1 + x + \frac{x^2}{2}$. First three terms of exponential series.

Flashcard 9: What is the Taylor polynomial approximation for $f(x) = \text{sin}(x)$ at $x = 0$ up to $x^2$?

Answer: $x$. Only first term survives since $x^2$ term has coefficient zero.

Flashcard 10: What is the remainder term $R_3(x)$ for Taylor polynomial of $f(x) = e^x$ at $x = 0$?

Answer: $\frac{e^c x^4}{4!}$ for some $c \in (0,x)$. Next derivative term after cubic polynomial.

Flashcard 11: What is the $1^{st}$ degree Taylor polynomial for $f(x) = \text{e}^x$ centered at $1$?

Answer: $e + e(x-1)$. $e^1=e$ with slope $e$ at $x=1$.

Flashcard 12: Calculate the $3^{rd}$ degree Taylor polynomial for $f(x) = \text{cos}(x)$ at $x = \pi$.

Answer: $-1 + (x-\pi)^2$. $\cos(\pi)=-1$, derivatives give quadratic term.

Flashcard 13: Find the Taylor polynomial of degree 2 for $f(x) = x^3$ at $a = 1$.

Answer: $1 + 3(x-1) + 3(x-1)^2$. Taylor expansion of $x^3$ around $x=1$.

Flashcard 14: What is the $4^{th}$ degree Taylor polynomial for $f(x) = x^2 \text{e}^x$ at $x = 0$?

Answer: $0 + 0 + x^2 + x^3 + \frac{x^4}{2}$. Product of $x^2$ and exponential series.

Flashcard 15: What is the radius of convergence for the Taylor series of $e^x$?

Answer: Radius of convergence is $\infty$. Exponential function converges everywhere.

Flashcard 16: Find the Taylor polynomial of degree 2 for $f(x) = \text{e}^{-x}$ at $x = 0$.

Answer: $1 - x + \frac{x^2}{2}$. Replace $x$ with $-x$ in exponential series.

Flashcard 17: What is the general form of the $n^{th}$ degree Taylor polynomial of $f(x)$ centered at $a$?

Answer: $P_n(x) = f(a) + f'(a)(x-a) + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n$. Uses successive derivatives divided by factorials with powers of $(x-a)$.

Flashcard 18: What is the Taylor series expansion of $e^x$ centered at $0$?

Answer: $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$. Each term is $\frac{x^n}{n!}$ for the exponential function.

Flashcard 19: Find the first three non-zero terms of the Taylor series for $\text{arctan}(x)$ centered at $0$.

Answer: $x - \frac{x^3}{3} + \frac{x^5}{5}$. Pattern follows $\frac{(-1)^n x^{2n+1}}{2n+1}$.

Flashcard 20: What is the Taylor series expansion for $\text{ln}(1-x)$ centered at $0$?

Answer: $-x - \frac{x^2}{2} - \frac{x^3}{3} - \cdots$. Replace $x$ with $-x$ in $\ln(1+x)$ series.

Flashcard 21: What is the $3^{rd}$ degree Taylor polynomial for $f(x) = \text{sin}(2x)$ centered at $0$?

Answer: $2x - \frac{8x^3}{3!}$. Replace $x$ with $2x$ in sine series.

Flashcard 22: What is the $2^{nd}$ degree Taylor polynomial for $f(x) = \text{sin}(x)$ centered at $\pi$?

Answer: $0 - (x-\pi)$. $\sin(\pi)=0$, $\sin'(\pi)=-1$ gives linear term.

Flashcard 23: What is the $3^{rd}$ degree Taylor polynomial for $f(x) = \text{arcsin}(x)$ at $x = 0$?

Answer: $x + \frac{x^3}{6}$. Arcsine has pattern $\frac{(2n)!x^{2n+1}}{4^n(n!)^2(2n+1)}$.

Flashcard 24: Find the $4^{th}$ degree Taylor polynomial for $f(x) = \text{ln}(x)$ centered at $1$.

Answer: $(x-1) - \frac{(x-1)^2}{2} + \frac{(x-1)^3}{3} - \frac{(x-1)^4}{4}$. Natural log series shifted to center at $x=1$.

Flashcard 25: Calculate the $2^{nd}$ degree Taylor polynomial for $f(x) = \text{cos}(3x)$ at $x = 0$.

Answer: $1 - \frac{9x^2}{2}$. Replace $x$ with $3x$ in cosine series.

Flashcard 26: What is the $4^{th}$ degree Taylor polynomial for $f(x) = \text{cos}(x)$ centered at $0$?

Answer: $1 - \frac{x^2}{2} + \frac{x^4}{24}$. Even powers with alternating signs through degree 4.

Flashcard 27: What is the formula for the remainder term $R_n(x)$ in Taylor polynomial?

Answer: $R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}$. Lagrange form with $c$ between $a$ and $x$.

Flashcard 28: State the first four terms of the Taylor series for $\cosh(x)$ centered at $0$.

Answer: $1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \frac{x^6}{6!}$. Even powers only, like cosine but hyperbolic.

Flashcard 29: What is the Taylor series for $\text{sin}(x)$ centered at $0$?

Answer: $\text{sin}(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$. Alternating signs with odd powers only.

Flashcard 30: What is the general form of the $n^{th}$ degree Taylor polynomial of $f(x)$ centered at $a$?

Answer: $P_n(x) = f(a) + f'(a)(x-a) + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n$. Uses successive derivatives divided by factorials with powers of $(x-a)$.