AP Calculus BC Flashcards: Taylor And Maclaurin Series

What this deck covers

This deck focuses on Taylor And Maclaurin Series, 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: Identify the interval of convergence for the Maclaurin series of $e^x$.

Answer: $(-\infty, \infty)$. Exponential function converges everywhere.

Flashcard 2: State the formula for the $n^{th}$ term of a Maclaurin series.

Answer: $\frac{f^{(n)}(0)x^n}{n!}$. Maclaurin term with center at $a = 0$.

Flashcard 3: What is the Maclaurin series for $e^x$?

Answer: $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \text{...}$. All derivatives of $e^x$ equal $e^x$, so $f^{(n)}(0) = 1$.

Flashcard 4: What is the Maclaurin series for $\text{sin}(x)$?

Answer: $\text{sin}(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \text{...}$. Alternating series with odd powers only.

Flashcard 5: Find $f^{(2)}(0)$ for $f(x) = x^4$ in its Maclaurin series.

Answer: $f^{(2)}(0) = 0$. Second derivative of $x^4$ is $12x^2$, so $f''(0) = 0$.

Flashcard 6: What is the Maclaurin series for $\text{tan}^{-1}(x)$?

Answer: $\text{tan}^{-1}(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \text{...}$. Same as $\arctan(x)$ series with alternating odd terms.

Flashcard 7: What is the Maclaurin series for $\text{cosh}(x)$?

Answer: $\text{cosh}(x) = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \frac{x^6}{6!} + \text{...}$. Hyperbolic cosine has only even powers.

Flashcard 8: What is the general form of the Maclaurin series for a function $f(x)$?

Answer: $f(x) = \frac{f(0)}{0!} + \frac{f'(0)x}{1!} + \frac{f''(0)x^2}{2!} + \frac{f'''(0)x^3}{3!} + \text{...}$. Maclaurin series is Taylor series centered at $a = 0$.

Flashcard 9: Find the first derivative of $f(x) = \text{sin}(x)$ for its Taylor series.

Answer: $f'(x) = \text{cos}(x)$. First derivative of $\sin(x)$ is $\cos(x)$.

Flashcard 10: Find $f^{(3)}(0)$ for $f(x) = \text{cos}(x)$ in its Maclaurin series.

Answer: $f^{(3)}(0) = 0$. Third derivative of $\cos(x)$ is $\sin(x)$, so $f^{(3)}(0) = 0$.

Flashcard 11: State the Taylor polynomial of degree 2 for $f(x) = \text{sin}(x)$ at $a = 0$.

Answer: $x - \frac{x^3}{6}$. First three terms of sine series (degree 2 has no $x^2$ term).

Flashcard 12: Find the third derivative $f'''(x)$ for $f(x) = x^3$ in its Taylor series.

Answer: $f'''(x) = 6$. Third derivative of $x^3$ is constant $6$.

Flashcard 13: What is $f^{(2)}(a)$ for $f(x) = x^2$ in its Taylor series centered at $a = 3$?

Answer: $f^{(2)}(3) = 2$. Second derivative of $x^2$ is constant $2$.

Flashcard 14: What is the Maclaurin series for $\text{sinh}(x)$?

Answer: $\text{sinh}(x) = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!} + \text{...}$. Hyperbolic sine has only odd powers.

Flashcard 15: Find $f'(0)$ for $f(x) = \text{arctan}(x)$ in its Maclaurin series.

Answer: $f'(0) = 1$. First derivative of $\arctan(x)$ is $\frac{1}{1+x^2}$, so $f'(0) = 1$.

Flashcard 16: What is the radius of convergence for the series of $f(x) = \frac{1}{1-x}$?

Answer: $R = 1$. Geometric series converges when $|x| < 1$.

Flashcard 17: Determine $f^{(4)}(0)$ for $f(x) = e^x$ in its Maclaurin series.

Answer: $f^{(4)}(0) = 1$. All derivatives of $e^x$ equal $1$ at $x = 0$.

Flashcard 18: Find the second derivative of $f(x) = e^x$ for its Taylor series.

Answer: $f''(x) = e^x$. All derivatives of $e^x$ equal $e^x$.

Flashcard 19: Evaluate $f^{(4)}(0)$ for $f(x) = x^4$ in its Maclaurin series.

Answer: $f^{(4)}(0) = 24$. Fourth derivative of $x^4$ is constant $24$.

Flashcard 20: What is the Maclaurin series for $\text{cos}(x)$?

Answer: $\text{cos}(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \text{...}$. Alternating series with even powers only.

Flashcard 21: What is the Maclaurin series for $\frac{1}{1-x}$?

Answer: $\frac{1}{1-x} = 1 + x + x^2 + x^3 + \text{...}$. Geometric series with first term $1$ and ratio $x$.

Flashcard 22: State the formula for the $n^{th}$ term of a Taylor series.

Answer: $\frac{f^{(n)}(a)(x-a)^n}{n!}$. Standard form for any term in Taylor expansion.

Flashcard 23: Evaluate $f'(0)$ for $f(x) = \text{cos}(x)$ in its Maclaurin series.

Answer: $f'(0) = 0$. First derivative of $\cos(x)$ is $-\sin(x)$, so $f'(0) = 0$.

Flashcard 24: What is the Maclaurin series for $\text{ln}(1+x)$?

Answer: $\text{ln}(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \text{...}$. Alternating series with coefficients $\frac{1}{n}$.

Flashcard 25: Evaluate $f(0)$ for $f(x) = \text{ln}(1+x)$ in its Maclaurin series.

Answer: $f(0) = 0$. Natural log of $1$ equals $0$.

Flashcard 26: What is the Taylor polynomial of degree 3 for $f(x) = \text{ln}(x)$ at $a = 1$?

Answer: $0 + (x-1) - \frac{(x-1)^2}{2} + \frac{(x-1)^3}{3}$. Taylor series for $\ln(x)$ using derivatives at $x = 1$.

Flashcard 27: What is the Maclaurin series for $\text{arctan}(x)$?

Answer: $\text{arctan}(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \text{...}$. Alternating series with odd powers and reciprocal coefficients.

Flashcard 28: Determine the interval of convergence for $\text{ln}(1+x)$ Maclaurin series.

Answer: $(-1, 1]$. Series converges for $-1 < x \leq 1$.

Flashcard 29: What is the general form of the Taylor series for a function $f(x)$ centered at $a$?

Answer: $f(x) = \frac{f(a)}{0!} + \frac{f'(a)(x-a)}{1!} + \frac{f''(a)(x-a)^2}{2!} + \frac{f'''(a)(x-a)^3}{3!} + \text{...}$. General Taylor series formula using derivatives at center $a$.

Flashcard 30: Calculate $f^{(3)}(a)$ for $f(x) = x^3$ in its Taylor series at $a = 1$.

Answer: $f^{(3)}(1)= 6$. Third derivative of $x^3$ is constant $6$.