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: 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 2: 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 3: State the Taylor polynomial of degree 2 for $f(x) = \sin(x)$ at $a = 0$.

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

Flashcard 4: 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 5: 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 6: 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 7: 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 8: 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 9: 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 10: Determine the interval of convergence for $\text{ln}(1+x)$ Maclaurin series.

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

Flashcard 11: 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 12: 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 $6$.

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 $\text{sin}(x)$?

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 15: 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 16: 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 17: 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 18: Determine the interval of convergence for $\text{ln}(1+x)$ Maclaurin series.

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

Flashcard 19: 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 20: 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 21: 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 22: 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 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 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 25: 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 26: 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 27: 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 28: 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 29: 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 30: What is the Taylor polynomial of degree 1 for $f(x) = x^3$ at $a = 2$?

Answer: $8 + 12(x-2)$. Using $f(2) = 8$ and $f'(2) = 12$ for linear approximation.

Flashcard 31: 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 32: 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 33: 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 34: 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 35: 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 36: 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 37: 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 38: 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 39: 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 40: 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 41: 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 42: 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 43: 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 44: 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 45: 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 46: 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 47: 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 48: 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 49: 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 50: 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 51: 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 52: Identify the interval of convergence for the Maclaurin series of $e^x$.

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

Flashcard 53: 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 54: 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 55: 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 56: What is the Taylor polynomial of degree 1 for $f(x) = x^3$ at $a = 2$?

Answer: $8 + 12(x-2)$. Using $f(2) = 8$ and $f'(2) = 12$ for linear approximation.

Flashcard 57: Identify the interval of convergence for the Maclaurin series of $e^x$.

Answer: $(-\text{\textinfty}, \text{\textinfty})$. Exponential function converges everywhere.

Flashcard 58: 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 59: 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 $6$.

Flashcard 60: 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 61: 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 62: Find the second derivative of $f(x) = e^x$ for its Taylor series.

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