Taylor and Maclaurin Series - AP Calculus BC

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

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

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

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

3. What is the Maclaurin series for $\text{cos}(x)$?

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

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

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

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

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

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

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

7. What is the Maclaurin series for $\text{arctan}(x)$?

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

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

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

9. What is the Maclaurin series for $\text{sin}(x)$?

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

10. What is the general form of the Maclaurin series for a function $f(x)$?

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

11. What is the Maclaurin series for $\text{tan}^{-1}(x)$?

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

12. What is the Maclaurin series for $\text{cosh}(x)$?

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

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

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

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

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

15. What is the Maclaurin series for $\text{ln}(1+x)$?

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

16. What is the Taylor polynomial of degree 1 for $f(x) = x^3$ at $a = 2$?

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

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

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

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

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

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

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

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

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

21. What is the general form of the Taylor series for a function $f(x)$ centered at $a$?

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

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

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

23. What is the Maclaurin series for $e^x$?

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

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

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

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

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

26. What is the Maclaurin series for $\text{sinh}(x)$?

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

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

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

28. Identify the interval of convergence for the Maclaurin series of $e^x$.

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

29. What is the Maclaurin series for $\frac{1}{1-x}$?

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

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

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

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

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

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

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

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