Self-Assessment Quiz: Maclaurin Series for Common Functions (e^x, sin(x), cos(x))

📚 Quick Study Guide

• 🔢 The Maclaurin series is a Taylor series expansion of a function about 0.
• ➗ General form: $f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + ... = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n$
• 📈 Maclaurin series for $e^x$: $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ... = \sum_{n=0}^{\infty} \frac{x^n}{n!}$
• 📐 Maclaurin series for $sin(x)$: $sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + ... = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!}$
• 🌀 Maclaurin series for $cos(x)$: $cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + ... = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!}$

🧪 Practice Quiz

1. What is the coefficient of the $x^3$ term in the Maclaurin series expansion of $e^x$?
1. A) 1
2. B) $\frac{1}{2}$
3. C) $\frac{1}{6}$
4. D) 0
2. Which of the following represents the Maclaurin series for $sin(x)$?
1. A) $1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...$
2. B) $x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + ...$
3. C) $1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + ...$
4. D) $x + \frac{x^3}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!} + ...$
3. What is the value of $cos(0)$ based on its Maclaurin series?
1. A) 0
2. B) 1
3. C) -1
4. D) Undefined
4. Which term is missing in the Maclaurin series of $sin(x)$: $x - \frac{x^3}{3!} + \frac{x^5}{5!} - ... $?
1. A) $\frac{x^2}{2!}$
2. B) $\frac{x^4}{4!}$
3. C) $\frac{x^6}{6!}$
4. D) No term is missing
5. What is the Maclaurin series for $e^{-x}$?
1. A) $1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...$
2. B) $1 - x + \frac{x^2}{2!} - \frac{x^3}{3!} + ...$
3. C) $1 - x - \frac{x^2}{2!} - \frac{x^3}{3!} - ...$
4. D) $-1 + x - \frac{x^2}{2!} + \frac{x^3}{3!} - ...$
6. What is the coefficient of $x^2$ in the Maclaurin series of $cos(2x)$?
1. A) -1
2. B) -2
3. C) 2
4. D) 1
7. Using the Maclaurin series of $e^x$, approximate $e^{0.1}$. Which of the following is closest to the true value?
1. A) 1.00
2. B) 1.01
3. C) 1.10
4. D) 1.11