Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![## Calculus Problems and Solutions
### Problem 005
**Find the sum of the series**
\[
\sum_{n=0}^{\infty} \frac{(-1)^n 2^{2n}}{n!}
\]
Options:
1. \( \text{sum} = e^{-2} \)
2. \( \text{sum} = 4 \)
3. \( \text{sum} = e^{2} \)
4. \( \text{sum} = e^{-4} \)
5. \( \text{sum} = e^{4} \)
6. \( \text{sum} = 2 \)
### Problem 006
**Use an appropriate Taylor series to evaluate**
\[
\lim_{x \to 0} \frac{\sin(3x) - 3x + 2x^3}{x^3}
\]
Options:
1. \( \text{limit} = -\frac{11}{6} \)
2. \( \text{limit} = -\frac{5}{2} \)
3. \( \text{limit} = -\frac{13}{6} \)
4. \( \text{limit} = -\frac{19}{6} \)
5. \( \text{limit} = -\frac{17}{6} \)
### Problem 007
**Use an appropriate Taylor series representation to evaluate**
\[
\lim_{x \to 0} \frac{3x - \tan^{-1}(3x)}{6x^3}
\]
Options:
1. \( \text{limit} = \frac{3}{4} \)
2. \( \text{limit} = \frac{5}{4} \)
3. \( \text{limit} = \frac{3}{2} \)
4. \( \text{limit} = \frac{9}{2} \)
5. \( \text{limit} = \frac{5}{2} \)
### Problem 008
**Find a power series representation for the function**
\[ f(x) = 2^x \]
**on** \( (-\infty, \infty) \).
Options:
1. \( f(x) = \sum_{n=0}^{\infty} \frac{(\ln(2))^n}{n](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcd27f8e2-62cc-4a99-bab5-627bf16ca968%2F91963193-4ab5-4fd2-b0c5-e694e85a75ff%2Fqbpou3s_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Calculus Problems and Solutions
### Problem 005
**Find the sum of the series**
\[
\sum_{n=0}^{\infty} \frac{(-1)^n 2^{2n}}{n!}
\]
Options:
1. \( \text{sum} = e^{-2} \)
2. \( \text{sum} = 4 \)
3. \( \text{sum} = e^{2} \)
4. \( \text{sum} = e^{-4} \)
5. \( \text{sum} = e^{4} \)
6. \( \text{sum} = 2 \)
### Problem 006
**Use an appropriate Taylor series to evaluate**
\[
\lim_{x \to 0} \frac{\sin(3x) - 3x + 2x^3}{x^3}
\]
Options:
1. \( \text{limit} = -\frac{11}{6} \)
2. \( \text{limit} = -\frac{5}{2} \)
3. \( \text{limit} = -\frac{13}{6} \)
4. \( \text{limit} = -\frac{19}{6} \)
5. \( \text{limit} = -\frac{17}{6} \)
### Problem 007
**Use an appropriate Taylor series representation to evaluate**
\[
\lim_{x \to 0} \frac{3x - \tan^{-1}(3x)}{6x^3}
\]
Options:
1. \( \text{limit} = \frac{3}{4} \)
2. \( \text{limit} = \frac{5}{4} \)
3. \( \text{limit} = \frac{3}{2} \)
4. \( \text{limit} = \frac{9}{2} \)
5. \( \text{limit} = \frac{5}{2} \)
### Problem 008
**Find a power series representation for the function**
\[ f(x) = 2^x \]
**on** \( (-\infty, \infty) \).
Options:
1. \( f(x) = \sum_{n=0}^{\infty} \frac{(\ln(2))^n}{n
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