0(3). Hint: く。 17< 3,メ23 and
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Problem 1RQ
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![**Transcription Explanation:**
**Title:** Validating Big-O Notation for Exponential Functions
**Content:**
When analyzing the function \(2^x + 17\), we aim to establish that it is \(O(3^x)\).
**Hint for Proof:**
1. **Basic Inequality:**
\[
2^x < 3^x \quad \text{for} \quad x > 1
\]
2. **Additional Condition:**
\[
17 < 3^x \quad \text{for} \quad x \geq 3
\]
**Explanation:**
To show that \(2^x + 17\) is bounded by \(3^x\), we need to verify that, eventually, both \(2^x\) and 17 are dominated by \(3^x\) as \(x\) increases.
- For \(x > 1\), the term \(2^x\) grows slower than \(3^x\) since the base of the exponential in \(2^x\) is smaller.
- For \(x \geq 3\), the constant 17 becomes negligible compared to the growth rate of \(3^x\).
Thus, for sufficiently large values of \(x\), \(2^x + 17\) will be less than some constant multiple of \(3^x\), satisfying the Big-O notation definition.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa1e1d114-cba9-4c4f-a9ab-1a346fb28105%2F4baa52d7-1858-4dd2-8a9c-3f436e0ab626%2Fhm5y6o_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Transcription Explanation:**
**Title:** Validating Big-O Notation for Exponential Functions
**Content:**
When analyzing the function \(2^x + 17\), we aim to establish that it is \(O(3^x)\).
**Hint for Proof:**
1. **Basic Inequality:**
\[
2^x < 3^x \quad \text{for} \quad x > 1
\]
2. **Additional Condition:**
\[
17 < 3^x \quad \text{for} \quad x \geq 3
\]
**Explanation:**
To show that \(2^x + 17\) is bounded by \(3^x\), we need to verify that, eventually, both \(2^x\) and 17 are dominated by \(3^x\) as \(x\) increases.
- For \(x > 1\), the term \(2^x\) grows slower than \(3^x\) since the base of the exponential in \(2^x\) is smaller.
- For \(x \geq 3\), the constant 17 becomes negligible compared to the growth rate of \(3^x\).
Thus, for sufficiently large values of \(x\), \(2^x + 17\) will be less than some constant multiple of \(3^x\), satisfying the Big-O notation definition.
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