›) Determine the big-theta estimate for the function below. Show all relevant working f(x) = (3x³ + 8x² — 7)log(12x7 + 6x³ + 9x + 1).
›) Determine the big-theta estimate for the function below. Show all relevant working f(x) = (3x³ + 8x² — 7)log(12x7 + 6x³ + 9x + 1).
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![**Problem Statement:**
Determine the big-theta (Θ) estimate for the function below. Show all relevant working.
\[ f(x) = (3x^3 + 8x^2 - 7) \log(12x^7 + 6x^3 + 9x + 1) \]
**Explanation:**
To find the big-theta estimate, focus on the highest degree terms in both the polynomial and the logarithmic function, which dominate the behavior as \( x \to \infty \).
1. **Polynomial Analysis:**
- The highest degree term in the polynomial part \( 3x^3 + 8x^2 - 7 \) is \( 3x^3 \).
2. **Logarithmic Analysis:**
- The highest degree term in the logarithmic part \( \log(12x^7 + 6x^3 + 9x + 1) \) is \( \log(x^7) \) because the term \( 12x^7 \) dominates as \( x \to \infty \).
- Using the logarithmic property, \(\log(ax^b) = \log(a) + b\log(x)\), we get \(\log(12x^7) \approx 7\log(x)\).
3. **Combine Results:**
- The function simplifies to: \( f(x) \approx 3x^3 \cdot 7\log(x) = 21x^3 \log(x) \).
4. **Big-Theta Estimate:**
- Hence, the big-theta estimate is \( \Theta(x^3 \log(x)) \).
This means that as \( x \to \infty \), \( f(x) \) grows at a rate proportional to \( x^3 \log(x) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F16a501d0-ad86-4e86-a9d4-08a716d78319%2Fca7dd341-f72f-483d-a344-6d4cb3cf7591%2Fxdvl32f_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Determine the big-theta (Θ) estimate for the function below. Show all relevant working.
\[ f(x) = (3x^3 + 8x^2 - 7) \log(12x^7 + 6x^3 + 9x + 1) \]
**Explanation:**
To find the big-theta estimate, focus on the highest degree terms in both the polynomial and the logarithmic function, which dominate the behavior as \( x \to \infty \).
1. **Polynomial Analysis:**
- The highest degree term in the polynomial part \( 3x^3 + 8x^2 - 7 \) is \( 3x^3 \).
2. **Logarithmic Analysis:**
- The highest degree term in the logarithmic part \( \log(12x^7 + 6x^3 + 9x + 1) \) is \( \log(x^7) \) because the term \( 12x^7 \) dominates as \( x \to \infty \).
- Using the logarithmic property, \(\log(ax^b) = \log(a) + b\log(x)\), we get \(\log(12x^7) \approx 7\log(x)\).
3. **Combine Results:**
- The function simplifies to: \( f(x) \approx 3x^3 \cdot 7\log(x) = 21x^3 \log(x) \).
4. **Big-Theta Estimate:**
- Hence, the big-theta estimate is \( \Theta(x^3 \log(x)) \).
This means that as \( x \to \infty \), \( f(x) \) grows at a rate proportional to \( x^3 \log(x) \).
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