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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I found the derivative of a limited growth function, and I need to find the second derivative. I know I need to take the natural log of both sides but I'm not sure how to do this. I have a photo attached of where the equation is at. Thank you!
![### Mathematical Expression Analysis
The expression given is:
\[
\ln (f'(t)) = \ln \left( \frac{(1.056 \times 10^9) e^{-0.55t}}{(1 + 600 e^{-0.55t})^2} \right)
\]
#### Explanation:
- **Natural Logarithm (\(\ln\))**: The expression involves the natural logarithm of the derivative of a function \(f\) with respect to \(t\), denoted as \(f'(t)\).
- **Exponential Term**: The function includes an exponential decay term \(e^{-0.55t}\), which suggests a process that decreases over time.
- **Numerator**: The term \((1.056 \times 10^9) e^{-0.55t}\) in the numerator suggests multiplication of a constant with an exponentially decaying function.
- **Denominator**: The denominator has \((1 + 600 e^{-0.55t})^2\), indicating the presence of an exponential term raised to the power of 2. The coefficient 600 also multiplies the exponential function.
This expression could represent a variety of real-world phenomena, such as modeling decay processes or population dynamics, where the growth or decay rate changes over time. The presence of a squared term in the denominator suggests some form of damping or friction effect that increases with time.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fddff5265-cb9b-49c9-95fd-ed295bd74659%2Fe2515f93-e4ec-461c-99be-e40a74b91f88%2Fonu1r3_processed.png&w=3840&q=75)
Transcribed Image Text:### Mathematical Expression Analysis
The expression given is:
\[
\ln (f'(t)) = \ln \left( \frac{(1.056 \times 10^9) e^{-0.55t}}{(1 + 600 e^{-0.55t})^2} \right)
\]
#### Explanation:
- **Natural Logarithm (\(\ln\))**: The expression involves the natural logarithm of the derivative of a function \(f\) with respect to \(t\), denoted as \(f'(t)\).
- **Exponential Term**: The function includes an exponential decay term \(e^{-0.55t}\), which suggests a process that decreases over time.
- **Numerator**: The term \((1.056 \times 10^9) e^{-0.55t}\) in the numerator suggests multiplication of a constant with an exponentially decaying function.
- **Denominator**: The denominator has \((1 + 600 e^{-0.55t})^2\), indicating the presence of an exponential term raised to the power of 2. The coefficient 600 also multiplies the exponential function.
This expression could represent a variety of real-world phenomena, such as modeling decay processes or population dynamics, where the growth or decay rate changes over time. The presence of a squared term in the denominator suggests some form of damping or friction effect that increases with time.
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