What happens to ( \frac{F(10C)}{F(C)} ) if c is small

To analyze what happens to \( \frac{F(10C)}{F(C)} \) when \( c \) is small, let's consider the behavior of the exponential terms when \( c \) approaches 0.

Given:
\[ F(c) = a \left(1 - e^{-bc}\right) \]
\[ \frac{F(10C)}{F(C)} = \frac{1 - e^{-10bc}}{1 - e^{-bc}} \]

When \( c \) is small, \( bc \) and \( 10bc \) will also be small. For small values of \( x \), \( e^{-x} \) can be approximated using the Taylor series expansion:
\[ e^{-x} \approx 1 - x + \frac{x^2}{2} - \cdots \]

Using this approximation:
\[ e^{-bc} \approx 1 - bc \]
\[ e^{-10bc} \approx 1 - 10bc \]

Now substitute these approximations into the ratio:
\[ \frac{F(10C)}{F(C)} \approx \frac{1 - (1 - 10bc)}{1 - (1 - bc)} \]
\[ \frac{F(10C)}{F(C)} \approx \frac{1 - 1 + 10bc}{1 - 1 + bc} \]
\[ \frac{F(10C)}{F(C)} \approx \frac{10bc}{bc} \]

Simplify the expression:
\[ \frac{F(10C)}{F(C)} \approx 10 \]

Thus, when \( c \) is small, the ratio \( \frac{F(10C)}{F(C)} \) approaches 10.

If you have any questions or need further details, feel free to ask!

Here are 5 numbered questions you might want to explore:

1. What happens to the ratio \( \frac{F(10C)}{F(C)} \) as \( c \) becomes very large?
2. How does the ratio change if the function is \( F(c) = a \left(1 - e^{-b \sqrt{c}}\right) \)?
3. Can you find the exact value of \( \frac{F(10C)}{F(C)} \) for a specific value of \( c \) and \( b \)?
4. How would this function \( F(c) \) be used in a practical application such as population growth or radioactive decay?
5. How can you approximate \( F(c) \) for other small values of \( c \) using a different method?

Tip: Approximations using Taylor series are powerful tools for understanding the behavior of functions near specific points, especially when dealing with exponential functions.

Explore the behavior of the ratio F(10C) / F(C) as c approaches 0 with detailed analysis using exponential function approximations and Taylor series. Understand how small values of c impact the ratio and the implications in mathematical modeling. Suitable for advanced mathematics enthusiasts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation