Math Problem Statement
Find the exponential equation
Find the exponential equation for the given data points.
Solution
To find the exponential equation of the form from the given data, we will analyze the values of and provided in the table.
Data provided:
\hline x & 3 & 6 & 9 & 15 & 21 \\ \hline f(x) & 100 & 103.3 & 106.709 & 113.868 & 121.507 \\ \hline \end{array}$$ ### Steps to find the equation: 1. **Identify the general form**: The equation is assumed to have the exponential form: $$f(x) = a \cdot e^{bx}$$ where $$a$$ and $$b$$ are constants we need to find. 2. **Linearize the equation**: Taking the natural logarithm (ln) of both sides of the equation: $$\ln(f(x)) = \ln(a \cdot e^{bx}) = \ln(a) + bx$$ This equation is now linear in terms of $$\ln(f(x))$$, which means we can treat this as a linear regression problem to determine $$a$$ and $$b$$. 3. **Perform a regression**: We will use the given data points for $$x$$ and $$f(x)$$ to compute a best fit using logarithmic transformation. I'll now perform the necessary calculations to find $$a$$ and $$b$$.The exponential equation that best fits the given data is: $$f(x) = 96.806 \cdot e^{0.01082x}$$ Here: - $$a = 96.806$$ - $$b = 0.01082$$ This represents the exponential relationship between $$x$$ and $$f(x)$$. Would you like further details or have any questions about this solution? ### Related Questions: 1. How can we verify if this equation accurately fits the data? 2. What does the constant $$b$$ represent in an exponential function? 3. Can this model be used for predicting values of $$f(x)$$ beyond the given range of $$x$$? 4. How would this model change if the growth rate were negative? 5. How can you apply this equation to practical problems, such as population growth? **Tip:** When dealing with exponential functions, always check the value of $$b$$. If $$b > 0$$, the function represents growth; if $$b < 0$$, it represents decay.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Exponential Functions
Regression Analysis
Formulas
f(x) = a * e^(bx)
Natural logarithm transformation ln(f(x)) = ln(a) + bx
Theorems
Linear regression method for finding best-fit parameters in exponential functions
Suitable Grade Level
Grades 10-12