To find the best fit exponential function for the given data set, we will follow these steps:
1. Data Table: We have the following data points:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
0 & 11.1 \\
\hline
1 & 4 \\
\hline
2 & 1.44 \\
\hline
3 & 0.52 \\
\hline
4 & 0.19 \\
\hline
5 & 0.07 \\
\hline
\end{array}
\][/tex]
2. Transform the Data: To linearize the exponential data, we transform [tex]\(y\)[/tex] using the natural logarithm ([tex]\(\ln\)[/tex]):
[tex]\[
\ln(y) \text{ for each } y
\][/tex]
Calculations:
[tex]\[
\begin{array}{|c|c|c|}
\hline
x & y & \ln(y) \\
\hline
0 & 11.1 & \ln(11.1) \approx 2.4079 \\
\hline
1 & 4 & \ln(4) \approx 1.3863 \\
\hline
2 & 1.44 & \ln(1.44) \approx 0.3646 \\
\hline
3 & 0.52 & \ln(0.52) \approx -0.6539 \\
\hline
4 & 0.19 & \ln(0.19) \approx -1.6607 \\
\hline
5 & 0.07 & \ln(0.07) \approx -2.6593 \\
\hline
\end{array}
\][/tex]
3. Perform Linear Regression: Perform a linear regression with [tex]\(x\)[/tex] as the independent variable and [tex]\(\ln(y)\)[/tex] as the dependent variable. We need to find the best fit line in the form:
[tex]\[
\ln(y) = mx + c
\][/tex]
where [tex]\(m\)[/tex] is the slope and [tex]\(c\)[/tex] is the y-intercept.
4. Extract Parameters: Once we have [tex]\(m\)[/tex] and [tex]\(c\)[/tex] from the regression:
- [tex]\(\ln(a) = c \Rightarrow a = e^c\)[/tex]
- [tex]\(b = e^m\)[/tex]
5. Calculate Parameters: Using a graphing calculator or statistical software:
- Slope ([tex]\(m\)[/tex]) and intercept ([tex]\(c\)[/tex]) for the data points are computed as follows:
Using the least squares method on the linear regression data, we get:
[tex]\[
m \approx -1.03850 \quad \text{and} \quad c \approx 2.40392
\][/tex]
6. Transform Back to Exponential Form:
[tex]\[
a = e^c \approx e^{2.40392} \approx 11.06 \quad (\text{rounded to 2 decimal places})
\][/tex]
[tex]\[
b = e^m \approx e^{-1.03850} \approx 0.35 \quad (\text{rounded to 2 decimal places})
\][/tex]
7. Best Fit Exponential Function:
Thus, the best fit exponential function is:
[tex]\[
y = 11.06 \times 0.35^x
\][/tex]
So, the final best fit exponential function in the form [tex]\( y = a \cdot b^x \)[/tex] is:
[tex]\[
y = 11.06 \cdot 0.35^x
\][/tex]
