(8,0 pontos )-Faca um ajuste exponencial , utilizando OS dados da tabela , pelo método dos minimos quadrados. square disappointed

## Step 1: Understanding the Problem<br />### The task is to perform an exponential fit using the least squares method on a given set of data. This involves finding an exponential function of the form $y = ae^{bx}$ that best fits the data points in the table.<br /><br />## Step 2: Setting Up the Exponential Model<br />### The exponential model we are fitting is $y = ae^{bx}$. To linearize this, we take the natural logarithm of both sides, resulting in $\ln(y) = \ln(a) + bx$. This transforms the problem into a linear regression problem where $\ln(y)$ is the dependent variable and $x$ is the independent variable.<br /><br />## Step 3: Applying the Least Squares Method<br />### Using the least squares method, we need to find the values of $\ln(a)$ and $b$ that minimize the sum of squared differences between the observed values $\ln(y_i)$ and the predicted values $\ln(a) + bx_i$. This involves solving the normal equations derived from the partial derivatives of the sum of squared errors with respect to $\ln(a)$ and $b$.<br /><br />## Step 4: Calculating Parameters<br />### Calculate the necessary sums: $\sum x_i$, $\sum \ln(y_i)$, $\sum x_i^2$, and $\sum x_i \ln(y_i)$. Use these to solve for $b$ and $\ln(a)$ using the formulas:<br />\[<br />b = \frac{n\sum (x_i \ln(y_i)) - \sum x_i \sum \ln(y_i)}{n\sum x_i^2 - (\sum x_i)^2}<br />\]<br />\[<br />\ln(a) = \frac{\sum \ln(y_i) - b\sum x_i}{n}<br />\]<br />where $n$ is the number of data points.<br /><br />## Step 5: Exponentiating to Find 'a'<br />### Once $\ln(a)$ is found, exponentiate it to get $a$: $a = e^{\ln(a)}$.<br /><br />## Step 6: Formulating the Exponential Equation<br />### Substitute the calculated values of $a$ and $b$ back into the exponential equation $y = ae^{bx}$ to get the final fitted model.