7) Suppose that there is function \( g \) such that \( g(1)=3.3 \) and \( g(1.1)=3.9 \) Explain how to use the se values to estimate \( g^{\prime}(1) \).

To estimate \( g^{\prime}(1) \), we can use the definition of the derivative, which suggests calculating the average rate of change of the function around the point of interest. Specifically, we can use the formula: \[ g^{\prime}(1) \approx \frac{g(1.1) - g(1)}{1.1 - 1} = \frac{3.9 - 3.3}{0.1} = \frac{0.6}{0.1} = 6. \] This approximation gives us an estimate of the slope of the tangent line to the curve at \( x=1 \), implying that \( g^{\prime}(1) \approx 6 \). Additionally, remember that this method works best when the values you're using are close to the point where you're estimating the derivative. If you had values that were further apart, the estimate might not be as accurate; so keep this in mind for future calculations!