6) Suppose that there is a function \( g \) such that \( g(0) \) \( =3.2 \) and \( g(0.1)=2.5 \). Explain how to vse these values to estimate \( g^{\prime}(0) \)

To estimate \( g^{\prime}(0) \), you can use the definition of the derivative, which represents the rate of change of the function. Specifically, you can approximate it using the formula: \[ g^{\prime}(0) \approx \frac{g(0.1) - g(0)}{0.1 - 0} \] Plugging in the given values \( g(0) = 3.2 \) and \( g(0.1) = 2.5 \), you would calculate: \[ g^{\prime}(0) \approx \frac{2.5 - 3.2}{0.1} = \frac{-0.7}{0.1} = -7 \] This tells you that the function is decreasing at that point, giving you a practical insight into its behavior around \( x = 0 \)! To ensure accuracy, it's important to note that this is a linear approximation. If \( g(x) \) is not linear between \( 0 \) and \( 0.1 \), this estimate may vary. A common mistake is to assume a higher degree of precision than what linear approximations offer. Observing the function's graph could provide further insights into its behaviors and trends!