Questions: Find the slope of the tangent line to the graph of the function at the given point. g(x)=20-x^2 ; (4,4) Note: Use the limit process given below to answer this question. lim (Δx → 0) (g(4+Δx)-g(4))/Δx

Solution Steps

To find the slope of the tangent line to the graph of the function \( g(x) = 20 - x^2 \) at the point (4, 4), we need to use the limit process. This involves calculating the derivative of the function at \( x = 4 \) using the definition of the derivative.

1. Substitute \( g(x) \) into the limit definition of the derivative.
2. Simplify the expression inside the limit.
3. Evaluate the limit as \( \Delta x \) approaches 0.

We are given the function \( g(x) = 20 - x^2 \) and the point \((4, 4)\). We need to find the slope of the tangent line to the graph of the function at this point.

To find the slope of the tangent line, we use the limit definition of the derivative: \[ \lim_{\Delta x \to 0} \frac{g(4 + \Delta x) - g(4)}{\Delta x} \]

Substitute \( g(x) = 20 - x^2 \) into the limit expression: \[ g(4 + \Delta x) = 20 - (4 + \Delta x)^2 \] \[ g(4) = 20 - 4^2 = 4 \] \[ \lim_{\Delta x \to 0} \frac{(20 - (4 + \Delta x)^2) - 4}{\Delta x} \]

Simplify the expression inside the limit: \[ \lim_{\Delta x \to 0} \frac{20 - (16 + 8\Delta x + (\Delta x)^2) - 4}{\Delta x} \] \[ = \lim_{\Delta x \to 0} \frac{20 - 16 - 8\Delta x - (\Delta x)^2 - 4}{\Delta x} \] \[ = \lim_{\Delta x \to 0} \frac{0 - 8\Delta x - (\Delta x)^2}{\Delta x} \] \[ = \lim_{\Delta x \to 0} \frac{-8\Delta x - (\Delta x)^2}{\Delta x} \] \[ = \lim_{\Delta x \to 0} (-8 - \Delta x) \]

Evaluate the limit as \(\Delta x\) approaches 0: \[ \lim_{\Delta x \to 0} (-8 - \Delta x) = -8 \]

Final Answer