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

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
Transcript text: Find the slope of the tangent line to the graph of the function at the given point. \[ g(x)=20-x^{2} ; \quad(4,4) \] Note: Use the limit process given below to answer this question. \[ \lim _{\Delta x \rightarrow 0} \frac{g(4+\Delta x)-g(4)}{\Delta x} \]
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Solution

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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.
Step 1: Define the Function and the Point

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.

Step 2: Apply the Limit Definition of the Derivative

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

Step 3: Substitute the Function into the Limit Expression

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

Step 4: Simplify the Expression Inside the Limit

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

Step 5: Evaluate the Limit

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

Final Answer

\(\boxed{-8}\)

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