Problem 18 - College Algebra and Calculus: An Applied Approach

Question

Use the limit definition to find the slope of the tangent line to the graph of \(f\) at the given point. $$ f(x)=6 ;(-2,6) $$

Step-by-Step Solution

Answer

The slope of the tangent line to the graph of the given function at the given point is 0.

Step 1: Identify \(f(x)\) and the point

In this problem, the given function is \(f(x) = 6\) and the given point is (-2,6). Note that the function is a constant function - its value is always 6 for any value of \(x\).

Step 2: Compute \(f'(x)\) using the limit definition of the derivative

We know that \[f'(x) = \lim_{h\to 0} \frac{f(x+h)-f(x)}{h}\]. But since \(f(x) = 6\), both \(f(x+h)\) and \(f(x)\) always give the value 6 regardless of the value of \(x\) or \(h\). So, \[f'(x) = \lim_{h\to 0} \frac{6-6}{h} = \lim_{h\to 0} 0 = 0.\]

Step 3: Result Interpretation

So, the derivative or the slope of the tangent line to the function \(f(x)\) at any point is 0. This makes sense because a constant function is a horizontal line and the slope of a horizontal line is 0.