Problem 19 - 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. $$ f(x)=x^{2}-1 ;(2,3) $$

Step-by-Step Solution

Answer

The slope of the tangent line to the graph of \(f(x) = x^{2} - 1\) at \(x = 2\) is 4

Step 1: Identify the Function and the Given Point

In the problem, we are given that the function is \(f(x) = x^{2} - 1\) and the point of tangency is (2,3).

Step 2: Substitute into the Limit Definition of the Derivative

Substitute the function \(f(x)\) and the x-coordinate of the point of tangency into the limit definition of the derivative: \( f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} \) to get \( f'(2) = \lim_{h \to 0} \frac{(2 + h)^{2} - 1 - ((2)^{2} - 1)}{h} \). Simplifying the numerator, we have \( f'(2) = \lim_{h \to 0} \frac{4 + 4h + h^{2} - 1 - 4 + 1}{h} = \lim_{h \to 0} \frac{4h + h^{2}}{h} \) which simplifies further to \( \lim_{h \to 0} 4 + h \)

Step 3: Evaluate the Limit

Evaluate the limit as \(h\) approaches 0: \( \lim_{h \to 0} 4 + h = 4 + 0 \)

Step 4: State the Slope of the Tangent Line

Since the derivative at \(x = 2\) is 4, then the slope of the tangent line to the graph of \(f(x) = x^{2} - 1\) at \(x = 2\) is 4