Problem 39 - College Algebra and Calculus: An Applied Approach

Question

Use the limit definition to find an equation of the tangent line to the graph of \(f\) at the given point. Then verify your results by using a graphing utility to graph the function and its tangent line at the point. $$ f(x)=\frac{1}{2} x^{2} ;(2,2) $$

Step-by-Step Solution

Answer

The equation of the tangent line to the graph of \(f(x)\) at the point (2,2) is \(y = 2x - 2\).

Step 1: Find the Derivative Using Limit Definition

The limit definition of the derivative is \(f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\). Plug in \(f(x) = \frac{1}{2}x^2\) into the limit definition and simplify to find the derivative. In this case, \(f'(x) = x\).

Step 2: Find the Slope of the Tangent Line

After finding the derivative, plug 2 into the derivative to find the slope of the tangent line. Doing this gives \(f'(2) = 2\). So, the slope of the tangent line at the point (2,2) is 2.

Step 3: Find the Equation of the Tangent Line

With the slope of the tangent line and the coordinates of the point where the line is tangent to the function, use the point-slope form of a linear equation, \(y - y_1 = m(x - x_1)\), to find the equation of the line. Substituting \(m = 2\), \(x_1 = 2\), and \(y_1 = 2\) gives the equation \(y - 2 = 2(x - 2)\), or \(y = 2x - 2\).

Step 4: Verify the Results

Verify these results by using a graphing utility to graph \(f(x)\) and the tangent line, and confirm they touch at the point (2,2).