Problem 45 - 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. $$ f(x)=\frac{1}{x} ;(1,1) $$

Step-by-Step Solution

Answer

The equation of the tangent line to the graph of the function \(f(x)=\frac{1}{x}\) at the point \((1,1)\) can be obtained by calculating the derivative of the function at \(x=1\) (using limit definition) and then using the point-slope form of a line's equation. The answer should be confirmed graphically by plotting the function and its tangent line at \((1,1)\) with a graphing utility.

Step 1: Determine the slope of the tangent line

To find the slope of the tangent line to the graph of \(f\) at the point \((1,1)\), it is required to compute the derivative of \(f\) at \(x=1\), denoted as \(f'(1)\). The derivative of a function based on the limit definition is given as \(f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\). Applying this definition for the function \(f(x)=\frac{1}{x}\), we get \(f'(x) = \lim_{h \to 0} \frac{\frac{1}{x+h} - \frac{1}{x}}{h}\). Now, simplify it and calculate \(f'(1)\), this will give us the slope of the tangent line at \((1,1)\).

Step 2: Find the equation of the line

The formula for the equation of a line in point-slope form is \(y - y_1 = m(x - x_1)\), where \(m\) is the slope of the line, and \((x_1, y_1)\) is a point on the line. Using the slope found in Step 1 and the given point \((1,1)\), substitute into the formula to get the equation of the tangent line.

Step 3: Verify graphically

With a graphing utility, plot the function \(f(x)=\frac{1}{x}\) and the tangent line obtained in Step 2 at \((1,1)\). Verify if the tangent line touches the curve exactly at the point \((1,1)\). This verifies the results.