Question

In Exercises \(81-88\), (a) find an equation of the tangent line to the graph of \(f\) at the indicated point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. \(\frac{\text { Function }}{f(x)=\tan ^{2} x} \quad \frac{\text { Point }}{\left(\frac{\pi}{4}, 1\right)}\)

Short answer

The equation of the tangent line at the point (\(\pi/4\), 1) to the graph of \(f(x) = \tan^2 x\) is \(y - 1 = 4(x - \pi/4)\).

Step-by-step solution

Differentiate the Function

The first derivative of the function \(f(x)\) is the slope of the tangent line at any given x-value. We differentiate \(f(x) = \tan^2 x\) using the chain rule, where derivative of \(\tan x\) is \(1 + \tan^2 x\), and the outer function is \(x^{2}\). The derivative \(f'(x)\) is then \(2 \tan x \cdot (1 + \tan^2 x)\).

Determine the Slope at the Given Point

The slope of the tangent at the point \((\pi/4, 1)\) is calculated by substituting \(\pi/4\) in \(f'(x)\). Given that \(\tan (\pi/4) = 1\), we find \(f'(\pi/4) = 2 \cdot 1 \cdot (1 + 1^2) = 4\).

Find Equation of the Tangent Line

We use the point-slope form of a line, which is \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the point on the line and \(m\) is the slope. Substituting the given point \((\pi/4, 1)\) and the slope \(4\), we get the equation of the tangent line as \(y - 1 = 4(x - \pi/4)\).