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. \(\frac{\text { Function }}{f(x)=\sqrt{3 x^{2}-2}} \quad \frac{\text { Point }}{(3,5)}\)

Short answer

The equation of the tangent line to the graph of the function at the point (3,5) is \(y = \frac{9}{5}x - \frac{7}{5}\).

Step-by-step solution

Find the derivative of the function

To find the derivative of the function \(f(x) = \sqrt{3x^2 - 2}\), apply the chain rule:\[f'(x) = \frac{1}{2}\left(3x^2 - 2\right)^{-\frac{1}{2}} \cdot (6x)\]Simplify to get:\[f'(x) = \frac{3x}{\sqrt{3x^2 - 2}}\]

Find the slope of the tangent line at the given point

Substitute \(x = 3\) into the derivative to get the slope of the tangent line at that point:\[\text {slope} = f'(3) = \frac{3(3)}{\sqrt{3(3)^2 - 2}} = \frac{9}{\sqrt{3(9) - 2}} = \frac{9}{\sqrt{25}} = \frac{9}{5}\]

Find the equation of the tangent line

Use the point-slope form of a line to form the equation of the line:\[\text{Line equation} = y - y1 = m (x - x1)\]Where \(m\) is the slope, and \((x1, y1)\) is the given point. Hence,\[y - 5 = \frac{9}{5} (x - 3)\]Simplify this to:\[y = \frac{9}{5}x - \frac{7}{5}\]

Graphing function and tangent line

Using a graphing utility, graph the function and the tangent line to visualize the results.

Confirm calculation with the graphing utility

Use the derivative feature of the graphing utility to confirm that the calculated values match with the graphical representation. This should demonstrate the correctness of the obtained tangent line equation.