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. \(\frac{\text { Function }}{f(x)=\frac{1}{3} x \sqrt{x^{2}+5}} \quad \frac{\text { Point }}{(2,2)}\)

Short answer

The derivative of the function \(f(x) = \frac{1}{3}x\sqrt{x^2 + 5}\) is \(f'(x) = \frac{1}{3}(\sqrt{x^2+5} + \frac{x^2}{\sqrt{x^2+5}})\). The equation of the tangent line to the function at the point (2,2) is \(y = 2x\).

Step-by-step solution

Find Derivative

The first task calls for finding the derivative of the function \(f(x) = \frac{1}{3}x\sqrt{x^2 + 5}\). Applying the product rule and the chain rule: \(f'(x) = \frac{1}{3}\left(\sqrt{x^2+5} + \frac{x^2}{\sqrt{x^2+5}}\right)\)

Evaluate Derivative at Specific Point

Substitute \(x = 2\) into the derivative to find the slope of the tangent line: \(f'(2) = \frac{1}{3}(\sqrt{4 + 5} + \frac{4}{\sqrt{4+5}}) = 2\)

Find Equation of Tangent Line

To find the equation of the tangent line, use the point-slope formula, \(y-y_1 = m(x-x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is a point on the line (in this case, (2,2)): \(y - 2 = 2(x - 2)\) which simplifies to \(y = 2x\)

Confirm results through graphing.

Refer to the individual graphing tool for instructions on: (1) graphing the function and its tangent line and (2) utilizing the derivative feature. The graph of the original function and the tangent line should align with the results derived manually.