Question

Finding an Equation of a Tangent Line In Exercises \(73-80,\) (a) find an equation of the tangent line to the graph of \(f\) at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of the graphing utility to confirm your results. $$ f(x)=\tan ^{2} x,\left(\frac{\pi}{4}, 1\right) $$

Short answer

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

Step-by-step solution

Find the derivative of the function

The derivative of \(f(x)=\tan^{2}(x)\) is found using the chain rule for derivatives and the derivative of \(\tan(x)\). The chain rule states that the derivative of a composed function is the derivative of the outer function times the derivative of the inner function. The derivative of \(\tan(x)\) is \(\sec^{2}(x)\). Thus the derivative \(f'(x) = 2\tan(x) \cdot \sec^{2}(x)\).

Evaluate the derivative at the point

The derivative at the point \(\left(\frac{\pi}{4}, 1\right)\) is found by substituting \(x=\frac{\pi}{4}\) into the derivative. So \(f'\left(\frac{\pi}{4}\right) = 2\tan\left(\frac{\pi}{4}\right) \cdot \sec^{2}\left(\frac{\pi}{4}\right) = 2\cdot1\cdot2 = 4\). Thus the slope of the tangent line at the point is 4.

Find the equation of the tangent line

The equation of the tangent line is of the form \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept. Since we have the slope and a point on the line, we can use the point slope equation \(y-y_1 = m(x-x_1)\) with \(m=4\) and \((x_1, y_1) = \left(\frac{\pi}{4}, 1\right)\) to find the line equation \(y=4x -\pi +4\).

Verification

The results should be verified with a graphing utility. The graph of the function \(f(x)=\tan^{2}(x)\) and the line \(y=4x - \pi +4\) should be plotted together. The graph of the line should appear to be tangent to the graph of the function at the point \(\left(\frac{\pi}{4},1\right)\). Additionally, the derivative feature of the graphing utility should be used to confirm that the slope of the function at \(\left(\frac{\pi}{4}\right)\) is 4.

Final Conclusion

Therefore, the equation of the tangent line is \(y=4x - \pi +4\).

Key concepts

Derivative

To find an equation of a tangent line at a specific point on a curve, we first need to understand the concept of the derivative. The derivative of a function at a point represents the rate at which the function is changing at that point, essentially capturing the idea of the slope of the tangent line to the curve. Think of the derivative as a way to formalize how steep the curve is. In the exercise, we are dealing with the function \(f(x) = \tan^2(x)\). By using calculus, specifically the chain rule, we find its derivative to be \(f'(x) = 2\tan(x) \cdot \sec^2(x)\). This derivative function helps us understand how \(f\) behaves at each point along its domain.

Chain Rule

The chain rule is a crucial tool in calculus for computing the derivative of composite functions. Here, our function \(f(x) = \tan^2(x)\) can be viewed as a composition of functions: first squaring and then taking the tangent of \(x\). The chain rule states that to differentiate \(f(g(x))\), we multiply the derivative of the outer function \(f\) by the derivative of the inner function \(g\).
In this case, the derivative of the outer function \(u^2\) is \(2u\), where \(u = \tan(x)\). The derivative of the inner \(\tan(x)\) is \(\sec^2(x)\). By applying the chain rule, we multiply these results to get \(f'(x) = 2\tan(x) \cdot \sec^2(x)\). This rule helps us navigate complex functions seamlessly.

Graphing Utility

A graphing utility is a powerful tool that allows us to visualize mathematical functions and verify our work. In this exercise, after calculating the derivative and finding the equation of the tangent line, a graphing utility can check our results. By plotting the function \(f(x) = \tan^2(x)\) and the tangent line equation \(y = 4x - \pi + 4\), we can visually confirm the correctness of our solution.
Using the derivative feature of the graphing utility, we can also verify whether the slope at the point \((\frac{\pi}{4}, 1)\) matches our calculated derivative value of 4. Graphing utilities thus serve as an essential component not only for visualization but also for ensuring accuracy of our calculus results.

Slope

The slope is an essential concept when dealing with tangent lines. It describes how steep a line is. For lines on a graph, the slope is the vertical change divided by the horizontal change. In the context of a tangent line, the slope at a given point is equivalent to the derivative of the function at that point.
In this problem, the slope of the tangent line was calculated by evaluating the derivative of \(f(x) = \tan^2(x)\) at \(x = \frac{\pi}{4}\). We found that \(f'\left(\frac{\pi}{4}\right) = 4\), meaning the tangent line at the point \((\frac{\pi}{4}, 1)\) has a slope of 4. This value determines the tilt of our tangent line, and along with a point, helps define its precise equation. Understanding slope helps us predict and model how curves behave on a graph.