\( \int^{f}(x)=\tan (x) \) vs \( g(x)=2 \tan (x) \) vs \( h(x)=\tan (2 x) \)

To analyze the functions \( f(x) = \tan(x) \), \( g(x) = 2 \tan(x) \), and \( h(x) = \tan(2x) \), we will look at their properties, including their derivatives, and how they relate to each other. ### Step 1: Derivatives of the Functions 1. **For \( f(x) = \tan(x) \)**: - The derivative is given by: \[ f'(x) = \sec^2(x) \] 2. **For \( g(x) = 2 \tan(x) \)**: - The derivative is: \[ g'(x) = 2 \sec^2(x) \] 3. **For \( h(x) = \tan(2x) \)**: - Using the chain rule, the derivative is: \[ h'(x) = 2 \sec^2(2x) \] ### Step 2: Summary of Derivatives - \( f'(x) = \sec^2(x) \) - \( g'(x) = 2 \sec^2(x) \) - \( h'(x) = 2 \sec^2(2x) \) ### Step 3: Comparison of the Functions - **Function \( g(x) \)** is simply a vertical stretch of \( f(x) \) by a factor of 2. This means that for every \( x \), \( g(x) \) will be twice the value of \( f(x) \). - **Function \( h(x) \)** represents a horizontal compression of \( f(x) \) by a factor of 2. This means that the function \( h(x) \) will reach its values faster than \( f(x) \). ### Step 4: Behavior of the Functions - **Asymptotes**: - \( f(x) \) has vertical asymptotes at \( x = \frac{\pi}{2} + n\pi \) for \( n \in \mathbb{Z} \). - \( g(x) \) will have the same vertical asymptotes as \( f(x) \) since it is just a scaled version. - \( h(x) \) will have vertical asymptotes at \( x = \frac{\pi}{4} + \frac{n\pi}{2} \) for \( n \in \mathbb{Z} \) due to the factor of 2 in the argument. ### Conclusion - The functions \( f(x) \) and \( g(x) \) are related by a vertical scaling, while \( h(x) \) is related to \( f(x) \) by a horizontal compression. - The derivatives indicate how steeply each function increases, with \( g(x) \) increasing the fastest due to its scaling factor. If you need further analysis or specific calculations, please let me know!