Questions: Find an equation for the line tangent to the graph of y=tan^(-1)(1/2 x) at the point (2, pi/4). What is the tangent line to y=tan^(-1)(1/2 x) at the point (2, pi/4) ? y=

Transcript text: Find an equation for the line tangent to the graph of $y=\tan ^{-1}\left(\frac{1}{2} x\right)$ at the point $\left(2, \frac{\pi}{4}\right)$. What is the tangent line to $y=\tan ^{-1}\left(\frac{1}{2} x\right)$ at the point $\left(2, \frac{\pi}{4}\right)$ ? \[ \mathrm{y}=\square \] (Type an exact answer, using $\pi$ as needed.)

Solution

Solution Steps

To find the equation of the tangent line to the graph of $ y = \tan^{-1}\left(\frac{1}{2} x\right) $ at the point $ (2, \frac{\pi}{4}) $, we need to follow these steps:

Differentiate the function: Find the derivative of $ y = \tan^{-1}\left(\frac{1}{2} x\right) $ with respect to $ x $ to get the slope of the tangent line.
Evaluate the derivative at the given point: Substitute $ x = 2 $ into the derivative to find the slope of the tangent line at that point.
Use the point-slope form of a line: With the slope from step 2 and the given point $ (2, \frac{\pi}{4}) $, use the point-slope form $ y - y_1 = m(x - x_1) $ to write the equation of the tangent line.

Step 1: Differentiate the Function

We start with the function $ y = \tan^{-1}\left(\frac{1}{2} x\right) $. The derivative of this function is given by: \[ \frac{dy}{dx} = \frac{0.5}{0.25x^2 + 1} \]

Step 2: Evaluate the Derivative at the Point

Next, we evaluate the derivative at the point $ x = 2 $: \[ \frac{dy}{dx}\bigg|_{x=2} = \frac{0.5}{0.25(2^2) + 1} = \frac{0.5}{0.25 \cdot 4 + 1} = \frac{0.5}{1 + 1} = \frac{0.5}{2} = 0.25 \] Thus, the slope of the tangent line at the point $ (2, \frac{\pi}{4}) $ is $ m = 0.25 $.

Step 3: Use the Point-Slope Form

Using the point-slope form of the line, we have: \[ y - y_1 = m(x - x_1) \] Substituting $ m = 0.25 $, $ x_1 = 2 $, and $ y_1 = \frac{\pi}{4} $: \[ y - \frac{\pi}{4} = 0.25(x - 2) \] Rearranging this gives: \[ y = 0.25x - 0.5 + \frac{\pi}{4} \] Simplifying further, we find: \[ y = \frac{1}{4}x - 0.5 + \frac{\pi}{4} \]