To find the equation of the tangent line to the curve [tex]\(y = 2x \sin(x)\)[/tex] at the point [tex]\(\left(\frac{7}{2}, \pi\right)\)[/tex], follow these steps:
1. Determine the function and its derivative:
The given function is [tex]\(y = 2x \sin(x)\)[/tex].
2. Find the derivative of the function:
To get the slope of the tangent line, we need the derivative of [tex]\(y\)[/tex] with respect to [tex]\(x\)[/tex], denoted as [tex]\( \frac{dy}{dx} \)[/tex].
Using the product rule, [tex]\(\frac{d}{dx} [u \cdot v] = u' \cdot v + u \cdot v' \)[/tex], where [tex]\( u = 2x \)[/tex] and [tex]\( v = \sin(x) \)[/tex]:
[tex]\[
\frac{dy}{dx} = \frac{d}{dx} [2x \sin(x)] = 2 \frac{d}{dx} [x \sin(x)]
\][/tex]
Let's find the derivative [tex]\( \frac{d}{dx} [x \sin(x)] \)[/tex]:
[tex]\[
\frac{d}{dx} [x \sin(x)] = x \cos(x) + \sin(x)
\][/tex]
Therefore, the derivative function becomes:
[tex]\[
\frac{dy}{dx} = 2 [x \cos(x) + \sin(x)] = 2x \cos(x) + 2 \sin(x)
\][/tex]
3. Evaluate the derivative at the given point [tex]\(x = \frac{7}{2}\)[/tex]:
Substitute [tex]\(x = \frac{7}{2}\)[/tex] into the derivative function to find the slope at this point:
[tex]\[
\left. \frac{dy}{dx} \right|_{x = \frac{7}{2}} = 2 \left( \frac{7}{2} \cos\left( \frac{7}{2} \right) + \sin \left( \frac{7}{2} \right) \right)
\][/tex]
After evaluating this expression, we find the slope [tex]\( m \)[/tex] of the tangent line at [tex]\( x = \frac{7}{2} \)[/tex] is approximately:
[tex]\[
m = -7.256763266414814
\][/tex]
4. Write the equation of the tangent line:
The equation of the tangent line in point-slope form is [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\((x_1, y_1)\)[/tex] is the point of tangency, and [tex]\(m\)[/tex] is the slope:
[tex]\[
y - \pi = -7.256763266414814 \left( x - \frac{7}{2} \right)
\][/tex]
Rearrange this equation to slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[
y = -7.256763266414814x + 7.256763266414814 \cdot \frac{7}{2} + \pi
\][/tex]
Calculate the constant term:
[tex]\[
y = -7.256763266414814x + 25.398671932451855 + \pi
\][/tex]
Simplify the constant term:
[tex]\[
-7.256763266414814x + 22.257078778862056 = y
\][/tex]
Therefore, the equation of the tangent line is:
[tex]\[
y = -7.256763266414814 x - (-22.257078778862056)
\][/tex]
