What is the equation of the line tangent to f(x)=xsin⁡2x at x=π12?

Explanation:To determine the equation of the tangent line, we will require a slope and a point, given that x=π12. The slope will come from the derivative f'(x), which is found with the Product Rule and Chain Rule:f′(x)=(x)⋅(cos⁡(2x)⋅2)+(sin⁡(2x))⋅(1)f′(x)=2xcos⁡(2x)+sin⁡(2x)At x=π12,f′(π12)=2(π12)⋅cos⁡(2⋅π12)+sin⁡(2⋅π12)=π6⋅cos⁡(π6)+sin⁡(π6)=π6⋅32+12=π⋅312+12=π⋅312+612=6+π⋅312To find the point of tangency, we use f(x) to find the y value when x=π12:f(π12)=(π12)⋅sin⁡(2⋅π12)=π12⋅sin⁡(π6)=π12⋅12=π24We can then use the point-slope formula to determine the equation of the tangent line:y−y1=m(x−x1)y−π24=(6+π⋅312)(x−π12)

"What is the equation of the line tangent..." was answered by our community

gabolzm6d

Answered question

2022-04-09

What is the equation of the line tangent to

f (x) = x \sin 2 x

x = \frac{π}{12}

Answer & Explanation

WigwrannyErarmbmk

Beginner2022-04-10Added 13 answers

Explanation:
To determine the equation of the tangent line, we will require a slope and a point, given that

x = \frac{π}{12}

. The slope will come from the derivative f'(x), which is found with the Product Rule and Chain Rule:

f^{'} (x) = (x) \cdot (\cos (2 x) \cdot 2) + (\sin (2 x)) \cdot (1)

f^{'} (x) = 2 x \cos (2 x) + \sin (2 x)

x = \frac{π}{12}, f^{'} (\frac{π}{12}) = 2 (\frac{π}{12}) \cdot \cos (2 \cdot \frac{π}{12}) + \sin (2 \cdot \frac{π}{12})

= \frac{π}{6} \cdot \cos (\frac{π}{6}) + \sin (\frac{π}{6}) = \frac{π}{6} \cdot \frac{\sqrt{3}}{2} + \frac{1}{2}

= \frac{π \cdot \sqrt{3}}{12} + \frac{1}{2} = \frac{π \cdot \sqrt{3}}{12} + \frac{6}{12} = \frac{6 + π \cdot \sqrt{3}}{12}

To find the point of tangency, we use f(x) to find the y value when

x = \frac{π}{12}

f (\frac{π}{12}) = (\frac{π}{12}) \cdot \sin (2 \cdot \frac{π}{12}) = \frac{π}{12} \cdot \sin (\frac{π}{6}) = \frac{π}{12} \cdot \frac{1}{2} = \frac{π}{24}

We can then use the point-slope formula to determine the equation of the tangent line:

y - y_{1} = m (x - x_{1})

y - \frac{π}{24} = (\frac{6 + π \cdot \sqrt{3}}{12}) (x - \frac{π}{12})

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