What is the equation of the line tangent to f(x)=xsin2(2x) at x=π12?

Explanation:We havef(x)=xsin⁡(2x)2thenf′(x)=sin2(2x)+x⋅2⋅sin⁡(2x)⋅cos⁡(2x)⋅2f′(x)=sin2(2x)+2x+cos⁡(2x)we getf′(π12)sin2(π6)+2⋅π12⋅cos⁡(π6)f′(π12)=312⋅π+14andf(π12)=π48and the Tangent line is given byy=(312⋅π+14)x+nwheren=−1144⋅π2⋅3

Question: "What is the equation of the line tangent..."

High School
Calculus and Analysis
Calculus 1
Derivatives

Ariel Collier

Answered question

2022-04-10

What is the equation of the line tangent to

f (x) = x \sin^{2} (2 x)

x = \frac{π}{12}

Answer & Explanation

madking711kzcb

Beginner2022-04-11Added 9 answers

Explanation:
We have

f (x) = x {\sin (2 x)}^{2}

then

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

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

we get

f^{'} (\frac{π}{12}) \sin^{2} (\frac{π}{6}) + 2 \cdot \frac{π}{12} \cdot \cos (\frac{π}{6})

f^{'} (\frac{π}{12}) = \frac{\sqrt{3}}{12} \cdot π + \frac{1}{4}

and

f (\frac{π}{12}) = \frac{π}{48}

and the Tangent line is given by

y = (\frac{\sqrt{3}}{12} \cdot π + \frac{1}{4}) x + n

where

n = - \frac{1}{144} \cdot π^{2} \cdot \sqrt{3}

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