What is the equation of the line tangent to f(x)=2xcos3xf(x)=2xcos3x at x=pi/3x=π3?

1 Answer

Tangent line:
y=-2xy=2x

Explanation:

From the given trigonometric equation

f(x)=2x*cos 3xf(x)=2xcos3x

Let y=2x*cos 3xy=2xcos3x
solve for the point (x_1, y_1)(x1,y1) of which x_1=pi/3x1=π3

y_1=2(pi/3)*cos 3(pi/3)=(2pi)/3*cos pi=(2pi)/3*(-1)y1=2(π3)cos3(π3)=2π3cosπ=2π3(1)

y_1=-(2pi)/3y1=2π3

The point is (pi/3, -(2pi)/3)(π3,2π3)

Solve for the slope m=y'(pi/3)

y'=2*[x*d/dx(cos 3x)+(cos 3x)d/dx(x)]
y'=2*[x*(-3*sin 3x)+(cos 3x)(1)]

slope m=y'(pi/3)=2*[(pi/3)*(-3*sin 3(pi/3))+(cos 3(pi/3))(1)]

m=2[-pi*sin pi+cos pi]
m=2[0-1]
m=-2

Use Point-Slope form to find the equation of the tangent line

y-y_1=m(x-x_1)

y--(2pi)/3=-2(x-pi/3)

y+(2pi)/3=-2x+(2pi)/3

y=-2x

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God bless....I hope the explanation is useful.