Let p ( x ) = 4 sin ⁡ x . An equation of the line tangent to the graph of p ( x ) at the point ( π / 6 , 2 ) can be written in the form y = m x + b . which are the values for m and b.

Answer 1

The tangent line is a straight line that goes through the point (π/6, 2).

First, write the equation in the point-slope form: y - 2 = m (x - π/6) (here m is a slope).

The slope of the tangent line to the curve p(x) at the point (π/6, 2) equals the the derivative of p(x) at x = π/6.

So, m = p' (π/6).

p' (x) = (4 sin x)' = 4 cos x

m = p' (π/6) = 4 cos(π/6) = 4 √3 / 2 = 2√3

The equation of the tangent line in point-slope form: y - 2 = 2√3 (x - π/6)

y - 2 = 2√3 x - 2π√3 / 6 = 2√3 x - π√3 / 3

The equation of the tangent line in slope-intercept form: y = 2√3 x + 2 - (π√3 / 3)

Finally, m = 2√3, b = 2 - (π√3 / 3)