Question

Find the tangent line equations for the given functions at the given point(s): f(x) = tan x + 9 sin x at (π, 0)

Answer 1

`{ \bf{f(x) = \tan x + 9 \sin x }}`

For gradient, differentiate f(x):

`{ \tt{ (dy)/(dx) = { \sec }^(2)x + 9 \cos x }}`

Substitute for x as π:

`{ \tt{gradient = { \sec }^(2) \pi + 9 \cos(\pi ) }} \\ { \tt{gradient = - 8 }}`

Gradient of tangent = -8

`{ \bf{y =mx + b }} \\ { \tt{0 = ( - 8\pi) + b}} \\ { \tt{b = 8\pi}} \\ y - intercept = 8\pi`

Equation of tangent:

`{ \boxed{ \bf{y = - 8x + 8\pi}}}`