How do you solve this linear approximation problem?

Question

Use differentials (linear approximation) to estimate tan(-1). Base the approximation at x= -pi / 3.

Answer 1

The linear approximation of f(x) at x=x0 is

L(x) = f(x0) + f'(x0)(x-x0)

where f(x) = tan x, f'(x) = sec^2 x, x0 = -pi/3, and x = -1.

Plugging all of this in, you have

L(-1) = tan(-pi/3) + sec^2(-pi/3)(pi/3 - 1)
= -sqrt(3) + 4(pi/3 - 1)
= -1.54326

compared to the actual value tan(-1) = -1.55741.