The curve f(x)=sec(x)/(x+2) passes through the point (0,1/2). Use the equation of the tangent line to the curve at the point (0,1/2) to approximate f(0.1).

Question

The curve f(x)=sec(x)/(x+2) passes through the point (0,1/2).  Use the equation of the tangent line to the curve at the point (0,1/2) to approximate f(0.1).

phi · Answer

f'(x)  is
(sec(x) (-1+(2+x) tan(x)))/(2+x)^2

evaluate at x=0 to find the slope is -1/4

now find the equation of the tangent line using the slope and a know point on the line:
y - 1/2  = -1/4 *(x-0) 
y = (-1/4) x + 1/2

we can approximate f(x) by using  f(x) ~ (-1/4) x + 1/2
f(x) ~ -0.25*0.1 + 0.5 = 0.475

the actual value  per wolfram is f(0.1)= 0.478581