# Linear Approximations and Differentials

1. Oct 27, 2006

Alright I just did the following question, and was hoping I did it right:

Use differentials(or, equivalently, a linear approximation) to estimate the given number)

cos 31.5* (* meaning degrees)

f(x) = cosx
f(31.5*) = ?

a is chosen to be the closest number to the number evaluating in the function of, such that the equation can be easily evaluated.
a= 30*, then dx = 31.5*-30*= 1.5*

delta y is approximately equal to dy
delta y = f(31.5*)-f(30*)
= cos31.5* - cos30*
= cos31.5* - root3/2

dy = f'(30*)dx

f'(x) = -sinx -> f'(30*) = -sin(30*) = -0.5

dy = (-1/2)(1.5*) = -1/2(pi/120) = -(pi)/240

Hence: cos31.5* - root3/2 is approximately equal to -(pi)/240 or:
cos(31.5*) = root3/2 - (pi)/240

I'm just checking to see if I followed the correct steps in solving this question. Thanks guys.