Question

find the linear approximation to square root(a+x) for x near o. a is a constant (pos.) L(x)= sqrt(a)+1/2(a+0)^-1/2 (x-0) is this correct?
use you approximation to estimate sqrt (4+0.1)
How would I do this part ?


Also confused with this?
find the linear approximation to f(x)=x* cos(x) at a = pi: I found L(x)= pi cos(pi) + (1*cos(pi)+pi*-sinpi)(x-pi) how would get my actual number answer? what am I missing?

use your approximation to estimate x+1/100cos(pi+1/100)? how would I do this ?

Answer 1

your approximation for √(x+a) is correct
∆y/∆x ≈ y' = 1/(2√(x+a))
That is,
y(x+∆x) - y(x) = y'(x) * ∆x
y(4+0.1) = √(4+0) + 1/(2√(4+0)) * 0.1
y(4+0.1) = 2 + 0.1/4 = 2.025
in fact, √4.1 = 2.0248

for the other,
y = x cosx
y' = cosx - x sinx
y'(π) = -1
y(π+.01) = y(π) + (-1)(.01) = -π-.01 = -3.15159265
In fact, (π+.01) cos(π+.01) = -3.1514351