Linear Approximation?

Question

Find the local linear approximation of the function
y= sqr(1+x) at x_0=1, and use it to approximate sqr0.9 and sqr1.1.

I got 1+1/2x for the linear function, but i didn't get the approximate number.

Answer 1

y=1/2sqr(1+x)

Answer 2

The linear approximation of f(x) at x0 is

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

In this case, f(x) = sqrt(x), x0=1, and x=1.1 or 0.9..

f'(x) = 1/(2*sqrt(x)), and f'(x0) = f'(1) = 1/2, so
L(x) = 1 + 1/2(x-1), so
L(1.1) = 1 + 1/2(0.1) = 1.05, and
L(0.9) = 1 + 1/2(-0.1) = 0.95.

Alternatively, f(x) = sqrt(1+x), x0=0, and x=0.1 or -0.1..