Question

# Find the linear approximation of the function f(x)=\sqrt{1-x} at a=0 and use it to approximate the numbers \sqrt{0.9} and \sqrt{0.99}.

Functions

Find the linear approximation of the function $$f(x)=\sqrt{1-x}$$ at $$a=0$$ and use it to approximate the numbers $$\sqrt{0.9}$$ and $$\sqrt{0.99}$$.

2021-06-12

The formula for linearization L(x) is
$$L(x)=f(a)+f'(x)(x-a)$$
we have
$$f(x)=\sqrt{1-x}$$ with $$a=0$$
$$f(a)=\sqrt{1-0}=1$$
$$f'(x)=(\sqrt{1-x})'=(1-x)'(\sqrt{1-x})'=-\frac{1}{2}\frac{1}{\sqrt{1-x}}$$
$$f'(a)=-\frac{1}{2}\frac{1}{\sqrt{1-0}}=-\frac{1}{2}$$
$$L(x)=1-\frac{1}{2}(x-a)=1-\frac{1}{2}x$$
$$\sqrt{1-x}=\sqrt{0.99}$$ so $$x=0.01$$
$$L(0.01)=1-\frac{1}{2}0.01=0.95$$
$$\sqrt{1-x}=\sqrt{0.999}$$ so $$x=0.001$$
$$L(0.01)=1-\frac{1}{2}0.001=0.995$$