Question

use linear approximation, i.e. the tangent line, to approximate \frac{1}{1.004} as follows: let f(x) = \frac{1}{x} and find the equation of the tangent line to f(x) at a
ice\ point near 1.004. then use this to approximate \frac{1}{1.004}.

Explanation:

Identify the function and the base point

$$ f(x) = \frac{1}{x},\quad a = 1 $$

Calculate the function value and derivative at the base point

$$
LATEXBLOCK0
$$

Construct the linear approximation and evaluate

$$
LATEXBLOCK1
$$

Answer:

Use linear approximation, i.e. the tangent line, to approximate \(\frac{1}{1.004}\) as follows: Let \(f(x) = \frac{1}{x}\) and find the equation of the tangent line to \(f(x)\) at a "nice" point near \(1.004\). Then use this to approximate \(\frac{1}{1.004}\). <blank>0.996</blank>