{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero.

b. Estimate f(0.2) and give a bound on the error in the approximation.

f(x) = tan x ≈ x

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Identify the function and the approximation given: the function is \(f(x) = \tan x\), and the approximation near zero is \(\tan x \approx x\).

To estimate \(f(0.2)\) using the approximation, substitute \(x = 0.2\) into the approximation: \(f(0.2) \approx 0.2\).

To find a bound on the error, recall that the error in approximating \(\tan x\) by \(x\) near zero can be analyzed using the remainder term from the Taylor series expansion of \(\tan x\) around 0.

The Taylor series of \(\tan x\) at 0 starts as \(\tan x = x + \frac{x^3}{3} + \cdots\). The error when approximating by \(x\) is roughly the size of the next term, which is about \(\frac{x^3}{3}\).

Calculate the error bound by evaluating \(\left| \frac{x^3}{3} \right|\) at \(x = 0.2\), which gives an estimate of the maximum error in the approximation.

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