{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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Small Angle Approximation

The small angle approximation states that for values of x near zero, certain functions like tan(x) can be approximated by simpler expressions, such as tan(x) ≈ x. This simplifies calculations by replacing complex functions with linear ones, but the accuracy depends on how close x is to zero.

Error Bound Using Taylor's Remainder

When approximating a function by its Taylor polynomial, the error bound quantifies the maximum difference between the true function and the approximation. For tan(x) ≈ x, the error can be bounded using the next term in the Taylor series or the Lagrange remainder formula, ensuring the approximation's reliability.

Taylor Series Expansion

The Taylor series expresses a function as an infinite sum of terms calculated from its derivatives at a point. For tan(x) around zero, the series starts as x + x^3/3 + ..., allowing approximations by truncating after the first term and understanding how higher-order terms affect accuracy.

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