{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) = sin x ≈ x

Verified step by step guidance

1. Identify the function and the approximation given: the function is \(f(x) = \sin x\), and the approximation near zero is \(\sin x \approx x\).
2. To estimate \(f(0.2)\) using the approximation, substitute \(x = 0.2\) into the approximation: \(f(0.2) \approx 0.2\).
3. To find a bound on the error, recall the Taylor remainder theorem. The error in approximating \(\sin x\) by \(x\) near zero is given by the next term in the Taylor series expansion, which involves \(\frac{\cos c}{2} x^2\) for some \(c\) between 0 and \(x\).
4. Since \(|\cos c| \leq 1\) for all real \(c\), the maximum error bound is \(\left| R_2 \right| \leq \frac{|x|^3}{6}\), because the next term in the Taylor series for \(\sin x\) after \(x\) is \(-\frac{x^3}{3!}\).
5. Calculate the error bound by substituting \(x = 0.2\) into \(\frac{|x|^3}{6}\) to get the maximum possible error in the approximation.

Key Concepts

Small Angle Approximation

The small angle approximation states that for values of x near zero, sin(x) can be approximated by x. This simplifies calculations by replacing the sine function with a linear expression, which is accurate for small angles measured in radians.

Taylor Series Expansion

The Taylor series expresses a function as an infinite sum of terms calculated from its derivatives at a single point. For sin(x) near zero, the series starts as x - x³/6 + ..., and truncating after the first term gives the small angle approximation.

Error Bound in Approximations

An error bound estimates the maximum difference between the true function value and its approximation. For Taylor approximations, the Lagrange remainder formula provides a way to calculate this bound, ensuring the approximation's accuracy is quantifiable.