{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) = ln (1 + x) ≈ x − x²/2

Verified step by step guidance

1. Identify the function and its approximation: The function given is \(f(x) = \ln(1 + x)\), and the approximation near zero is \(f(x) \approx x - \frac{x^2}{2}\).
2. Calculate the approximate value at \(x = 0.2\) by substituting into the approximation formula: \(f(0.2) \approx 0.2 - \frac{(0.2)^2}{2}\).
3. Understand that the error bound can be estimated using the remainder term from the Taylor series expansion of \(\ln(1+x)\) around \(x=0\).
4. Recall that the Taylor series for \(\ln(1+x)\) is \(x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots\), so the next term after the approximation is \(\frac{x^3}{3}\), which helps in bounding the error.
5. Use the Lagrange form of the remainder to bound the error: the error \(R_2(x)\) satisfies \(|R_2(x)| \leq \max_{c \in [0, x]} \left| \frac{f^{(3)}(c)}{3!} x^3 \right|\), where \(f^{(3)}(x)\) is the third derivative of \(\ln(1+x)\).

Key Concepts

Taylor Series Approximation

A Taylor series approximates a function near a point by a polynomial whose terms involve derivatives of the function at that point. For small values of x near zero, ln(1 + x) can be approximated by the first few terms of its Taylor series, such as x − x²/2, simplifying calculations.

Error Bound in Taylor Approximations

The error bound quantifies the maximum difference between the actual function value and its Taylor polynomial approximation. It is often estimated using the remainder term, which depends on higher-order derivatives evaluated within the interval, ensuring the approximation's accuracy is understood.

Properties of the Natural Logarithm Function

The natural logarithm function ln(1 + x) is defined for x > -1 and is smooth and differentiable near zero. Understanding its behavior and derivatives helps in constructing accurate approximations and estimating errors when x is close to zero.