{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) =√(1+x) ≈ 1 + x/2

Verified step by step guidance

1. Identify the function and its approximation: The function given is \(f(x) = \sqrt{1+x}\), and the approximation near \(x=0\) is \(f(x) \approx 1 + \frac{x}{2}\).
2. Calculate the approximate value at \(x=0.2\) using the linear approximation: Substitute \(x=0.2\) into the approximation to get \(f(0.2) \approx 1 + \frac{0.2}{2}\).
3. Understand the error bound concept: The error in the approximation can be estimated using the remainder term from Taylor's theorem, which involves the second derivative of \(f(x)\).
4. Find the second derivative of \(f(x)\): First, compute \(f'(x) = \frac{1}{2\sqrt{1+x}}\), then find \(f''(x) = -\frac{1}{4(1+x)^{3/2}}\).
5. Use the error bound formula for the linear approximation: The error \(R_2\) satisfies \(|R_2| \leq \frac{M}{2} |x|^2\), where \(M\) is the maximum value of \(|f''(x)|\) on the interval between 0 and 0.2. Determine \(M\) and calculate the bound.

Key Concepts

Linear Approximation (Tangent Line Approximation)

Linear approximation uses the tangent line at a point to estimate function values near that point. For f(x) near x=0, f(x) ≈ f(0) + f'(0)(x - 0). Here, √(1+x) is approximated by 1 + x/2, which is the tangent line at x=0.

Error Bound Using the Remainder Term in Taylor's Theorem

The error bound estimates how far the approximation is from the true value. Using Taylor's theorem, the remainder term involves higher derivatives evaluated at some point between 0 and x, providing a maximum possible error for the approximation.

Derivative of the Square Root Function

Understanding the derivative of f(x) = √(1+x) is essential for both approximation and error estimation. The first derivative is f'(x) = 1/(2√(1+x)), which determines the slope of the tangent line used in the linear approximation.