{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

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Identify the function and its approximation: The function given is \(f(x) = \sqrt{1+x}\), and the approximation near \(x=0\) is \(f(x) \approx 1 + \frac{x}{2}\).

Calculate the approximate value at \(x=0.2\) using the linear approximation: Substitute \(x=0.2\) into the approximation to get \(f(0.2) \approx 1 + \frac{0.2}{2}\).

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 \(f(x)\).

Find the second derivative of \(f(x)\): First, compute \(f'(x) = \frac{1}{2\sqrt{1+x}}\), then find \(f''(x) = -\frac{1}{4(1+x)^{3/2}}\).

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.

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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.

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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.

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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.

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