{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero. 


a. Estimate f(0.1) and give a bound on the error in the approximation.


f(x) = eˣ ≈ 1 + x

Verified step by step guidance

1. Identify the function and the approximation given: the function is \(f(x) = e^{x}\) and the approximation near zero is \(f(x) \approx 1 + x\).
2. Calculate the approximate value of \(f(0.1)\) using the linear approximation: substitute \(x = 0.1\) into \(1 + x\) to get the estimate.
3. Recall that the error in using the linear approximation can be bounded using the remainder term from Taylor's theorem. For \(e^{x}\), the remainder after the linear term is given by \(R_2 = \frac{e^{c} x^{2}}{2}\) for some \(c\) between 0 and \(x\).
4. To find an error bound, note that \(e^{c}\) is increasing, so the maximum value of \(e^{c}\) on the interval \([0, 0.1]\) is \(e^{0.1}\). Use this to bound the error: \(|R_2| \leq \frac{e^{0.1} (0.1)^{2}}{2}\).
5. Summarize the result: the approximate value of \(f(0.1)\) is \(1 + 0.1\), and the error in this approximation is at most \(\frac{e^{0.1} (0.1)^{2}}{2}\). This gives a clear estimate and a guaranteed bound on the error.

Key Concepts

Taylor Series Approximation

The Taylor series expresses a function as an infinite sum of terms calculated from its derivatives at a single point. Near zero, eˣ can be approximated by 1 + x, which is the first-order Taylor polynomial. This approximation simplifies calculations for small x values by using only the first two terms.

Error Bound in Taylor Approximations

The error bound estimates how far the approximation is from the true function value. For the first-order Taylor approximation of eˣ, the remainder term involves the second derivative and can be bounded using the Lagrange form of the remainder. This helps quantify the maximum possible error when approximating f(0.1).

Exponential Function Properties

The exponential function eˣ is smooth and infinitely differentiable, with all derivatives equal to eˣ. This property ensures the Taylor series converges for all real x and allows easy computation of derivatives needed for approximations and error bounds.