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When calculating numerical approximations, one crucial aspect is to determine the accuracy of the result. Absolute error is the measure used to express the difference between the approximate value and the exact value of a quantity. Specifically, for a function approximation, the absolute error is calculated using the formula:
\[ \text{Absolute Error} = | \text{exact value} - \text{approximate value} | \]
In the context of approximating \( e^{-0.15} \) with a Taylor polynomial \( p_{2}(x) \), after obtaining the approximate value from the polynomial, you would find the absolute error by comparing this result with the exact value provided by a scientific calculator. This process is essential for understanding the reliability of the approximation and allows us to gauge the margin of error inherent to the polynomial's estimation.

Exponential functions are key players in various areas of mathematics and science, often serving to model growth and decay processes. The exponential function with base \( e \), denoted by \( f(x) = e^x \), is particularly important due to its unique mathematical properties. It is its own derivative, which means it changes at a rate proportional to its value—a property that manifests in natural phenomena like radioactive decay and population growth.
In the given exercise, we are dealing with an exponential function in the form of \( f(x) = e^{-x} \). This function describes a decreasing exponential trend, which is common in processes like cooling and discharging capacitors. By approximating this function using Taylor polynomials, we can efficiently compute values of the function for small inputs \( x \), which is useful when precise calculations become computationally demanding or when the function forms part of a larger, more complex model.

Taylor polynomials provide a means to approximate functions around a specific point, often near zero, known as the expansion point. The Taylor polynomial of degree \( n \) for a function \( f \) centered at \( x = a \) is given by the formula:
\[ p_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x - a)^k \]
In this formula, \( f^{(k)}(a) \) denotes the \( k \)-th derivative of \( f \) evaluated at \( a \), and \( k! \) stands for \( k \) factorial, the product of all positive integers up to \( k \). To calculate the Taylor polynomial for our exponential function \( f(x) = e^{-x} \), one must evaluate its derivatives at the expansion point, which in this case could be zero for convenience. Then, these derivatives are plugged into the Taylor polynomial formula.
For the exercise in question, the Taylor polynomial approximation involves calculating the polynomial up to the second-degree term. The given second-degree polynomial \( p_{2}(x) = 1 - x + x^2 / 2 \) helps us approximate our function \( f(x) \) near the point \( x = 0 \), which is a straightforward process involving algebraic substitution of the value into the polynomial.