Final Answer:
the polynomial of least degree that will approximate f(x) throughout the given interval with an error of magnitude less than 10^-4 is (option d) Approximation.
Step-by-step explanation:
In mathematical analysis, the Taylor series is used for polynomial approximation of a function around a specific point. To find the polynomial of least degree that approximates a function f(x) throughout a given interval with an error less than 10^-4, the process involves selecting the appropriate number of terms from the Taylor series expansion. This aligns with the concept of approximation, making option d) Approximation the correct choice (option d).
The Taylor series expansion is a method that allows us to represent a function as an infinite sum of terms involving its derivatives. By considering a sufficient number of terms, we can create a polynomial that closely approximates the original function within a specified interval. The error in the approximation decreases as more terms are included. In this context, option d) Approximation captures the essence of using Taylor series for the purpose of achieving a desired level of accuracy.
In conclusion, when employing Taylor series to find a polynomial of least degree for approximating a function within a given interval, the goal is to minimize the error. Option d) Approximation accurately represents this mathematical process, emphasizing the use of polynomial approximations to closely match the behavior of a function while maintaining a specified level of precision.