P5(x) refers to the 5th degree polynomial approximation of a function at a specific point x, using the 4th Order Taylor Series. It is calculated by taking the 4th derivative of the function at x and dividing it by 4!, then adding the 3rd derivative at x divided by 3!, the 2nd derivative at x divided by 2!, and the 1st derivative at x, and finally adding the function value at x.
The 4th Order Taylor Series for Sin(x) is derived by taking the 4th derivative of Sin(x) at a specific point x, which is equal to -Sin(x), and dividing it by 4!, then adding the 3rd derivative of Sin(x) at x, which is equal to -Cos(x), divided by 3!, the 2nd derivative of Sin(x) at x, which is equal to -Sin(x), divided by 2!, and the 1st derivative of Sin(x) at x, which is equal to Cos(x). This results in the familiar series expansion of Sin(x) = x - (x^3/3!) + (x^5/5!) - (x^7/7!) + ...
The 4th Order Taylor Series for Sin(x) allows us to approximate the value of Sin(x) at a specific point x with a 5th degree polynomial. This is useful in numerical analysis and allows for more accurate calculations than using the actual Sin(x) function.
The 4th Order Taylor Series of xSin(2x) is a more complex series compared to the 4th Order Taylor Series of Sin(x). This is because xSin(2x) is a composite function and requires the use of the chain rule in taking derivatives. The resulting series is a combination of x^2, x^4, x^6, and higher powers of x, whereas the 4th Order Taylor Series of Sin(x) only includes x, x^3, x^5, and higher powers of x.
The 4th Order Taylor Series is commonly used in engineering and scientific fields to approximate the behavior of a system or function. It is particularly useful in situations where the actual function is difficult to calculate, and an approximation is needed. For example, it can be used in calculating trajectories of objects in motion, predicting the behavior of electrical circuits, and in numerical methods for solving differential equations.