Taylor's Theorem

Taylor's theorem is a mathematical result in calculus that allows us to approximate a function using a polynomial. It provides a way to expand a function around a point in terms of its derivatives at that point.

Function and Expansion

We are given the function f(x) = 40 + 53(x - 2) and we need to expand it using Taylor's theorem. The expansion will be centered around the point x = 2.

Taylor's Theorem Formula

The general formula for Taylor's theorem is:

f(x) = f(a) + f'(a)(x - a) + f''(a)(x - a)^2/2! + f'''(a)(x - a)^3/3! + ...

where f(a) represents the value of the function at the point a, f'(a) represents the first derivative of the function at the point a, f''(a) represents the second derivative of the function at the point a, and so on.

Expansion of f(x)

Let's apply Taylor's theorem to expand f(x) = 40 + 53(x - 2) around x = 2.

- Evaluate f(2) at the center point:
- f(2) = 40 + 53(2 - 2) = 40

- Calculate the first derivative of f(x):
- f'(x) = 53

- Evaluate f'(2) at the center point:
- f'(2) = 53

- Plug these values into the Taylor's theorem formula:
- f(x) = f(2) + f'(2)(x - 2) = 40 + 53(x - 2)

Final Expansion

The expanded form of the function f(x) using Taylor's theorem is:

f(x) = 40 + 53(x - 2)

Summary

In summary, we have used Taylor's theorem to expand the function f(x) = 40 + 53(x - 2) around the point x = 2. The expansion results in the equation f(x) = 40 + 53(x - 2). Taylor's theorem provides a powerful tool for approximating functions using polynomials, allowing us to gain insights into the behavior of functions and make useful approximations in various mathematical and scientific applications.