Taylor Polynomials For F(x) = 6e^x + 7e^{-x} At A = 0

by Alex Johnson 54 views

Let's dive into finding the first few Taylor polynomials for the function f(x) = 6e^x + 7e^{-x} centered at a = 0. This means we're looking for approximations of this function using polynomials, which are particularly accurate near the point x = 0. These approximations are incredibly useful in various fields, from physics to engineering, where we often need to simplify complex functions for easier calculations. We will focus on finding the Taylor linear function, T_1(x), and the Taylor quadratic function, T_2(x).

Understanding Taylor Polynomials

Before we get into the specifics, let's quickly recap what Taylor polynomials are all about. A Taylor polynomial is essentially a polynomial approximation of a function at a specific point. The n-th degree Taylor polynomial of a function f(x) centered at x = a is given by:

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

Where:

  • f'(a), f''(a), ... , f^(n)(a) are the first, second, and n-th derivatives of f(x) evaluated at x = a, respectively.
  • n! denotes the factorial of n.

In simpler terms, we are using the function's value and its derivatives at a single point to construct a polynomial that closely mimics the function's behavior around that point. The higher the degree of the polynomial, the better the approximation, at least in a neighborhood around a.

Calculating the Derivatives

To find the Taylor polynomials, we first need to calculate the derivatives of our function, f(x) = 6e^x + 7e^{-x}. Let's find the first two derivatives, as we are interested in T_1(x) and T_2(x).

First Derivative

The first derivative, f'(x), is found by differentiating f(x) with respect to x:

f'(x) = d/dx (6e^x + 7e^{-x}) = 6e^x - 7e^{-x}

Second Derivative

The second derivative, f''(x), is found by differentiating f'(x) with respect to x:

f''(x) = d/dx (6e^x - 7e^{-x}) = 6e^x + 7e^{-x}

Notice that the second derivative, f''(x), is the same as the original function, f(x). This is a neat property of this particular function, which will simplify our calculations.

Evaluating at a = 0

Now, we need to evaluate the function and its derivatives at the center point, a = 0.

Evaluating f(0)

f(0) = 6e^0 + 7e^{-0} = 6(1) + 7(1) = 13

Evaluating f'(0)

f'(0) = 6e^0 - 7e^{-0} = 6(1) - 7(1) = -1

Evaluating f''(0)

f''(0) = 6e^0 + 7e^{-0} = 6(1) + 7(1) = 13

So, we have f(0) = 13, f'(0) = -1, and f''(0) = 13. These values are crucial for constructing our Taylor polynomials.

Constructing the Taylor Polynomials

Now that we have the function and its derivatives evaluated at a = 0, we can construct the Taylor polynomials.

Taylor Linear Function: T_1(x)

The Taylor linear function, T_1(x), is the first-degree Taylor polynomial. It is given by:

T_1(x) = f(0) + f'(0)(x - 0) = f(0) + f'(0)x

Substituting the values we found earlier:

T_1(x) = 13 + (-1)x = 13 - x

So, the Taylor linear function for f(x) = 6e^x + 7e^{-x} centered at a = 0 is T_1(x) = 13 - x. This is a linear approximation of the function near x = 0.

Taylor Quadratic Function: T_2(x)

The Taylor quadratic function, T_2(x), is the second-degree Taylor polynomial. It is given by:

T_2(x) = f(0) + f'(0)(x - 0) + (f''(0)/2!)(x - 0)^2 = f(0) + f'(0)x + (f''(0)/2)x^2

Substituting the values we found earlier:

T_2(x) = 13 + (-1)x + (13/2)x^2 = 13 - x + (13/2)x^2

So, the Taylor quadratic function for f(x) = 6e^x + 7e^{-x} centered at a = 0 is T_2(x) = 13 - x + (13/2)x^2. This is a quadratic approximation of the function near x = 0, and it is generally a better approximation than the linear one.

Visualizing the Approximations

To truly appreciate the accuracy of these Taylor polynomials, it's helpful to visualize them. Imagine plotting the original function, f(x) = 6e^x + 7e^{-x}, along with T_1(x) = 13 - x and T_2(x) = 13 - x + (13/2)x^2. Near x = 0, you'll notice that both T_1(x) and T_2(x) closely follow the curve of f(x). However, as you move further away from x = 0, the approximations start to deviate from the original function. The quadratic approximation, T_2(x), typically remains closer to f(x) for a larger interval around x = 0 compared to the linear approximation, T_1(x).

Applications and Significance

Taylor polynomials aren't just theoretical exercises; they have numerous practical applications. They are widely used in:

  • Approximating Function Values: When dealing with complex functions that are difficult to evaluate directly, Taylor polynomials provide a simple way to estimate function values, especially near the center point a.
  • Simplifying Complex Equations: In physics and engineering, Taylor expansions are used to simplify complex equations, making them easier to solve and analyze.
  • Numerical Analysis: Taylor polynomials are fundamental in numerical methods for solving differential equations and performing other numerical computations.
  • Optimization: They can be used to approximate functions in optimization problems, allowing for efficient solutions.
  • Understanding Function Behavior: By examining the Taylor polynomial, we can gain insights into the local behavior of a function, such as its rate of change and concavity.

Conclusion

In summary, we have successfully found the Taylor linear function, T_1(x) = 13 - x, and the Taylor quadratic function, T_2(x) = 13 - x + (13/2)x^2, for the function f(x) = 6e^x + 7e^{-x} centered at a = 0. These polynomials provide increasingly accurate approximations of the original function near x = 0. Taylor polynomials are a powerful tool in mathematics, science, and engineering, allowing us to simplify complex problems and gain a deeper understanding of function behavior. This exploration highlights the importance of calculus in providing practical solutions to real-world problems.

For more information on Taylor series and their applications, you can visit Khan Academy's article on Taylor Series.