Question

In Exercises \(55-58,\) use a computer algebra system to find the fifth-degree Taylor polynomial (centered at \(c\) ) for the function. Graph the function and the polynomial. Use the graph to determine the largest interval on which the polynomial is a reasonable approximation of the function. $$ f(x)=x \cos 2 x, \quad c=0 $$

Short answer

The final answer will be the 5th degree Taylor polynomial for the function \(f(x) = x \cos 2x\) centered at \(c=0\), and an interval on the real number line where this polynomial approximates the given function reasonably well. You will obtain these results after carrying out the calculations and graphical analysis in the steps outlined.

Step-by-step solution

Compute the derivatives

First, find the first five derivatives of the function \(f(x)=x \cos 2x\).

Evaluate derivatives at \(c=0\)

After obtaining the derivatives, you need to evaluate each of them at \(c = 0\).

Determine the fifth degree Taylor polynomial

Now that you have the derivatives and their values at \(c=0\), you can use them to define the Taylor polynomial of degree five. The general expression of a Taylor polynomial is given by \(P(x) = f(c) + f'(c)(x - c) + \frac{{f''(c)(x - c)^2}}{{2!}} + \frac{{f'''(c)(x - c)^3}}{{3!}} + \frac{{f''''(c)(x - c)^4}}{{4!}} + \frac{{f'''''(c)(x - c)^5}}{{5!}}\).

Visualize the function and its Taylor polynomial

Next, plot the original function and its Taylor approximation on the same graph. This will greatly help in understanding how well the polynomial approximates the function.

Determine the interval of approximation

By analyzing the graph, you are required to identify the interval where the function and its Taylor polynomial coincide reasonably well. This is usually done by visually comparing the two curves and noting where they are very close.

Key concepts

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This mathematical framework is essential for analyzing change and can be used to solve complex problems in physics, engineering, economics, and more. The study of calculus is divided into two main branches: differential calculus, which concerns the rate of change and slopes of curves, and integral calculus, which focuses on the accumulation of quantities and the areas under and between curves.

In solving our Taylor polynomial problem, we primarily use concepts from differential calculus. The process involves taking derivatives, which measure how a function changes as its input changes. This is foundational for understanding how to approximate complex functions with simpler, polynomial functions, which is exactly what a Taylor polynomial does.

Derivatives

A derivative represents the rate at which a function's output changes as its input changes. In formal terms, the derivative of a function at a point is the slope of the tangent line to the function's graph at that point. Derivatives can be calculated for all different orders: first derivatives, second derivatives, and so on. Each higher-order derivative provides more nuanced information about the function's behavior.

For instance, the first derivative tells us about the function's rate of change, whereas the second derivative provides insights into the function's concavity, or the rate at which the rate of change itself changes. In the Taylor polynomial exercise, we compute multiple derivatives of the given function, as these are crucial to constructing the polynomial that will approximate the function around a specific point.

Function Approximation

Function approximation is a technique used in mathematics to find a simpler function that closely matches the values of a more complex function within a certain range. A well-known method of function approximation is the Taylor series, which uses polynomials to approximate a function.

The main idea behind Taylor polynomials is that you can take a complicated function and approximate it with a polynomial, which is much easier to work with. By increasing the degree of the polynomial, you can achieve a higher level of accuracy in the approximation. In the exercise provided, we use a fifth-degree polynomial, suggesting that we're including terms up to the function's fifth derivative for our approximation. A Taylor polynomial is particularly good at approximating a function near the center point (in this case, given as c=0), and it's one of the reasons why Taylor polynomials have great importance in both theoretical and applied mathematics.

Computer Algebra System

A Computer Algebra System (CAS) is a software program that facilitates symbolic mathematics. These powerful systems can manipulate mathematical expressions in a way that mimics hand-written algebra performed by humans. CAS tools are used for a variety of purposes, from simple calculations like solving equations to more complicated tasks such as symbolic integrations, differentiations, simplifying algebraic expressions, and even generating function approximations.

In the exercise we're discussing, a CAS would be instrumental in computing the derivatives required for the Taylor polynomial. It can quickly handle the often tedious calculations involved in finding higher-order derivatives, significantly reducing the potential for human error. Furthermore, a CAS can graph both the original function and the Taylor polynomial, which helps in visualizing and determining the accuracy of the approximation over a particular interval. When using a computer algebra system, the process of finding and working with Taylor polynomials becomes more accessible, allowing students and professionals to focus on analysis rather than computation.