Question

Graph \(y=\cosh x\) and the Taylor polynomial $$ T_{2 M}(x)=\sum_{n=0}^{M} \frac{x^{2 n}}{(2 n) !} $$ on the interval (-5,5) for \(M=1,2,3, \ldots,\) until you find a value of \(M\) for which there's no perceptible difference between the two graphs. Choose the scale on the \(y\) -axis so that \(0 \leq y \leq 75\).

Short answer

#Answer# The smallest value of \(M\) for which there is no appreciable difference between the graphs of \(y = \cosh x\) and its Taylor polynomial of order \(2M\) within the interval \((-5, 5)\) is \(M = 3\).

Step-by-step solution

Define the function y = \cosh x

The hyperbolic cosine function is defined as: $$ \cosh x = \frac{e^x + e^{-x}}{2} $$ where \(e^x\) denotes the exponential function. Furthermore, the Taylor polynomial of order \(2M\) for hyperbolic cosine function is given as: $$ T_{2M}(x) = \sum_{n=0}^M \frac{x^{2n}}{(2n)!} $$

Define the Taylor polynomial of order 2M

To find the Taylor polynomial of order \(2M\), we can write: $$ T_{2M}(x) = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \cdots + \frac{x^{2M}}{(2M)!} $$

Decrease the interval range of x and y within the given limits

The given limits for the graphs are: - \(x \in (-5, 5)\) - \(y \in [0, 75]\)

Compare the graphs of both functions for different values of M

Plot the graphs of \(y = \cosh x\) and \(T_{2M}(x)\) for \(M = 1, 2, 3, \ldots\) and compare the graphs to determine the smallest value of \(M\) for which there is no appreciable difference between the graphs. For \(M=1\): $$ T_{2}(x) = 1 + \frac{x^2}{2!} $$ There is a visible difference between the graphs of \(\cosh x\) and \(T_{2}(x)\) within the given interval. For \(M=2\): $$ T_{4}(x) = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} $$ The difference between the graphs of \(\cosh x\) and \(T_{4}(x)\) becomes much smaller, but there is still some difference visible. For \(M=3\): $$ T_{6}(x) = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \frac{x^6}{6!} $$ For \(M = 3\), there is almost no difference between the graphs of \(\cosh x\) and \(T_{6}(x)\) within the interval \((-5, 5)\), and the graphs seem to be overlapping each other. Therefore, we can conclude that \(M = 3\) is the smallest value of \(M\) for which there is no perceptible difference between the graphs of \(y = \cosh x\)和\(T_{2M}(x)\) within the given interval.

Key concepts

Understanding the Hyperbolic Cosine Function

The hyperbolic cosine function, denoted as \(\cosh x\), is an even function, similar in shape to the usual cosine function but defined for the set of real numbers using the exponential function \(e^x\). The beauty of \(\cosh x\) lies in its definition: \[ \cosh x = \frac{e^x + e^{-x}}{2} \.\]
This function describes the shape of a hanging cable or chain, called a catenary, and it is not periodic like the trigonometric cosine function. When graphing \(y = \cosh x\), you’ll notice that as \(x\) increases in absolute value, the growth of \(\cosh x\) also accelerates due to the influence of \(e^x\).

For students, a common challenge with \(\cosh x\) is its unfamiliarity compared to its trigonometric counterparts. The key to understanding it is in recognizing the function's symmetry and its unbounded nature as \(x\) moves away from zero. The hyperbolic cosine function is integral in the realm of hyperbolic geometry and finds applications in various physics problems, like the description of relativistic motion.

Graphing Functions with Precision

Graphing functions is a fundamental skill in mathematics that allows students to visualize the behavior of functions across different domains. When graphing functions like \(y = \cosh x\) and its Taylor polynomial approximations, precision is crucial. The goal is not only to plot the points but to understand how the function behaves as \(x\) changes.

Here are some tips to ensure precision when graphing:

• Ascertain the function's domain and range, and use appropriate scales on the axes.
• Notice the function's symmetry or periodicity, which could provide insights into the function's behavior beyond the plotted range.
• When graphing a series approximation, like a Taylor polynomial, check the convergence against the actual function by plotting them on the same grid to see how well they align.
• Adjust the graph's viewing window to capture the essence of the function. For instance, the exercise suggests graphing with \(y\) values up to 75 to capture the significant details of \(\cosh x\).

For understanding the hyperbolic cosine function, graphing provides immediate visual feedback and deepens comprehension of how the function grows and where its approximation becomes indistinguishably close to the true function.

Convergence of Series and Approximations

The convergence of series is a central concept in calculus and analysis. It is the idea that as more terms are added to a series, it approaches a certain value, called the limit. In the context of Taylor polynomial approximations, convergence means that adding more terms of the series results in a polynomial that more closely resembles the function it approximates.

For the hyperbolic cosine function, the Taylor polynomial of order \(2M\) given by \[ T_{2M}(x) = \sum_{n=0}^M \frac{x^{2n}}{(2n)!} \] converges to the actual function \(\cosh x\) as \(M\) increases.

Understanding convergence helps students appreciate why we can approximate functions using polynomials and how this approximation improves with more terms. When graphing the function and its Taylor polynomial, it is the visual representation of this convergence we are looking for. This is exemplified in the exercise where no noticeable difference is seen between the actual function and the polynomial when \(M = 3\), indicating that the polynomial has converged sufficiently to the function. Such convergence is paramount in many practical applications, from engineering designs to algorithmic estimates in computer science.