Problem 77 - Algebra and Trigonometry Real Mathematics, Real People

Question

Using calculus, it can be shown that the tangent function can be approximated by the polynomial \(\tan x \approx x+\frac{2 x^{3}}{3 !}+\frac{16 x^{5}}{5 !}\) where \(x\) is in radians. Use a graphing utility to graph the tangent function and its polynomial approximation in the same viewing window. How do the graphs compare?

Step-by-Step Solution

Answer

Considering the viewing window (the same interval which is valid for tangent function), both the tangent function and its polynomial approximation graphs look quite similar. However, one is a continuous function while the other includes vertical asymptotes. The small differences that can be noticed reflect the approximation error.

Step 1: Graph of the tangent function

Using a graphing utility, plot the function \(\tan (x)\). Make sure to consider the nature of this function which includes asymptotes at \(x = \pm \frac{\pi}{2}\) + \(n\pi\), where \(n\) is an integer.

Step 2: Graph of the polynomial function

In the same viewing window, graph the given polynomial function \(x + \frac{2x^3}{3!} + \frac{16x^5}{5!}\). Remember that the domain for the tangent function is \(-\frac{\pi}{2} < x < \frac{\pi}{2}\), so graph this function over the same interval.

Step 3: Observations and Comparison of Graphs

Observe the graph of both the functions. Notice how one graph is a smooth curve while the other one has sharp increases and decreases. The tangent function has vertical asymptotes while the polynomial function does not. However, over the interval \(-\frac{\pi}{2} < x < \frac{\pi}{2}\), the two graphs appear to be a very close approximation of each other. This can be attributed to the Taylor series approximation for the tangent function.

Key Concepts

Tangent Function

The tangent function, commonly denoted as \(\tan(x)\), is a fundamental trigonometric function that relates the angle of a right triangle to the ratios of two of its sides. In a Cartesian coordinate system, you can think of it as the ratio of the sine and cosine functions, or specifically, \(\tan(x) = \frac{\sin(x)}{\cos(x)}\).
This function is periodic with a period of \(\pi\), meaning it repeats its values every \(\pi\) units.
One distinctive feature of the tangent function is its vertical asymptotes, which occur at \(x = \frac{\pi}{2} + n\pi\), where \(n\) is an integer. These asymptotes result in the tangent function going to infinity, which translates into sharp jumps or breaks in its graph. As such, when graphing \(\tan(x)\), we must pay special attention to these features, which can help guide us in understanding its behavior and its limitations within certain intervals.

Polynomial Approximation

Polynomial approximation is a technique where we use polynomials to estimate more complex functions. This method is beneficial as polynomials are smoother and easier to handle mathematically. The Taylor series is one such polynomial approximation that effectively approximates a function around a specific point with the use of multiple terms.
In the case of \(\tan(x)\), we use the polynomial \(x + \frac{2x^3}{3!} + \frac{16x^5}{5!}\) to approximate the tangent function around \(x = 0\). This polynomial comprises a series of terms based on the derivatives of the function at a single point. Each added term enhances the approximation's accuracy, especially when \(x\) is near the center of the expansion.

• The linear term \(x\) accounts for the most basic approximation.
• Higher-order terms \(\frac{2x^3}{3!}\) and \(\frac{16x^5}{5!}\) add more curvature and accuracy to the approximation as they consider the higher derivatives of \(\tan(x)\).

When dealing with such approximations, it’s important to remember that they are most accurate near the point of expansion and less so further away.

Graphing Utility

A graphing utility is a powerful tool used to visualize mathematical functions. It allows us to graph functions quickly and easily while making real-time comparisons between different types of functions, such as a primary function and its polynomial approximation.
In our example, using a graphing utility to plot the tangent function \(\tan(x)\) and its polynomial approximation, \(x + \frac{2x^3}{3!} + \frac{16x^5}{5!}\), in the same view can give us significant insights into their similarities and differences.

• First, plot \(\tan(x)\), paying attention to its periodicity and its asymptotes at specific intervals.
• Second, overlay the polynomial approximation and observe how closely it matches the tangent function for small values of \(x\).

Comparing the graphs side by side can reveal how well the polynomial captures the behavior of the tangent function over the interval \(-\frac{\pi}{2} < x < \frac{\pi}{2}\). This visual representation helps us see where the polynomial most accurately reflects the tangent function and where deviations occur.

Calculus

Calculus is a branch of mathematics that deals with continuous change. It provides the tools needed to understand the behavior of functions, such as the tangent function, especially concerning their rate of change and curvature. By employing calculus, we can derive Taylor series, which are pivotal for creating polynomial approximations.
The Taylor series formula is an infinite sum of terms calculated from a function's derivatives at a single point. In simpler terms, it rests on the idea of building up a function's profile using polynomials, which are infinitely differentiable.

• Each term in the Taylor series adds more precision to the approximation, deriving from higher-order derivatives.
• The approximations are generally valid within a certain range, known as the radius of convergence.

Using calculus to approximate functions through Taylor's theorem allows us to understand how a function behaves around a specific point, providing key insights and enabling easier calculation, especially for functions that are otherwise difficult to compute directly.