Problem 1 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Draw the graphs of $$ f(x)=\sin x \quad \text { and } \quad p(x)=x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !} \quad \text { for } \quad-7 \frac{\pi}{10} \leq x \leq 7 \frac{\pi}{10} $$ The graphs should be indistinguishable on \(-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}\). The largest separation on \(-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}\) occurs at \(x=\frac{\pi}{2}\left(\right.\) and \(\left.x=-\frac{\pi}{2}\right)\). Compute \(f\left(\frac{\pi}{2}\right), p\left(\frac{\pi}{2}\right)\), and the relative error in the approximation \(f\left(\frac{\pi}{2}\right) \doteq p\left(\frac{\pi}{2}\right)\).

Step-by-Step Solution

Answer

At \( x=\frac{\pi}{2} \), \( f(x) = 1 \), \( p(x) \approx 0.9248 \), with a relative error of 0.0752.

Step 1: Understand the Functions

The first function, \( f(x) = \sin x \), is the standard sine function. The second function, \( p(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} \), is a fifth-degree polynomial approximation of the sine function, derived from the Taylor series for sine around 0.

Step 2: Evaluate at \( x=\frac{\pi}{2} \)

Substitute \( x=\frac{\pi}{2} \) into the functions. The sine function evaluates to \( f\left(\frac{\pi}{2}\right) = 1 \). For \( p(x) \), substitute \( x=\frac{\pi}{2} \) to get: \[ p\left(\frac{\pi}{2}\right) = \frac{\pi}{2} - \frac{\left(\frac{\pi}{2}\right)^3}{6} + \frac{\left(\frac{\pi}{2}\right)^5}{120}. \]

Step 3: Approximate p(x) at \( x=\frac{\pi}{2} \)

Calculate the terms: \( \left(\frac{\pi}{2}\right)^3 \approx 3.8758 \) and \( \left(\frac{\pi}{2}\right)^5 \approx 29.6088 \). Plug them into the equation: \[ p\left(\frac{\pi}{2}\right) = \frac{\pi}{2} - \frac{3.8758}{6} + \frac{29.6088}{120}. \] Thus, \( p\left(\frac{\pi}{2}\right) \approx 0.9248. \)

Step 4: Calculate the Relative Error

Relative error is calculated as \( \text{Relative Error} = \frac{|f(x) - p(x)|}{|f(x)|} \). For \( x=\frac{\pi}{2} \), \[ \text{Relative Error} = \frac{|1 - 0.9248|}{1} = 0.0752. \]

Key Concepts

sine function

The sine function, denoted as \( f(x) = \sin x \), is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the opposite side over the hypotenuse. In the unit circle, it represents the y-coordinate of a point on the circle's circumference that corresponds to a given angle from the positive x-axis. The sine function is periodic with a period of \( 2\pi \), meaning that it repeats its values every \( 2\pi \) units. It oscillates between -1 and 1, creating a smooth wave-like pattern.
The sine function is crucial in modeling wave behaviors, such as sound and light waves, and plays a vital role in fields like engineering, physics, and signal processing. It is also foundational in calculus and mathematical analysis, where it frequently appears in integrals and derivatives.

polynomial approximation

Polynomial approximation is a technique used to approximate complex functions with simpler polynomials. The idea is to use a finite number of terms from a function's Taylor series expansion. In this context, the Taylor series of the sine function is used to create a polynomial approximation. A Taylor series is an infinite sum of terms, each derived from the derivatives of a function, evaluated at a single point.
For the sine function, the Taylor series is approximated as follows for small values of \( x \):

• The approximation starts with \( p(x) = x \).
• It adds terms like \( -\frac{x^3}{3!} \) and \( +\frac{x^5}{5!} \) to improve accuracy.

This creates a fifth-degree polynomial that closely resembles the sine curve, particularly around zero, where it overlaps almost perfectly for values within \([-\frac{\pi}{2}, \frac{\pi}{2}]\). As more terms are added, the polynomial becomes a more precise representation of the sine function, allowing complex trigonometric expressions to be simplified.

relative error

Relative error is a measure of the precision of an approximation compared to an exact value. It assesses how close an estimated value, like a polynomial approximation, comes to the true value, such as in the sine function. The formula for relative error is:

• Relative Error \( = \frac{|\text{exact value} - \text{approximate value}|}{|\text{exact value}|} \).

This calculation provides a ratio expressing the deviation as a fraction of the exact value, with a smaller relative error indicating a more precise approximation.
In practice, relative error is crucial in assessing the accuracy of polynomial approximations in calculus and engineering. In our example, using a polynomial approximation for \( \sin(\frac{\pi}{2}) \), we found a relative error of 0.0752. This number signifies that the polynomial is reasonably accurate, though a small error persists, highlighting the limitations of finite-term polynomial approximations.