Problem 7 - Calculus for Biology and Medicine

Question

In Problems \(6-10\), compute the Taylor polynomial of degree \(n\) about \(a=0\) for the indicated functions. $$ f(x)=\cos x, n=5 $$

Step-by-Step Solution

Answer

The Taylor polynomial of degree 5 for \(\cos x\) about \(a=0\) is \(P_5(x) = 1 - \frac{1}{2}x^2 + \frac{1}{24}x^4\).

Step 1: Understand the Taylor Polynomial Formula

The Taylor polynomial of degree \(n\) about \(a=0\) for a function \(f(x)\) is given by: \( P_n(x) = f(a) + f'(a)x + \frac{f''(a)}{2!}x^2 + ... + \frac{f^{(n)}(a)}{n!}x^n \). For this exercise, \(a=0\) and \(n=5\).

Step 2: Find the Derivatives of \(f(x) = \cos x\)

First, compute successive derivatives of \(f(x) = \cos x\):1. \(f(x) = \cos x\)2. \(f'(x) = -\sin x\)3. \(f''(x) = -\cos x\)4. \(f'''(x) = \sin x\)5. \(f''''(x) = \cos x\)6. \(f'''''(x) = -\sin x\)

Step 3: Evaluate the Derivatives at \(x = 0\)

Substitute \(x = 0\) into each derivative to find its value:- \(f(0) = \cos 0 = 1\)- \(f'(0) = -\sin 0 = 0\)- \(f''(0) = -\cos 0 = -1\)- \(f'''(0) = \sin 0 = 0\)- \(f''''(0) = \cos 0 = 1\)- \(f'''''(0) = -\sin 0 = 0\)

Step 4: Construct the Taylor Polynomial

Using the derivatives evaluated at \(x=0\), we create the polynomial:\[ P_5(x) = 1 - \frac{1}{2!}x^2 + \frac{1}{4!}x^4 \]Given that the coefficients for the first, third, and fifth-degree terms are zero, they are omitted.

Key Concepts

Derivatives

Derivatives are a fundamental concept in calculus. They allow us to understand how a function is changing at any given point.
This is important for constructing a Taylor polynomial, as derivatives help us capture the behavior of the function around a point.When we differentiate a function, we are essentially finding its rate of change or slope at a particular point.

• The first derivative, denoted as \(f'(x)\), represents the slope of the tangent line to the function at any point \(x\).
• The second derivative, \(f''(x)\), gives us information about the concavity of the function; whether the curve is opening upwards or downwards.

For the cosine function, calculating multiple derivatives garners the same cyclic behavior due to the inherent trigonometric properties:

• \(f(x) = \cos x\)
• \(f'(x) = -\sin x\)
• \(f''(x) = -\cos x\)

This cyclicity repeats every four derivatives, showing how essential derivatives are in understanding the function's periodicity.

Cosine Function

The cosine function \(f(x) = \cos x\) is one of the fundamental trigonometric functions. It frequently appears in math and physics, particularly in problems involving waves or oscillations.The cosine function is periodic with a period of \(2\pi\), meaning it repeats its values every \(2\pi\) units.

• At \(x = 0\), the cosine function has a value of \(1\).
• The maximum value of the cosine function is \(1\), and the minimum value is \(-1\).

This behavior is important when considering the Taylor polynomial, as the cosine function’s values at specific points can determine the coefficients of the polynomial.When you calculate derivatives of the cosine function, these periodic values come in handy in assessing the polynomial components:

• Even derivatives (e.g., \(f'', f\''\''\)) will be either \(\pm 1\) when evaluated at \(x = 0\).
• Odd derivatives (e.g., \(f', f'''\)) will evaluate to \(0\) at \(x = 0\), simplifying the Taylor polynomial.

Maclaurin Series

A Maclaurin series is a special version of a Taylor series where we approximate a function about \(x = 0\). It's an extremely useful tool in mathematical analysis for approximating functions.In essence, the Maclaurin series allows us to express complex functions as an infinite sum of polynomial terms, which can be more manageable to analyze or compute.For any function \(f(x)\), the series is given by:\[ f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \ldots \]The Maclaurin series of the cosine function captures its behavior around zero perfectly since:

• The even power terms carry the significant values, because at \(x=0\), only even derivatives contribute with non-zero values.
• The resulting series for the cosine function becomes an alternating series: \(1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \ldots\)

This approach helps in approximating the cosine function using a polynomial, which is easier to compute or evaluate in various applications.