Question

In the following exercises, find the Taylor polynomials of degree two approximating the given function centered at the given point. $$ f(x)=\sin (2 x) \text { at } a=\frac{\pi}{2} $$

Short answer

The Taylor polynomial of degree two is \( P_2(x) = -2x + \pi \).

Step-by-step solution

Understand the Taylor Polynomial Formula

The Taylor polynomial of degree two for a function \( f(x) \) centered at \( a \) is given by:\[P_2(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2\]We need to find \( f(a) \), \( f'(a) \), and \( f''(a) \) to construct the Taylor polynomial.

Calculate \( f(a) \)

First, substitute \( a = \frac{\pi}{2} \) into the function:\[f\left( \frac{\pi}{2} \right) = \sin\left( 2 \times \frac{\pi}{2} \right) = \sin(\pi) = 0\]So, \( f(a) = 0 \).

Find the First Derivative and Evaluate at \( a \)

The first derivative of \( f(x) = \sin(2x) \) is:\[f'(x) = 2\cos(2x)\]Next, evaluate at \( a = \frac{\pi}{2} \):\[f'\left( \frac{\pi}{2} \right) = 2\cos\left( 2 \times \frac{\pi}{2} \right) = 2\cos(\pi) = -2\]Therefore, \( f'(a) = -2 \).

Find the Second Derivative and Evaluate at \( a \)

The second derivative of \( f(x) = \sin(2x) \) is:\[f''(x) = -4\sin(2x)\]Now, evaluate at \( a = \frac{\pi}{2} \):\[f''\left( \frac{\pi}{2} \right) = -4\sin\left( 2 \times \frac{\pi}{2} \right) = -4\sin(\pi) = 0\]Thus, \( f''(a) = 0 \).

Construct the Taylor Polynomial of Degree Two

Now, substitute \( f(a) = 0 \), \( f'(a) = -2 \), and \( f''(a) = 0 \) into the Taylor polynomial formula:\[P_2(x) = 0 + (-2)(x - \frac{\pi}{2}) + \frac{0}{2!}(x - \frac{\pi}{2})^2\]Simplify the expression:\[P_2(x) = -2(x - \frac{\pi}{2}) = -2x + \pi\]

Write the Final Result

The Taylor polynomial of degree two for \( f(x) = \sin(2x) \) centered at \( a = \frac{\pi}{2} \) is:\[P_2(x) = -2x + \pi\]

Key concepts

Degree Two Polynomial

A degree two polynomial is essentially a quadratic expression. It takes the form

• \( ax^2 + bx + c \)

where \( a \), \( b \), and \( c \) are constants. In the context of Taylor polynomials, the degree corresponds to how many terms of the derivative you consider when approximating a function at a particular point, called the 'center' or 'expansion point'.
This is particularly useful in calculus because it allows us to approximate complex functions using simpler polynomials. For a degree two Taylor polynomial, we use a quadratic formula, which involves the function value at the center and its first two derivatives. This gives a parabola-like approximation.In our exercise with \( f(x) = \sin (2x) \), we centered the polynomial at \( a = \frac{\pi}{2} \). The Taylor polynomial of degree two for this function was derived using the formula

• \( P_2(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 \)

The polynomial \( P_2(x) \) turned out to be a line rather than a parabola because the second derivative, \( f''(a) \), equaled zero in this case, simplifying the polynomial to a linear expression-reducing the term with \( (x - a)^2 \).

Function Approximation

Function approximation is the process of finding a simpler function that closely mimics the behavior of a more complex one. This is particularly beneficial in the analysis of mathematical and real-world functions, where precise calculations are not feasible.
In calculus, Taylor polynomials are a powerful tool for approximating functions near a point. They allow us to estimate the value of functions that could otherwise be cumbersome or impossible to compute directly.Using Taylor polynomials for approximation offers several advantages:

• Simplifies calculations by using basic algebraic expressions instead of complex functions.
• Maintains the function’s essential features close to the expansion point, like slope and concavity.
• Provides a method to evaluate functions where using exact values might be overly time-consuming or difficult.

While approximate, these polynomials give a snapshot of how the function behaves around a given point. For \( f(x) = \sin(2x) \) at \( a = \frac{\pi}{2} \), the exercise helped us derive an approximation in the form of a degree two polynomial. This linear form, \( P_2(x) = -2x + \pi \), provides insight into the function’s behavior near \( x = \frac{\pi}{2} \).

Taylor Series Expansion

The Taylor series expansion is a method for expressing a function as an infinite sum of terms calculated from its derivatives at a single point. This concept extends the Taylor polynomial idea to an infinite degree, aiming for ever-closer approximation.A Taylor series can be written as:

• \( f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \ldots \)

The idea behind the Taylor series is that by considering more terms, the polynomial approximation becomes increasingly accurate for a wider range of values around \( a \).In our exercise, only the first three terms, constituting a degree two polynomial, were used. This provided a local approximation around the center \( a = \frac{\pi}{2} \).
For many functions and applications, especially when \( x \) is very close to \( a \), a Taylor polynomial with a few terms can offer significant insight.
The approximation's accuracy improves with both more terms and smaller intervals from the center.
Although, for most practical purposes, the first few derivatives can be sufficient if you're only interested in a local approximation or if computational efficiency is a consideration.