Problem 49 - Thomas Calculus

Question

Quadratic Approximations The Taylor polynomial of order 2 generated by a twice-differentiable function \(f(x)\) at \(x=a\) is called the quadratic approximation of \(f\) at \(x=a .\) Find the (a) linearization (Taylor polynomial of order 1 ) and (b) quadratic approximation of \(f\) at \(x=0\). \(f(x)=\sin x\)

Step-by-Step Solution

Answer

Linearization: \( x \); Quadratic approximation: \( x \).

Step 1: Understand the Linearization Concept

The linearization of a function at a point is essentially its Taylor polynomial of order 1. It is represented by the equation: \( f(x) \approx f(a) + f'(a)(x-a) \) where \( a \) is the point of approximation. Here, we approximate around \( x = 0 \).

Step 2: Calculate the Function Value and Derivative at x=0

For \( f(x) = \sin x \), calculate its first derivative: \( f'(x) = \cos x \). Now, at \( x = 0 \), \( f(0) = \sin 0 = 0 \) and \( f'(0) = \cos 0 = 1 \).

Step 3: Find the Linearization

Using the formula for linearization, we have: \( f(x) \approx f(0) + f'(0)(x - 0) = 0 + 1 \cdot x = x \). Therefore, the linearization of \( \sin x \) at \( x=0 \) is \( x \).

Step 4: Understand the Quadratic Approximation Concept

The quadratic approximation is the Taylor polynomial of order 2. It is given by: \( f(x) \approx f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 \).

Step 5: Calculate the Second Derivative at x=0

The second derivative of \( f(x) = \sin x \) is \( f''(x) = -\sin x \). Therefore, at \( x = 0 \), \( f''(0) = -\sin 0 = 0 \).

Step 6: Find the Quadratic Approximation

Using the Taylor polynomial of order 2: \( f(x) \approx f(0) + f'(0)x + \frac{f''(0)}{2}x^2 \). Replacing the values, we get: \( 0 + x + \frac{0}{2}x^2 = x \). Thus, the quadratic approximation of \( \sin x \) at \( x=0 \) is \( x \).

Key Concepts

Understanding Quadratic Approximation

Quadratic approximation revolves around using a Taylor polynomial of order 2 to approximate the behavior of a function near a particular point. It extends linear approximation by including a second-degree term, which provides a more accurate depiction of the function's curvature at that point.
When you use a quadratic approximation, you're essentially saying, "I want to model this function with a parabola that is closely aligned with the original function at a certain point." This method becomes incredibly useful when dealing with functions where their first derivative does not capture all of their essential features, or when enhanced precision is required.
The mathematical representation of a quadratic approximation is:
\[ f(x) \approx f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 \]
Here, \( f(a) \) is the function value at \( x = a \), \( f'(a) \) represents the first derivative, reflecting the slope of the function at that point, and \( f''(a) \) is the second derivative, indicating the nature of the function’s curvature.
This additional second-order term, \( \frac{f''(a)}{2!}(x-a)^2 \), allows the quadratic approximation to conform more closely to the original function's graph by aligning its concavity with that of the function.

Basics of Linearization

Linearization is the process of approximating a function with its first-order Taylor polynomial, resulting in a linear function that closely intersects the curve of the original function at a certain point. This technique is highly advantageous when you're working with functions that are complicated or non-linear and need a straightforward representation in a localized area.
The core idea here is to create a tangent line that represents the rate of change of the function at a particular point. The mathematical formula used is:
\[ f(x) \approx f(a) + f'(a)(x-a) \]
In this equation, \( f(a) \) is the function's value at the point \( a \), and \( f'(a) \) is the slope of the tangent, essentially giving the rate at which the function value changes around \( a \).
Think of linearization as zooming in on a very small area of a curve and depicting it as straight. It provides a useful tool for when you need to convey initial behavior or trends of a function without delving into more complex or higher-order changes.
Utilizing the concept of a tangent line, linearization efficiently simplifies larger and intricate functions while maintaining accuracy over a short span around the point of approximation.

Exploring Twice-Differentiable Functions

A twice-differentiable function is one that allows you to take not just the first derivative but also the second derivative. This level of differentiability reveals far more about a function's behavior than the first derivative alone.
For a function \( f(x) \), being twice-differentiable implies that \( f'(x) \) is well-defined and continuously smooth, meaning the function shouldn’t have any abrupt turns or points that aren’t smooth.
Understanding both first and second derivatives is crucial because:

• The first derivative \( f'(x) \) provides information about the slope or gradient - essentially how steep or flat the function is at any point, indicating increase or decrease.


• The second derivative \( f''(x) \) provides insights into the function's concavity or convexity. If \( f''(x) > 0 \), the function is concave up (like a cup), and if \( f''(x) < 0 \), it's concave down (like a frown).

These components are critical especially when performing tasks like quadratic approximations, as they allow for higher accuracy by capturing the function’s bending at a close range to the point of interest.
Thus, ensuring a function is twice-differentiable confirms it is well-suited for advanced approximations and analyses.