Problem 42 - 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 \(x=0\). \(f(x)=e^{\sin x}\)

Step-by-Step Solution

Answer

Linearization is \(1 + x\); Quadratic approximation is \(1 + x + \frac{x^2}{2}\).

Step 1: Define the Taylor Series

The Taylor series of a function \(f(x)\) at \(x=a\) is given by: \[ T_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n \] For the purpose of solving the problem, we need the first two terms for the linearization and the first three terms for the quadratic approximation.

Step 2: Evaluate Function and Derivatives at x=0

Given \(f(x) = e^{\sin x}\), we find the value of \(f(0)\), \(f'(0)\), and \(f''(0)\). 1. \(f(0) = e^{\sin 0} = e^0 = 1\).2. First derivative: \(f'(x) = e^{\sin x} \cos x\). - \(f'(0) = e^{0} \cdot \cos 0 = 1\).3. Second derivative: \(f''(x) = e^{\sin x} (\cos^2 x - \sin x)\). - \(f''(0) = e^0 (1 - 0) = 1\).

Step 3: Find Linearization (Order 1 Polynomial)

The linearization of \(f(x)\) at \(x=0\) is given by the first two terms of the Taylor series:\[ T_1(x) = f(0) + f'(0) \cdot x = 1 + 1 \cdot x = 1 + x \]

Step 4: Compute Quadratic Approximation (Order 2 Polynomial)

The quadratic approximation of \(f(x)\) at \(x=0\) is given by the first three terms of the Taylor series:\[ T_2(x) = f(0) + f'(0) \cdot x + \frac{f''(0)}{2} \cdot x^2 = 1 + 1 \cdot x + \frac{1}{2} \cdot x^2 = 1 + x + \frac{x^2}{2} \]

Key Concepts

Quadratic Approximation

Quadratic approximation is a method used in calculus to approximate a differentiable function near a specific point. This approximation is also known as the Taylor polynomial of degree 2.
When we use quadratic approximation, we consider the value of the function at a point, its first derivative, and its second derivative to create a polynomial that closely resembles the function around that point.

• The formula for the quadratic approximation of a function at a point \(x=a\) is:\[ T_2(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2}(x-a)^2 \]
• This formula includes terms up to the second degree in \(x-a\), which makes it more accurate than linear approximations, especially when we move further from the point \(a\).

In our exercise, for the function \(f(x) = e^{\sin x}\), the quadratic approximation at \(x = 0\) would be\[ T_2(x) = 1 + x + \frac{x^2}{2} \] This polynomial gives a simple way to represent \(f(x)\) around \(x=0\), capturing not only the slope but also how the curve bends.

Linearization

Linearization is a simpler method compared to quadratic approximation, aimed at finding a linear polynomial that approximates a function near a given point. Often, linear approximations are used for a quick estimate or when the changes in \(x\) are relatively small around the point of interest.

• The linearization of a function at a point \(x=a\) involves the first-order Taylor polynomial:\[ T_1(x) = f(a) + f'(a)(x-a) \]
• This focuses solely on the value of the function at \(a\) and its first derivative, ignoring the curvature (second and higher derivatives).

For \(f(x) = e^{\sin x}\) at \(x=0\), the linearization is: \[ T_1(x) = 1 + x \] This result captures the immediate change of the function as \(x\) varies slightly from 0, offering a quick approximation of \(f(x)\) near this point.

Derivatives

In the context of Taylor series, derivatives play a critical role in approximating functions. They represent the rate at which a function changes and, consequently, help us determine the shape and direction of the curve.

• The first derivative, \(f'(x)\), is the slope or rate of change of the function at any given point. It is used in linearization.
• The second derivative, \(f''(x)\), measures how the rate of change itself is changing. It helps in determining the curvature of the function, crucial for quadratic approximation.

For the given function \(f(x) = e^{\sin x}\):- The first derivative \( f'(x) = e^{\sin x} \cos x \) tells us how \(f(x)\) directionally adjusts as \(x\) changes.- The second derivative \( f''(x) = e^{\sin x} (\cos^2 x - \sin x) \) provides insights into the function's concavity or convexity.
These derivatives at \(x=0\), specifically \(f'(0) = 1\) and \(f''(0) = 1\), are directly used in building both the linear and quadratic approximations, demonstrating their significance when simplifying complex functions.