Problem 45 - 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

Both linearization and quadratic approximation of \(\sin x\) at \(x=0\) are \(x\).

Step 1: Recall the Formula for Taylor Polynomials

The Taylor polynomial of order \(n\) for a function \(f\) at the point \(x=a\) is given by:\[ P_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \ldots + \frac{f^{(n)}(a)}{n!}(x-a)^n \] Since we are interested in the linearization (order 1) and the quadratic approximation (order 2), we will use this formula for \(n=1\) and \(n=2\).

Step 2: Calculate the Linearization (Order 1)

The linearization involves using the Taylor polynomial of order 1. For \(f(x) = \sin x\), we first find:1. \(f(0) = \sin(0) = 0\)2. \(f'(x) = \cos x\), thus \(f'(0) = \cos(0) = 1\)Plugging these into the formula gives:\[ P_1(x) = f(0) + f'(0) \, (x - 0) = 0 + 1 \cdot x = x \]So, the linearization at \(x = 0\) is \(x\).

Step 3: Calculate the Quadratic Approximation (Order 2)

Now, for the quadratic approximation, calculate up to the second derivative.1. We've already calculated \(f'(0) = 1\).2. Calculate \(f''(x) = -\sin x\), hence \(f''(0) = -\sin(0) = 0\).The quadratic Taylor polynomial is:\[ P_2(x) = f(0) + f'(0)(x-0) + \frac{f''(0)}{2!}(x-0)^2 \]Substituting the values, we get:\[ P_2(x) = 0 + 1 \cdot x + \frac{0}{2}x^2 = x \]Thus, the quadratic approximation at \(x=0\) is also \(x\).

Step 4: Summarize the Results

Both the linearization and the quadratic approximation of \(f(x)=\sin x\) at \(x=0\) yield the function \(x\). This indicates that up to the second derivative, the behavior of \(\sin x\) at 0 is well-represented by \(y = x\).

Key Concepts

Quadratic Approximation

Quadratic approximation is a method used to estimate the value of a function near a specific point by using a second-order polynomial. This polynomial is called the quadratic Taylor polynomial. We can consider it as a tool to understand how the function behaves very close to the chosen point. To perform quadratic approximation, we use the Taylor polynomial of order 2. This is given by the formula:\[ P_2(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2}(x-a)^2 \]When applying this to a twice-differentiable function, we need to find the value of the function, its first derivative, and its second derivative, all evaluated at the point of interest. This approximation includes the curvature of the function, which makes it more accurate than a linearization.For example, in the original exercise, the quadratic approximation of the function \(f(x) = \sin x\) at \(x = 0\) resulted in \(x\). This is because the second derivative \(f''(0)\) equals zero, which simplifies our expression. Whether for study or practical purposes, understanding quadratic approximations helps in grasping the more fine-tuned behavior of functions.

Linearization

Linearization is the process of approximating a function near a specific point using a linear function. It is essentially a first-order Taylor polynomial, which means it considers the value and the first derivative of the function at a particular point.To linearize a function, the formula is:\[ P_1(x) = f(a) + f'(a)(x-a) \]This generates a straight line that "touches" the function at the point \(x = a\), giving us an immediate way to see how the function changes around that point. The main advantage of linearization is its simplicity, making calculations straightforward.In the case of \(f(x) = \sin x\) at \(x = 0\), after calculation, we found that the linearization is simply \(x\), as \(f(0) = 0\) and \(f'(0) = 1\). Hence, linearization gives us a simple yet effective method to approximate \(\sin x\) close to zero.

Twice-differentiable Function

A twice-differentiable function, as the name suggests, is a function for which we can take the derivative twice. This property is crucial for creating quadratic approximations. It allows us to accurately describe the function not only in terms of its slope at a point but also in terms of its curvature.Key points about a twice-differentiable function:- The first derivative \(f'(x)\) tells us how the function's value changes with small variations in \(x\).- The second derivative \(f''(x)\) provides insights into the function's accelerating or decelerating change, essentially describing the curvature or "bend" of the function graph.When constructing Taylor polynomials, particularly for quadratic approximations, having a twice-differentiable function ensures that we can include those terms up to the second order, which allows for a more precise approximation of the function near the point of interest.In our initial example using \(f(x) = \sin x\), the function being twice-differentiable allowed us to effectively calculate both linear and quadratic approximations at \(x = 0\).