Problem 2 - Analysis 1: Ein Lehr- und Arbeitsbuch für Studienanfänger

Question

Zeigen sie, daß die Funktion \(x \mapsto f(x) ;=x^{2}\), \(x \in \mathcal{R}\), in jedew Punkt \(x_{0} \in \mathbb{R}\) linear approximierbar ist. Geben sie jeweils das approximierende Polynom an. Skizzieren sie den Fall \(x_{0}=-2\). Sie werden auf seite 183 sehen, daB nicht jede reelle Funktion in allen Punkten ihres Definitionsberelches linear approximierbar ist.

Step-by-Step Solution

Answer

The function \(f(x) = x^2\) is linearly approximable at any point \(x_0\), with the approximating polynomial at \(x_0 = -2\) being \(L(x) = -4x - 4\).

Step 1: Understanding Linear Approximation

Linear approximation at a point \(x_0\) involves finding a linear function (a line) that closely approximates the function near \(x_0\). For any function \( f(x) \), the linear approximation at a point \( x_0 \) is given by its tangent line at that point. The formula for the tangent line is \( L(x) = f(x_0) + f'(x_0)(x-x_0) \).

Step 2: Finding the Derivative

For the function \(f(x) = x^2\), the first step is to find its derivative, \(f'(x)\). Calculating the derivative, we apply the power rule: \( f'(x) = 2x \).

Step 3: Applying the Derivative at a Point

We are given a point \(x_0 = -2\). Substitute this into the derivative to find the slope of the tangent line at \(x_0\): \( f'(-2) = 2(-2) = -4 \).

Step 4: Writing the Linear Approximation

Substitute \(x_0 = -2\) and \( f'(-2) = -4 \) into the tangent line formula. Also, calculate \(f(-2) = (-2)^2 = 4\). The linear approximation is: \(L(x) = 4 + (-4)(x + 2)\). Simplify this to: \(L(x) = -4x - 4 \).

Step 5: Graphing the Function and its Approximation

To sketch the graph for \(x_0 = -2\): draw the parabola \(f(x) = x^2\) and the tangent line \(L(x) = -4x - 4\) which passes through the point \((-2, 4)\). The line should closely match the curve of \(f(x)\) around the point \(x_0 = -2\).

Step 6: Conclusion: Confirming Linear Approximability

The function \(f(x) = x^2\) is linearly approximable at every point \(x_0\), as the tangent line can be determined for any real number \(x_0\). The situation described on page 183 likely points out that not every function behaves this well at all points.

Key Concepts

Tangent Line

The concept of a tangent line is central to understanding linear approximation. A tangent line touches a curve at a specific point, just like it lightly rests on the surface of the curve without cutting through it. This line is unique in that it has the same direction as the curve at that point. In terms of equations, for a function at a specific point \( x_0 \), the equation of the tangent line can be represented as \( L(x) = f(x_0) + f'(x_0)(x - x_0) \). This expression is formed using both the value of the function \( f(x_0) \) at that point and its derivative \( f'(x_0) \).

• The derivative gives us the slope of this tangent line, telling us how steep it is.
• The \((x - x_0)\) part indicates how the line varies as you move away from \( x_0 \).
• The equation helps in approximating the function's behavior near \( x_0 \).

Understanding the tangent line helps make complex curves manageable by approximating them with something as simple as a line, at least nearby a point.

Derivative

Derivatives form the backbone of calculus and unlock the ability to analyze a function's rate of change at any point. In simple terms, the derivative of a function provides the slope of the tangent line to the function at any given point. Think of it as a tool that captures how quickly something changes.
For a function such as \( f(x) = x^2 \), the derivative is found using calculus rules. In this case, the power rule is employed to effortlessly compute it. The derivative, \( f'(x) = 2x \), reveals the slope at any point \( x \) on the curve.

• If \( x = i \), then the slope of the tangent at \( x \) is \( 2i \).
• The technique simplifies predicting how the function behaves near any point \( x \).
• Knowing \( f'(x) \) allows us to approximate the function as a straight line near \( x \).

Understanding derivatives enables the transformation of seemingly intricate problems into more straightforward approximations, key for further calculus applications.

Power Rule

An essential derivative rule, particularly when dealing with polynomials, is the power rule. The power rule simplifies the process of finding derivatives and is widely used thanks to its straightforward nature. The rule states that the derivative of a term \( ax^n \) is \( anx^{n-1} \).

• Apply it directly: With \( f(x) = x^n \), the derivative becomes \( nx^{n-1} \).
• This rule is especially helpful when dealing with polynomial functions which often involve terms like \( x^2, x^3, \) etc.
• For our function \( f(x) = x^2 \), applying the power rule yields \( f'(x) = 2x \).

The power rule saves time and effort in manual computation and helps quickly pinpoint the slope for polynomial functions, offering a quicker pathway to finding tangent lines for linear approximation.

Polynomial Approximations

Polynomial approximations aim to mimic complex functions using simpler polynomial expressions. They are integral to numerical methods used in calculus to estimate function values.
Linear approximation is a form of polynomial approximation which uses a polynomial of degree one, essentially a line. At a point \( x_0 \), the linear approximation gives the equation of the tangent line as its simplest degree one polynomial: \( L(x) = f(x_0) + f'(x_0)(x - x_0) \).

• This approximation aims to maintain proximity to the supposed function as close to \( x_0 \) as possible.
• When sketched, this tangent line aligns well with the curve of \( f(x) \) close to \( x_0 \).
• Other polynomial degrees can also be used for higher accuracy approximations.

Utilizing polynomial approximations like these facilitates understanding and computation, turning complex functions into manageable linear forms for many applications in calculus.