Problem 62 - Thomas Calculus

Question

Quadratic approximations Let \(Q(x)=b_{0}+b_{1}(x-a)+b_{2}(x-a)^{2}\) be a quadratic approximation to \(f(x)\) at \(x=a\) with the properties: i. \(Q(a)=f(a)\) ii. \(Q^{\prime}(a)=f^{\prime}(a)\) iii. \(Q^{\prime \prime}(a)=f^{\prime \prime}(a)\) Determine the coefficients \(b_{0}, b_{1},\) and \(b_{2}\) b. Find the quadratic approximation to \(f(x)=1 /(1-x)\) at \(x=0\) . c. Graph \(f(x)=1 /(1-x)\) and its quadratic approximation at \(x=0 .\) Then zoom in on the two graphs at the point \((0,1)\) . Comment on what you see. d. Find the quadratic approximation to \(g(x)=1 / x\) at \(x=1\) . Graph \(g\) and its quadratic approximation together. Comment on what you see. e. Find the quadratic approximation to \(h(x)=\sqrt{1+x}\) at \(x=0 .\) Graph \(h\) and its quadratic approximation together. Comment on what you see. f. What are the linearizations of \(f, g,\) and \(h\) at the respective points in parts (b), (d), and (e)?

Step-by-Step Solution

Answer

The quadratic approximations are: - For \( f(x) \): 1 + x + x^2. - For \( g(x) \): 2x - x^2. - For \( h(x) \): 1 + \frac{1}{2}x - \frac{1}{8}x^2.

Step 1: Understanding the Problem

We need to find a quadratic approximation of a function \( f(x) \) using the form \( Q(x)=b_{0}+b_{1}(x-a)+b_{2}(x-a)^{2} \). Given the conditions: \( Q(a)=f(a) \), \( Q'(a)=f'(a) \), and \( Q''(a)=f''(a) \), we'll determine the coefficients \( b_0, b_1, \) and \( b_2 \) and apply this to various functions.

Step 2: Determine Coefficients

1. **Substitute into first condition:** Set \( Q(a)=f(a) \), giving \( b_0 = f(a) \).2. **Find the first derivative:** \( Q'(x) = b_1 + 2b_2(x-a) \), and set \( Q'(a)=f'(a) \), giving \( b_1 = f'(a) \).3. **Find the second derivative:** \( Q''(x) = 2b_2 \), and set \( Q''(a)=f''(a) \), giving \( b_2 = \frac{f''(a)}{2} \).

Step 3: Calculate Approximation for f(x)

For \( f(x) = \frac{1}{1-x} \) at \( x = 0 \):- \( f(0) = 1 \)- \( f'(x) = \frac{1}{(1-x)^2} \), so \( f'(0) = 1 \)- \( f''(x) = \frac{2}{(1-x)^3} \), so \( f''(0) = 2 \).Thus, the quadratic approximation is \( Q(x) = 1 + x + x^2 \).

Step 4: Graph and Analyze f(x)

Graph \( f(x) = \frac{1}{1-x} \) and its quadratic approximation \( Q(x) = 1 + x + x^2 \). When zoomed in at \( (0,1) \), the graphs closely overlap, showing that the approximation is accurate near \( x = 0 \).

Step 5: Calculate Approximation for g(x)

For \( g(x) = \frac{1}{x} \) at \( x = 1 \):- \( g(1) = 1 \)- \( g'(x) = -\frac{1}{x^2} \), so \( g'(1) = -1 \)- \( g''(x) = \frac{2}{x^3} \), so \( g''(1) = 2 \).Thus, \( Q(x) = 1 - (x-1) + (x-1)^2 = 2x - x^2 \).

Step 6: Graph and Analyze g(x)

Graph \( g(x) = \frac{1}{x} \) and its quadratic approximation. The graphs coincide closely near \( x = 1 \), indicating a good approximation in this region.

Step 7: Calculate Approximation for h(x)

For \( h(x) = \sqrt{1+x} \) at \( x = 0 \):- \( h(0) = 1 \)- \( h'(x) = \frac{1}{2\sqrt{1+x}} \), so \( h'(0) = \frac{1}{2} \)- \( h''(x) = -\frac{1}{4(1+x)^{3/2}} \), so \( h''(0) = -\frac{1}{4} \).Thus, \( Q(x) = 1 + \frac{1}{2}x - \frac{1}{8}x^2 \).

Step 8: Graph and Analyze h(x)

Graph \( h(x)=\sqrt{1+x} \) and its quadratic approximation. They match closely near \( x = 0 \), indicating the approximation's reliability.

Step 9: Find Linearizations

Linearization involves the first two terms from the quadratic approximation:- For \( f(x) \), linearization is \( 1 + x \).- For \( g(x) \), linearization is \( 2x - x^2 \) simplifies to \( 1 + (x-1) = x \).- For \( h(x) \), linearization is \( 1 + \frac{1}{2}x \).

Key Concepts

Taylor series

The Taylor series is a powerful tool in mathematics used to approximate functions by polynomials. It's named after the mathematician Brook Taylor. The basic idea is to express a function as an infinite sum of its derivatives at a single point. This allows for a polynomial approximation that converges to the function in a neighborhood around that point. To construct a Taylor series, you need:

• The function and its derivatives at the point of expansion
• The formula: \[ T(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots \]

In practice, when working with approximations, only the first few terms are used, as higher order terms become significantly smaller.A quadratic approximation is a truncated Taylor series that includes terms up to the second derivative. This approach can effectively approximate functions within a small range of the expansion point, offering insights into the behavior of functions using relatively simple mathematics.

First derivative

The first derivative of a function, often denoted as \( f'(x) \), indicates how the function's output changes concerning a small change in input. It can tell us about the slope of the tangent line to the function at any given point. In the context of quadratic approximations and Taylor series, the first derivative provides the linear term of the approximation.For a function \( f(x) \), its first derivative at a point \( x = a \) gives:

• The slope of the function at that point
• Information about whether the function is increasing or decreasing

In a quadratic approximation, the first derivative ensures that the slope of the approximating polynomial matches the original function at the point \( x = a \). This maintains the linear behavior of the function around that point, crucial for creating an accurate approximation over a limited range.

Second derivative

The second derivative of a function, denoted as \( f''(x) \), provides information on how the rate of change of the function is changing—it tells us about the "curvature" or concavity of the function. In Taylor series and quadratic approximations, the second derivative is instrumental in shaping the quadratic term.The significance of the second derivative includes:

• Indicating concavity: whether the graph of the function is curving upwards or downwards
• Identifying points of inflection where the concavity changes

In the quadratic approximation, the second derivative's value at a particular point \( x = a \) is divided by 2, \[ \frac{f''(a)}{2} \], to form the coefficient of the quadratic term. This term adds a parabolic component to the approximation, allowing the polynomial to account for the function's curvature at that point. Thus, incorporating the second derivative helps in capturing the behavior of the function to a greater extent than the linear term alone.

Function approximation

Function approximation is a broad concept where complex functions are represented by simpler ones, such as polynomials. This is extremely useful in calculations and analysis, especially when dealing with functions that are hard to evaluate directly. Quadratic approximation is a specific type of function approximation using a second-degree polynomial. It is particularly valuable because:

• It provides a balance between simplicity and accuracy
• It can approximate a wide range of functions locally near the chosen expansion point
• It's computationally less demanding than higher-order polynomial approximations

Function approximation is fundamental in various fields, including numerical analysis, curve fitting, and real-world applications like signal processing and data modeling. By approximating functions with polynomials, complex mathematical and computational challenges become more manageable.