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

Step by step solution

01

Determine coefficients of Q(x)

Given the quadratic approximation form \( Q(x) = b_0 + b_1(x-a) + b_2(x-a)^2 \) and the conditions:1. \( Q(a) = f(a) \): Substituting \( x = a \) gives \( b_0 = f(a) \).2. \( Q'(a) = f'(a) \): Differentiating \( Q(x) \) we get \( Q'(x) = b_1 + 2b_2(x-a) \), substituting \( x = a \) gives \( b_1 = f'(a) \).3. \( Q''(a) = f''(a) \): Differentiating again \( Q'(x) \) gives \( Q''(x) = 2b_2 \), substituting \( x = a \) gives \( 2b_2 = f''(a) \) so \( b_2 = \frac{f''(a)}{2} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Taylor Series

The Taylor Series is a powerful tool for approximating functions using polynomials. It's especially useful when you need to make complex functions simpler near a specific point. Essentially, a Taylor Series expands a function into an infinite sum of terms, which are derived from the function's derivatives at a single point. This approach relies heavily on the idea of function continuity and differentiability. By finding derivatives up to a certain order and evaluating them at a point, you create a polynomial that approximates the function around that area.

• The quadratic approximation is a special case of the Taylor Series, using only up to the second derivative of the function.
• It helps understand how the function behaves near a particular point by considering its slope and curvature.
• This approximation is expressed as: \[Q(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2}(x-a)^2\]

So, when you see a function like a Taylor Series expansion, think of it as peeling back layers of a function to know its behavior near a point.

Derivatives

Understanding derivatives is crucial for grasping quadratic approximations and Taylor Series. A derivative represents the rate at which a function is changing at any given point, effectively acting as a function's instant slope.For quadratic approximations, the first and second derivatives play a key role:

• The first derivative, \(f'(x)\), informs us about the linear behavior of the function, like its slope.
• The second derivative, \(f''(x)\), tells us about the function's curvature, i.e., how the slope itself is changing.

By ensuring the approximation meets the function at the same value, slope, and curvature at a point, we create a polynomial that closely mirrors the function's behavior near that point. A savvy grasp of derivatives empowers you to dissect and predict function behavior effectively.

Function Graphing

Graphing a function and its approximation can reveal a lot about their relationship. By visualizing these functions, you can see how well an approximation fits the original function at the point of interest. Let's consider some helpful tips when comparing graphs:

• Graph the original function first to understand its basic shape and behavior.
• Then, plot the quadratic approximation alongside the original. Notice how close the two are within a small neighborhood around the approximation point.
• As you "zoom in" on the point of approximation, observe that the two graphs should almost overlap if the approximation is precise.

This observation aligns with the underlying mathematics, where achieving identical values, slopes, and curvatures at the point of interest ensures a good fit for the quadratic approximation. Regular practice of graphing these relationships strengthens your understanding of function behaviors and approximation accuracy.