Quadratic approximations

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

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To find the quadratic approximation of g(x) = 1/x at x = 1, we start by using the Taylor series expansion formula. The quadratic approximation is given by: f(x) ≈ f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2, where a is the point of approximation, which is x = 1 in this case.

First, calculate g(1) = 1/1 = 1. This is the value of the function at x = 1.

Next, find the first derivative of g(x). The derivative of g(x) = 1/x is g'(x) = -1/x^2. Evaluate this at x = 1 to get g'(1) = -1.

Now, find the second derivative of g(x). The derivative of g'(x) = -1/x^2 is g''(x) = 2/x^3. Evaluate this at x = 1 to get g''(1) = 2.

Substitute these values into the quadratic approximation formula: g(x) ≈ 1 - (x-1) + (x-1)^2. This gives the quadratic approximation of g(x) at x = 1. To graph g(x) and its quadratic approximation, plot both functions on the same set of axes and observe how the quadratic approximation closely follows g(x) near x = 1.

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주요 개념

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Taylor Series

The Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. For quadratic approximation, we use the first three terms of the Taylor series, which include the function value, its first derivative, and its second derivative at the point of approximation.

Derivatives

Derivatives measure how a function changes as its input changes. For quadratic approximation, the first and second derivatives of the function at the point of interest are crucial. They provide the slope and the curvature of the function, respectively, which are used to construct the quadratic approximation.