Ch. 3 - Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 3 - Derivatives

Problem 55d

Chapter 3, Problem 55d

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

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

Quadratic Approximation

Quadratic approximation is a method of estimating a function using a quadratic polynomial. It is derived from the Taylor series and provides a simple way to approximate a function near a specific point. This approximation is particularly useful for functions that are difficult to compute directly, offering a balance between accuracy and simplicity.

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Related Practice

Textbook Question

Quadratic approximations

[Technology Exercise] e. Find the quadratic approximation to h(x) = √(1 + x) at x = 0. Graph h and its quadratic approximation together. Comment on what you see.

Textbook Question

A weight is attached to a spring and reaches its equilibrium position (x = 0). It is then set in motion resulting in a displacement of x = 10 cos t, where x is measured in centimeters and t is measured in seconds. See the accompanying figure.

Find the spring’s displacement when t = 0, t = π/3, and t = 3π/4.

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Textbook Question

Quadratic approximations

[Technology Exercise] 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.

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Textbook Question

Assume that a particle’s position on the x-axis is given by

x = 3 cos t + 4 sin t,

where x is measured in feet and t is measured in seconds.

a. Find the particle’s position when t = 0, t = π/2, and t = π.

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Textbook Question

Assume that a particle’s position on the x-axis is given by

x = 3 cos t + 4 sin t,

where x is measured in feet and t is measured in seconds.

b. Find the particle’s velocity when t = 0, t = π/2, and t = π.