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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 3.9.55b
Chapter 3, Problem 3.9.55b

Quadratic approximations


b. Find the quadratic approximation to f(x) = 1/(1 − x) at x = 0.

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1
Identify the function f(x) = \( \frac{1}{1-x} \) and note that we want to find its quadratic approximation at x = 0.
Recall that the quadratic approximation of a function f(x) at a point a is given by the formula: \( f(a) + f'(a)(x-a) + \frac{f''(a)}{2}(x-a)^2 \).
Calculate the first derivative f'(x) of the function f(x) = \( \frac{1}{1-x} \). Use the derivative rule for \( \frac{1}{u} \), which is \( -\frac{u'}{u^2} \).
Calculate the second derivative f''(x) by differentiating f'(x). This involves applying the quotient rule or chain rule again.
Evaluate f(x), f'(x), and f''(x) at x = 0, and substitute these values into the quadratic approximation formula to find the quadratic approximation of f(x) at x = 0.

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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 a function f(x) centered at x = a, the series is f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + ..., which provides a polynomial approximation of the function near a.
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Intro to Transformations

Quadratic Approximation

Quadratic approximation is a specific case of the Taylor series where the function is approximated by a polynomial of degree two. It involves using the first three terms of the Taylor series: f(a), f'(a)(x-a), and f''(a)(x-a)^2/2. This approximation is useful for estimating the function's behavior near the point a.
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Derivatives Applied To Acceleration Example 2

Derivatives

Derivatives measure how a function changes as its input changes. The first derivative, f'(x), represents the rate of change or slope of the function, while the second derivative, f''(x), indicates the curvature or concavity. Calculating these derivatives at a specific point is essential for constructing the quadratic approximation.
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Related Practice
Textbook Question

Motion Along a Coordinate Line


Exercises 1–6 give the positions s = f(t) of a body moving on a coordinate line, with s in meters and t in seconds.


b. Find the body’s speed and acceleration at the endpoints of the interval.


s = 25/t² − 5/t, 1 ≤ t ≤ 5

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Textbook Question

A sliding ladder


A 13-ft ladder is leaning against a house when its base starts to slide away. By the time the base is 12 ft from the house, the base is moving at the rate of 5 ft/sec.


b. At what rate is the area of the triangle formed by the ladder, wall, and ground changing then?


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Textbook Question

Suppose that the functions f and g and their derivatives with respect to x have the following values at x = 0 and x = 1.


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Find the derivatives with respect to x of the following combinations at the given value of x.


b. f(x)g³(x), x = 0

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Textbook Question

Hauling in a dinghy A dinghy is pulled toward a dock by a rope from the bow through a ring on the dock 6 ft above the bow. The rope is hauled in at the rate of 2 ft/sec.


b. At what rate is the angle θ changing at this instant (see the figure)?

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Textbook Question

By computing the first few derivatives and looking for a pattern, find the following derivatives.


b. d¹¹⁰/dx¹¹⁰ (sin x − 3 cos x)

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Textbook Question

Temperature The given graph shows the outside temperature T in °F, between 6 a.m. and 6 p.m.

b. At what time does the temperature increase most rapidly? Decrease most rapidly? What is the rate for each of those times?

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