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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 3.9.14a
Chapter 3, Problem 3.9.14a

Use the linear approximation (1 + x)ᵏ ≈ 1 + kx to find an approximation for the function f(x) for values of x near zero.


a. f(x) = (1 − x)⁶

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1
Identify the function f(x) = (1 - x)⁶ and recognize that it is in the form of (1 + x)ᵏ with k = 6 and x replaced by -x.
Apply the linear approximation formula (1 + x)ᵏ ≈ 1 + kx to the function. Here, substitute x with -x and k with 6.
The linear approximation becomes: (1 - x)⁶ ≈ 1 + 6(-x).
Simplify the expression: 1 + 6(-x) becomes 1 - 6x.
Thus, the linear approximation for f(x) = (1 - x)⁶ near x = 0 is f(x) ≈ 1 - 6x.

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Key Concepts

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

Linear Approximation

Linear approximation is a method used to estimate the value of a function near a given point using the tangent line at that point. For a function f(x), the linear approximation at x = a is given by f(a) + f'(a)(x - a). This technique is particularly useful for simplifying complex functions near a specific point, often x = 0.
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Binomial Expansion

The binomial expansion is a way of expressing powers of binomials, such as (1 + x)ᵏ, as a series. For small values of x, the expansion can be approximated by the first few terms, often just 1 + kx for linear approximation. This simplification is useful for estimating the behavior of functions like (1 - x)⁶ near x = 0.
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Function Behavior Near Zero

Understanding the behavior of functions near zero involves analyzing how the function changes as x approaches zero. This often involves using approximations or expansions to simplify the function, making it easier to evaluate or estimate. For f(x) = (1 - x)⁶, using linear approximation helps predict its value when x is close to zero.
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Related Practice
Textbook Question

Differentiability and Continuity on an Interval


Each figure in Exercises 45–50 shows the graph of a function over a closed interval D. At what domain points does the function appear to be


a. differentiable?


Give reasons for your answers.

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.


a. Find the body’s displacement and average velocity for the given time interval.


s = (t⁴/4) − t³ + t², 0 ≤ t ≤ 3

Textbook Question

Tolerance


a. About how accurately must the interior diameter of a 10-m-high cylindrical storage tank be measured to calculate the tank’s volume to within 1% of its true value?

Textbook Question

In Exercises 47 and 48, find an equation for


(a) the tangent line to the curve at P


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Textbook Question

Quadratic approximations


a. Let Q(x) = b₀ + b₁(x − a) + b₂(x − a)² be a quadratic approximation to f(x) at x = a with these properties:


i. Q(a) = f(a)

ii. Q′(a) = f′(a)

iii. Q″(a) = f″(a).


Determine the coefficients b₀, b₁, and b₂.

Textbook Question

Diagonals If x, y, and z are lengths of the edges of a rectangular box, then the common length of the box’s diagonals is s = √(x² + y² + z²).

a. Assuming that x, y, and z are differentiable functions of t, how is ds/dt related to dx/dt, dy/dt, and dz/dt?

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