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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 3.9.14c
Chapter 4, Problem 3.9.14c

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


c. f(x) = 1/√(1 + x)

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1
Identify the function f(x) = 1/√(1 + x) and recognize that it can be rewritten as (1 + x)^(-1/2).
Use the linear approximation formula (1 + x)ᵏ ≈ 1 + kx, where k is the exponent of the expression. Here, k = -1/2.
Substitute k = -1/2 into the linear approximation formula to get (1 + x)^(-1/2) ≈ 1 - (1/2)x.
This approximation is valid for values of x near zero, providing a simpler expression to estimate f(x) without complex calculations.
Thus, the linear approximation for f(x) = 1/√(1 + x) near x = 0 is approximately 1 - (1/2)x.

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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 L(x) = 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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Linearization

Derivative

The derivative of a function measures how the function's output value changes as its input changes. It is a fundamental concept in calculus, representing the slope of the tangent line to the function at any given point. For the function f(x) = 1/√(1 + x), finding the derivative is crucial for applying linear approximation, as it provides the rate of change needed for the approximation.
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Derivatives

Binomial Approximation

The binomial approximation (1 + x)ᵏ ≈ 1 + kx is a simplification used when x is near zero. It is derived from the binomial series expansion and is particularly useful for approximating expressions involving powers of binomials. In the context of the given problem, this approximation helps simplify the function f(x) = 1/√(1 + x) by treating it as a binomial expression with k = -1/2.
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Determining Error and Relative Error
Related Practice
Textbook Question

Finding Indefinite Integrals


In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.


∫(x + 1) dx

Textbook Question

In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.

y = 1 / (x² - 1)

Textbook Question

In Exercises 53 and 54, find both dy/dx (treating y as a differentiable function of x) and dx/dy (treating x as a differentiable function of y). How do dy/dx and dx/dy seem to be related?


53. xy³ + x²y = 6

Textbook Question

Finding Functions from Derivatives


Suppose that f(0) = 5 and that f'(x) = 2 for all x. Must f(x) = 2x + 5 for all x? Give reasons for your answer.