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Chapter 4: Problem 28

Calculate the linear approximation for \(f(x)\) : $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ \(f(x)=(1-x)^{-n}\) at \(a=0 .\) (Assume that \(n\) is a positive integer.)

Short Answer

Expert verified
The linear approximation is \( f(x) \approx 1 + nx \).

Step by step solution

01

Identify the Function Components

First, notice that the function we are working with is \( f(x) = (1-x)^{-n} \). We need to identify the function and its derivative at the point \( a = 0 \) for the linear approximation formula.
02

Evaluate the Function at a

Next, calculate \( f(a) \), where \( a = 0 \). This is simply \( f(0) = (1-0)^{-n} = 1 \).
03

Find the Derivative of the Function

The derivative of \( f(x) = (1-x)^{-n} \) is found using the chain rule. The derivative is \( f'(x) = n(1-x)^{-n-1} \).
04

Evaluate the Derivative at a

Calculate \( f'(a) \) where \( a = 0 \). Substituting in, we find \( f'(0) = n(1-0)^{-n-1} = n \).
05

Substitute into the Linear Approximation Formula

Replace \( f(a) \) and \( f'(a) \) into the linear approximation formula: \[ f(x) \approx f(0) + f'(0)(x-0) = 1 + n \cdot (x - 0) = 1 + nx \].
06

Write the Final Linear Approximation

The linear approximation of \( f(x) = (1-x)^{-n} \) at \( a = 0 \) is \( f(x) \approx 1 + nx \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Calculus
Calculus is a branch of mathematics that deals with change. It provides tools to find how things change, accumulate, grow, or shrink. Central to calculus are two fundamental ideas: differentiation and integration. These concepts allow us to:
  • Understand rates of change, such as velocity and acceleration.
  • Calculate areas under curves and volumes of objects.
In the case of linear approximation, calculus helps us find a linear function that closely resembles a more complex function near a specific point. This is particularly useful because linear functions are simple, making calculations easier. In the given exercise, the task is to use calculus to approximate a complex function, using its behavior understood through derivatives, to provide a linear representation around a particular point.
Delving into Derivatives
Derivatives represent how a function changes as the input changes, calculated as the "slope" of the function at any given point. Imagine a car driving on a curvy road. At any instant, the speedometer tells you how fast the car is going, which is akin to finding the derivative. For the function \( f(x) = (1-x)^{-n} \), the derivative helps us understand how rapidly it changes as \( x \) shifts.
  • Finding the derivative of a function involves applying rules and formulas of differentiation.
  • A critical utility of derivatives is in linear approximation 鈥� determining how a function's change can be approximated linearly at a point.
In the exercise, the derivative \( f'(x) = n(1-x)^{-n-1} \) was calculated to find the slope of the tangent at \( a = 0 \). This slope is then used to construct a linear approximation.
Exploring the Chain Rule in Differentiation
The chain rule is an essential tool in calculus for finding derivatives of composed functions. It states that to differentiate a composite function, you multiply the derivative of the outer function by the derivative of the inner function. For a function like \( (1-x)^{-n} \), which is a composition of a basic power function and a linear function, the chain rule becomes quite handy.
  • First, identify the outer function \( y = (1-x)^{-n} \) and the inner function \( u = 1-x \).
  • The derivative of the outer function with respect to the inner is \( -n(1-x)^{-n-1} \), while the derivative of the inner function \( 1-x \) is \(-1\).
By applying the chain rule, you can derive that \( f'(x) = n(1-x)^{-n-1} \). The chain rule's systematic approach simplifies finding the derivative of functions involving compositions, which is a cornerstone practice in calculus, particularly when linear approximations are involved.

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Most popular questions from this chapter

The growth rate of a fungus varies over the course of one day. You find that the size of the fungus is given as a function of time by: $$L(t)=3.6 t+1.2 \cos (2 \pi t / 24)$$ where \(t\) is the time in hours, and \(L(t)\) is the size in millimeters. (a) Calculate the growth rate \(d L / d t\) (b) What is the largest growth rate of the fungus? What is the smallest growth rate?

Calculate the linear approximation for \(f(x)\) : $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ \(f(x)=(1+x)^{-n}\) at \(a=0 .\) (Assume that \(n\) is a positive integer.)

Calculate the linear approximation for \(f(x)\) : $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ \(f(x)=\frac{1}{(1+x)^{2}}\) at \(a=0\)

The speed \(v\) of blood flowing along the central axis of an artery of radius \(R\) is given by Poiseuille's law, $$v(R)=c R^{2}$$ where \(c\) is a constant. If you can determine the radius of the artery to within an accuracy of \(5 \%\), how accurate is your calculation of the speed?

Calculate the linear approximation for \(f(x)\) : $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ \(f(x)=\ln (1+2 x)\) at \(a=0\)
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