/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 Find the linearization of \(f(x)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 6

Find the linearization of \(f(x)\) at \(x=a\). \(f(x)=\sqrt{x}, a=100\)

Short Answer

Expert verified
The linearization is \(L(x) = \frac{1}{20}x + 5\).

Step by step solution

01

Identify the function and point of approximation

Here, the function is given as \(f(x) = \sqrt{x}\) and we need to find its linearization at \(x = 100\).
02

Find the derivative of the function

Differentiate \(f(x) = \sqrt{x}\) with respect to \(x\). The derivative \(f'(x)\) is found using the power rule. \[f'(x) = \frac{d}{dx}(x^{1/2}) = \frac{1}{2} x^{-1/2} = \frac{1}{2 \sqrt{x}}\]
03

Evaluate the function and its derivative at \(x = 100\)

Substitute \(x = 100\) into \(f(x)\) and \(f'(x)\). \[f(100) = \sqrt{100} = 10\] \[f'(100) = \frac{1}{2 \sqrt{100}} = \frac{1}{2 \cdot 10} = \frac{1}{20}\]
04

Write the linearization formula

The linearization \(L(x)\) of \(f(x)\) at \(x = a\) is given by: \[L(x) = f(a) + f'(a)(x - a)\] Substitute \(a = 100\), \(f(100) = 10\), and \(f'(100) = \frac{1}{20}\) into the formula.
05

Substitute values and simplify

Substitute the values into the linearization formula: \[L(x) = 10 + \frac{1}{20}(x - 100)\] Simplify the expression: \[L(x) = 10 + \frac{1}{20}(x - 100) = 10 + \frac{1}{20}x - 5 = \frac{1}{20}x + 5\]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Differentiation
Differentiation is one of the fundamental concepts in calculus. It involves finding the derivative of a function, which represents the rate at which the function's value changes. To differentiate a function, we use various rules and techniques, such as the power rule, product rule, and chain rule.
Power Rule
The Power Rule is a straightforward method for finding the derivative of a function of the form \(f(x) = x^n\), where \(n\) is any real number. The rule states that the derivative of \(x^n\) is \(nx^{n-1}\). For example, to find the derivative of \(f(x) = x^{1/2}\), we apply the power rule:
Linear Approximation
Linear approximation, or linearization, is a technique used to approximate the value of a function near a given point. It's useful when you need a simple expression that is close to the actual function. The linearization of a function \(f(x)\) at \(x = a\) is given by the formula:

Function Derivatives
Understanding how to find and use function derivatives is crucial in calculus. The derivative of a function explains how the function's value changes as the input changes. For any given function \(f(x)\), we denote its derivative as \(f'(x)\) or \(\frac{df}{dx}\). Knowing a function's derivative enables us to use techniques like linear approximation to make complex functions more manageable.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The temperature \(T\) of a person during an illness is given by \(T(t)=-0.1 t^{2}+1.2 t+98.6, \quad 0 \leq t \leq 12\) where \(T\) is the temperature \(\left({ }^{\circ} \mathrm{F}\right)\) at time \(t\), in days. Find the maximum value of the temperature and when it occurs.

Graph the function $$ f(x)=\frac{x^{2}-3}{2 x-4}. $$ Using only the TRACE and ZOOM features: a) Find all the \(x\) -intercepts. b) Find the \(y\) -intercept. c) Find all the asymptotes.

Differentiate implicily to find \(d y / d x\). Then find the slope of the curve at the given point. $$ 4 x^{3}-y^{4}-3 y+5 x+1=0 ; \quad(1,-2) $$

Graph the function. $$ f(x)=\left|\frac{1}{x}-2\right| $$

Differentiate implicily to find \(d y / d x\). Then find the slope of the curve at the given point. $$ \sin y+x^{2}=\cos y ; \quad(1,2 \pi) $$
See all solutions

Recommended explanations on Biology Textbooks

Ecology

Read Explanation

Communicable Diseases

Read Explanation

Plant Biology

Read Explanation

Biology Experiments

Read Explanation

Biological Molecules

Read Explanation

Cell Communication

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.