/*! 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 1 \(1-4\) Find the linearization \... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 1

\(1-4\) Find the linearization \(L(x)\) of the function at a. \(f(x)=x^{4}+3 x^{2}, \quad a=-1\)

Short Answer

Expert verified
The linearization is \(L(x) = 4 - 10(x + 1)\).

Step by step solution

01

Understand Linearization Formula

The linearization of a function \(f(x)\) at a point \(a\) is given by the formula \(L(x) = f(a) + f'(a) (x - a)\). This approximation uses the function's value and its derivative at \(a\) to create a linear approximation.
02

Compute f(a)

Calculate the value of the function at \(a = -1\). The function is \(f(x) = x^4 + 3x^2\). Substitute \(x = -1\) into \(f(x)\):\[ f(-1) = (-1)^4 + 3(-1)^2 = 1 + 3 = 4 \].
03

Determine f'(x)

Find the derivative of \(f(x)\). Using the power rule, the derivative is:\[ f'(x) = 4x^3 + 6x \].
04

Compute f'(a)

Evaluate the derivative at \(a = -1\). Substitute \(x = -1\) into \(f'(x)\):\[ f'(-1) = 4(-1)^3 + 6(-1) = -4 - 6 = -10 \].
05

Formulate the Linearization Equation

Use the values calculated in the previous steps to form \(L(x)\):\[ L(x) = f(a) + f'(a)(x - a) = 4 - 10(x + 1) \].

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.

Derivative
The concept of a derivative is a fundamental part of calculus. It measures how a function changes as its input changes, basically telling us the rate at which the function's value is changing at any point. Let's break it down a bit further:
  • A derivative expressed as \( f'(x) \) for a function \( f(x) \) gives us the slope of the tangent line to the function at any point \( x \).
  • It helps in determining things like velocity and acceleration in physics, or the rate of change in any mathematical model.
In our example, the derivative \(f'(x)\) was computed to be \(4x^3 + 6x\) using the power rule. This derivative tells us how the function \(f(x) = x^4 + 3x^2\) changes when \(x\) changes. At \(a = -1\), the derivative value is \(-10\), indicating a negative slope at that point.
Linear approximation
Linear approximation, also known as linearization, is a way to approximate a function using a linear function. The idea is to find a line that closely follows the behavior of the function near a certain point. Here's what you need to know:
  • The formula for linearization is \(L(x) = f(a) + f'(a)(x - a)\), where \(a\) is the point at which you're approximating.
  • This formula uses the function's value and derivative at \(a\) to craft a linear representation that approximates \(f(x)\) nearby \(a\).
In the exercise provided, we found \(f(a) = 4\) and \(f'(a) = -10\) at \(a = -1\). Thus, the linear approximation or linearization of our function at \(a\) is \(L(x) = 4 - 10(x + 1)\). This provides an easier-to-handle linear expression instead of the more complex original fourth-degree polynomial.
Power rule
The power rule is a handy tool in calculus for finding derivatives. It gives a quick way to differentiate functions of the form \(x^n\). Here's a closer look:
  • The power rule states that the derivative of \(x^n\) is \(nx^{n-1}\). For example, the derivative of \(x^4\) is \(4x^3\).
  • It simplifies the process of taking derivatives, especially when you have polynomial functions with several terms.
In the original exercise, the power rule was used to determine the derivative \(f'(x) = 4x^3 + 6x\). The first term, \(x^4\), was differentiated to \(4x^3\) and the second term, \(3x^2\), to \(6x\). This demonstrates how efficient the power rule can be for quickly finding derivatives without going into complex calculations.

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

\(37-48\) Use logarithmic differentiation to find the derivative of the function. $$y=(\ln x)^{\cos x}$$

Find a formula for \(f^{(n)}(x)\) if \(f(x)=\ln (x-1)\)

Find \(\frac{d^{9}}{d x^{9}}\left(x^{8} \ln x\right)\)

Find the derivative. Simplify where possible. $$y=x \operatorname{coth}\left(1+x^{2}\right)$$

A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 \(\mathrm{m}\) higher than the bow of the boat. If the rope is pulled in at a rate of \(1 \mathrm{m} / \mathrm{s},\) how fast is the boat approaching the dock when it is 8 \(\mathrm{m}\) from tho dock?
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Probability and Statistics

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.