Ch. 4 - Applications of the Derivative

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

Ch. 4 - Applications of the Derivative

Problema 4.R.8

Capitolo 4, Problema 4.R.8

Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).
g(x) = x⁴ - 50x² on [-1, 5]

Guida verificata passo dopo passo

To find the critical points of the function \( g(x) = x^4 - 50x^2 \), first compute the derivative \( g'(x) \).

The derivative is \( g'(x) = 4x^3 - 100x \). Set \( g'(x) = 0 \) to find the critical points.

Solve the equation \( 4x^3 - 100x = 0 \). Factor out the common term: \( 4x(x^2 - 25) = 0 \).

This gives \( x = 0 \) or \( x^2 - 25 = 0 \). Solving \( x^2 - 25 = 0 \) gives \( x = 5 \) or \( x = -5 \).

Evaluate \( g(x) \) at the critical points \( x = 0, 5, -5 \) and at the endpoints of the interval \( x = -1 \) and \( x = 5 \) to determine the absolute maximum and minimum values.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Durata del video:

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Critical Points

Critical points of a function occur where its derivative is zero or undefined. These points are essential for identifying local maxima and minima, as they represent potential locations where the function's behavior changes. To find critical points, one must first compute the derivative of the function and solve for the values of x that satisfy the condition.

Video consigliato:

04:50

Critical Points

Absolute Maximum and Minimum

The absolute maximum and minimum values of a function on a closed interval are the highest and lowest values that the function attains within that interval. To determine these values, one must evaluate the function at its critical points and at the endpoints of the interval. The largest and smallest of these values will be the absolute maximum and minimum, respectively.

Video consigliato:

03:22

Finding Extrema Graphically Example 4

Endpoints of an Interval

Endpoints of an interval are the boundary values that define the limits of the interval in which a function is analyzed. In the context of finding absolute extrema, it is crucial to evaluate the function at these endpoints, as they can potentially yield the highest or lowest values alongside the critical points. For the function g(x) = x⁴ - 50x² on [-1, 5], the endpoints are x = -1 and x = 5.

Video consigliato:

Percorso guidato

05:59

Estimating the Area Under a Curve with Right Endpoints & Midpoint

Pratica correlata

Domanda del libro di testo

Use the graphs of ƒ' and ƒ" to complete the following steps. <IMAGE>

a. Find the critical points of f and determine where f is increasing and where it is decreasing.

235

views

Domanda del libro di testo

45–46. Linear approximation

a. Find the linear approximation to f at the given point a.

b. Use your answer from part (a) to estimate the given function value. Does your approximation underestimate or overestimate the exact function value?

ƒ(x) = x²⸍³ ; a =27; ƒ(29)

303

views

Domanda del libro di testo

24–34. Curve sketching Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.

ƒ(x) = 4cos (π (x-1)) on [0, 2]

253

views

Domanda del libro di testo

90–103. Indefinite integrals Determine the following indefinite integrals.

∫ (⁴√x³ + √x⁵) dx

views

Domanda del libro di testo

104–107. Functions from derivatives Find the function f with the following properties.

ƒ'(t) = sin t + 2t; ƒ(0) = 5

views

Domanda del libro di testo