Skip to main content
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
Tutti i libri di testoBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblema 3
Capitolo 4, Problema 3

Sketch the graph of a function continuous on the given interval that satisfies the following conditions.
ƒ is continuous on the interval [-4, 4] ; f'(x) = 0 for x = -2, 0, and 3; ƒ has an absolute minimum at x = 3; ƒ has a local minimum at x = -2 ; ƒ has a local maximum at x = 0; ƒ has an absolute maximum at x = -4.

Guida verificata passo dopo passo
1
Identify the critical points where the derivative f'(x) = 0, which are x = -2, 0, and 3. These points are where the function could have local maxima, minima, or points of inflection.
Determine the nature of each critical point using the given conditions: x = -2 is a local minimum, x = 0 is a local maximum, and x = 3 is an absolute minimum.
Consider the endpoints of the interval [-4, 4]. The function has an absolute maximum at x = -4, which means the function value at this point is greater than or equal to the function values at all other points in the interval.
Sketch the graph by plotting the critical points and endpoints, ensuring that the function is continuous across the interval. Start at the absolute maximum at x = -4, decrease to the local minimum at x = -2, increase to the local maximum at x = 0, decrease again to the absolute minimum at x = 3, and then continue to x = 4.
Ensure the graph reflects the continuity of the function across the interval [-4, 4] and that the behavior at each critical point matches the specified conditions (local and absolute extrema).

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:
7m

Concetti chiave

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

Continuity of Functions

A function is continuous on an interval if there are no breaks, jumps, or holes in its graph within that interval. This means that for every point in the interval, the function approaches a limit that equals the function's value at that point. Understanding continuity is crucial for sketching graphs, as it ensures that the function behaves predictably across the specified range.
Video consigliato:
05:34
Intro to Continuity

Critical Points and Extrema

Critical points occur where the derivative of a function is zero or undefined, indicating potential local maxima or minima. In this question, the critical points at x = -2, 0, and 3 are essential for determining the function's behavior. An absolute minimum or maximum refers to the lowest or highest value of the function over the entire interval, while local extrema are the highest or lowest points in a small neighborhood around the critical points.
Video consigliato:
04:50
Critical Points

Graphing Techniques

Graphing techniques involve plotting points, identifying key features such as intercepts, and understanding the shape of the function based on its derivatives. For this question, sketching the graph requires incorporating the identified critical points and the specified conditions of local and absolute extrema, ensuring that the graph reflects the function's continuity and the behavior at those points.
Video consigliato:
Percorso guidato
06:15
Graphing The Derivative
Pratica correlata
Domanda del libro di testo

Interpreting the derivative Find the derivative of each function at the given point and interpret the physical meaning of this quantity. Include units in your answer.

When a faucet is turned on to fill a bathtub, the volume of water in gallons in the tub after t minutes is V(t)=3t. Find V′(12).

266
views
Domanda del libro di testo

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_x→ π/2⁻ (tanx ) / (3 / (2x - π))

219
views
Domanda del libro di testo

Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.


f(x) = (x+4)/(4-x)

261
views
Domanda del libro di testo

13-26 Implicit differentiation Carry out the following steps.

a. Use implicit differentiation to find dy/dx.

x⁴+y⁴ = 2;(1,−1)

219
views