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
Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 4, Problema 4.9.111a
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. F(x) = x³ - 4x + 100 and G(x) = x³ - 4x - 100 are antiderivatives of the same function.
Guida verificata passo dopo passo1
Recall that two functions are antiderivatives of the same function if their derivatives are equal, which means they differ by only a constant.
Find the derivative of \( F(x) = x^3 - 4x + 100 \). Using the power rule and constant rule, \( F'(x) = 3x^2 - 4 \).
Find the derivative of \( G(x) = x^3 - 4x - 100 \). Similarly, \( G'(x) = 3x^2 - 4 \).
Since \( F'(x) = G'(x) = 3x^2 - 4 \), both \( F(x) \) and \( G(x) \) are antiderivatives of the same function \( f(x) = 3x^2 - 4 \).
The difference between \( F(x) \) and \( G(x) \) is a constant (\( 200 \)), which confirms they are antiderivatives of the same function.
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:
2mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Antiderivative (Indefinite Integral)
An antiderivative of a function f(x) is another function F(x) whose derivative is f(x). It represents the reverse process of differentiation and is expressed as an indefinite integral, including an arbitrary constant C since differentiation eliminates constants.
Video consigliato:
Percorso guidato
05:04
Introduction to Indefinite Integrals
Constant of Integration
When finding antiderivatives, any constant term disappears upon differentiation. Therefore, all antiderivatives of a function differ by a constant, called the constant of integration, which accounts for all possible vertical shifts of the antiderivative graph.
Video consigliato:
Percorso guidato
06:18Integration by Parts for Definite Integrals
Verifying Antiderivatives by Differentiation
To determine if two functions are antiderivatives of the same function, differentiate both and compare their derivatives. If the derivatives are identical, the functions differ only by a constant and are antiderivatives of the same function.
Video consigliato:
Percorso guidato
05:50Antiderivatives
Pratica correlata
Domanda del libro di testo
Optimal popcorn box A small popcorn box is created from a 12" x 12" sheet of paperboard by first cutting out four shaded rectangles, each of length x and width x/2 (see figure). The remaining paperboard is folded along the solid lines to form a box. What dimensions of the box maximize the volume of the box? <IMAGE>
355
views
Domanda del libro di testo
Rectangles beneath a line
a. A rectangle is constructed with one side on the positive x-axis, one side on the positive y-axis, and the vertex opposite the origin on the line y = 10 - 2x. What dimensions maximize the area of the rectangle? What is the maximum area?
212
views
Domanda del libro di testo
Population models The population of a species is given by the function P(t) = Kt²/(t² + b) , where t ≥ 0 is measured in years and K and b are positive real numbers.
a. With K = 300 and b = 30, what is lim_t→∞ P(t), the carrying capacity of the population?
333
views
Domanda del libro di testo
Sketch a graph of a function f with the following properties.
f' < 0 and f" < 0, for x < -1
266
views
Domanda del libro di testo
Suppose the objective function P= xy is subject to the constraint 10x + y = 100, where x and y are real numbers.
a. Eliminate the variable y from the objective function so that P is expressed as a function of one variable x.
276
views