Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

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

Briggs 3rd Edition

Ch. 3 - Derivatives

Problema 69c

Capitolo 3, Problema 69c

The following equations implicitly define one or more functions.
c. Use the functions found in part (b) to graph the given equation.
x+y³−xy=1 (Hint: Rewrite as y³−1=xy−x and then factor both sides.)

Guida verificata passo dopo passo

Start by rewriting the given equation as suggested: \( y^3 - 1 = xy - x \). This sets up the equation for factoring.

Factor the left side of the equation \( y^3 - 1 \) using the difference of cubes formula: \( y^3 - 1 = (y - 1)(y^2 + y + 1) \).

Factor the right side of the equation \( xy - x \) by taking out the common factor \( x \): \( xy - x = x(y - 1) \).

Set the factored forms equal to each other: \( (y - 1)(y^2 + y + 1) = x(y - 1) \).

To find the functions, consider the case where \( y - 1 \neq 0 \) and divide both sides by \( y - 1 \), resulting in \( y^2 + y + 1 = x \). This gives the function \( x = y^2 + y + 1 \).

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.

Implicit Functions

Implicit functions are defined by equations where the dependent variable is not isolated on one side. In the context of calculus, understanding how to manipulate these equations is crucial for finding explicit forms of functions or for analyzing their properties. The equation given, x + y³ - xy = 1, is an example where y is implicitly defined in terms of x.

Video consigliato:

Percorso guidato

05:14

Finding The Implicit Derivative

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. In the context of the given equation, rewriting it as y³ - 1 = xy - x allows for easier manipulation and understanding of the relationship between x and y, which is essential for graphing the function.

Video consigliato:

06:11

Limits of Rational Functions: Denominator = 0

Graphing Functions

Graphing functions involves plotting points on a coordinate system to visually represent the relationship between variables. For the equation derived from the implicit function, understanding how to graph y in terms of x after factoring is key to visualizing the behavior of the function. This process often requires identifying key features such as intercepts and asymptotes.

Video consigliato:

Percorso guidato

5:53

Graph of Sine and Cosine Function

Pratica correlata

Domanda del libro di testo

Given the function f and the point Q, find all points P on the graph of f such that the line tangent to f at P passes through Q. Check your work by graphing f and the tangent lines.

f(x) = 1/x; Q (-2, 4)

223

views

Domanda del libro di testo

The following equations implicitly define one or more functions.

b. Solve the given equation for y to identify the implicitly defined functions y=f₁(x), y = f₂(x), ….

x+y³−xy=1 (Hint: Rewrite as y³−1=xy−x and then factor both sides.)

277

views

Domanda del libro di testo

Calculate the derivative of the following functions.

y = e^2x(2x-7)⁵