Skip to main content
Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 4 - Applications of DerivativesProblema 30
Capitolo 4, Problema 30

In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.
30. y = (x² - 4) / (x² - 2)

Guida verificata passo dopo passo
1
Step 1: Analyze the domain of the function y = (x² - 4) / (x² - 2). Identify any restrictions on x by setting the denominator equal to zero. Solve x² - 2 = 0 to find the values of x where the function is undefined.
Step 2: Determine the critical points by finding the derivative of the function y = (x² - 4) / (x² - 2). Use the quotient rule: \( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^2} \), where \( u = x^2 - 4 \) and \( v = x^2 - 2 \). Set the derivative equal to zero to solve for x.
Step 3: Identify local extreme points by analyzing the critical points found in Step 2. Use the first derivative test or second derivative test to determine whether each critical point corresponds to a local maximum, local minimum, or neither.
Step 4: Find inflection points by calculating the second derivative of the function. Set the second derivative equal to zero and solve for x. Verify changes in concavity by testing intervals around these points.
Step 5: Determine absolute extreme points by evaluating the function at critical points and endpoints of the domain (if applicable). Compare the values of the function at these points to identify the absolute maximum and minimum.

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:
14m

Concetti chiave

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

Graphing Rational Functions

Graphing rational functions involves analyzing the function's behavior by identifying key features such as intercepts, asymptotes, and the overall shape of the graph. For the function y = (x² - 4) / (x² - 2), one must determine where the function is undefined (vertical asymptotes) and where it crosses the axes (x and y intercepts) to sketch an accurate graph.
Video consigliato:
Percorso guidato
5:53
Graph of Sine and Cosine Function

Local and Absolute Extrema

Local extrema refer to points where a function reaches a maximum or minimum value within a specific interval, while absolute extrema are the highest or lowest points over the entire domain of the function. To find these points, one typically uses the first derivative test to identify critical points and the second derivative test to determine their nature.
Video consigliato:
05:58
Finding Extrema Graphically

Inflection Points

Inflection points occur where the concavity of a function changes, which can be identified by analyzing the second derivative. For the function y = (x² - 4) / (x² - 2), finding inflection points involves determining where the second derivative equals zero or is undefined, indicating a change in the curvature of the graph.
Video consigliato:
04:50
Critical Points
Pratica correlata
Domanda del libro di testo

Finding Functions from Derivatives


In Exercises 31–36, find all possible functions with the given derivative.


a. y′ = x

187
views
Domanda del libro di testo

Calculate the first derivatives of ƒ(𝓍) = 𝓍²/ (𝓍² + 1) and g(𝓍) = ―1/ (𝓍² + 1) . What can you conclude about the graphs of these functions?

208
views
Domanda del libro di testo

Finding Functions from Derivatives


Suppose that f'(x) = 2x for all x. Find f(2) if


a. f(0) = 0

190
views
Domanda del libro di testo

In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.

33. y = (x² - x + 1) / (x - 1)

155
views
Domanda del libro di testo

Finding Functions from Derivatives


In Exercises 31–36, find all possible functions with the given derivative.


b. y′ = x²

204
views
Domanda del libro di testo

Finding Functions from Derivatives


Suppose that f'(x) = 2x for all x. Find f(2) if


b. f(1) = 0

247
views