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
Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 4, Problema 33
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)
Guida verificata passo dopo passo1
Step 1: Begin by analyzing the function y = (x² - x + 1) / (x - 1). Identify any points where the function is undefined. The denominator x - 1 is zero when x = 1, indicating a potential vertical asymptote at x = 1.
Step 2: Determine the behavior of the function as x approaches the vertical asymptote from both sides. Evaluate the limits as x approaches 1 from the left and right to understand the behavior near the asymptote.
Step 3: Find the first derivative of the function to identify local extreme points. Use the quotient rule: if y = u/v, then y' = (u'v - uv')/v². Apply this to y = (x² - x + 1)/(x - 1) to find y'. Set y' = 0 to find critical points.
Step 4: Calculate the second derivative to find inflection points. Use the derivative obtained in Step 3 and differentiate again. Set the second derivative equal to zero to find potential inflection points.
Step 5: Evaluate the function at critical points and endpoints (if any) to find absolute extreme points. Compare the values of the function at these points 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:
12mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Graphing Rational Functions
Graphing rational functions involves identifying key features such as asymptotes, intercepts, and behavior at infinity. For the function y = (x² - x + 1) / (x - 1), vertical asymptotes occur where the denominator is zero, and horizontal asymptotes are determined by the degrees of the numerator and denominator. Understanding these features helps in sketching the graph accurately.
Video consigliato:
Percorso guidato
5:53
Graph of Sine and Cosine Function
Local Extreme Points
Local extreme points are points where the function reaches a local maximum or minimum. These can be found by taking the derivative of the function and setting it to zero to find critical points. Analyzing the second derivative or using the first derivative test helps determine the nature of these points, whether they are maxima, minima, or neither.
Video consigliato:
04:50Critical Points
Inflection Points
Inflection points occur where the function changes concavity, which can be identified by analyzing the second derivative. For y = (x² - x + 1) / (x - 1), finding where the second derivative equals zero or is undefined helps locate these points. Inflection points are crucial for understanding the overall shape and behavior of the graph.
Video consigliato:
04:50Critical 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' = 1 / 2√x
196
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
In Exercises 31–36, find all possible functions with the given derivative.
a. y′ = −1 / x²
207
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.
30. y = (x² - 4) / (x² - 2)
97
views
Domanda del libro di testo
Finding Functions from Derivatives
In Exercises 31–36, find all possible functions with the given derivative.
c. y' = sin (2t) + cos (t/2)
196
views