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 4.R.29
Capitolo 4, Problema 4.R.29

24–34. Curve sketching Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.
ƒ(x) = ln( x² + 3) / (x -1)

Guida verificata passo dopo passo
1
Identify the domain of the function \( f(x) = \frac{\ln(x^2 + 3)}{x - 1} \). The natural logarithm function \( \ln(x^2 + 3) \) is defined for all real numbers since \( x^2 + 3 > 0 \) for all \( x \). However, the denominator \( x - 1 \) cannot be zero, so \( x \neq 1 \). Thus, the domain is all real numbers except \( x = 1 \).
Find the vertical asymptote by setting the denominator equal to zero: \( x - 1 = 0 \). This gives \( x = 1 \) as a vertical asymptote.
Determine the behavior of the function as \( x \to \infty \) and \( x \to -\infty \). As \( x \to \infty \), \( \ln(x^2 + 3) \approx \ln(x^2) = 2\ln(x) \), and the function behaves like \( \frac{2\ln(x)}{x} \), which approaches 0. Similarly, as \( x \to -\infty \), the function also approaches 0.
Find the first derivative \( f'(x) \) to determine critical points and intervals of increase or decrease. Use the quotient rule: \( f'(x) = \frac{(x - 1) \cdot \frac{d}{dx}[\ln(x^2 + 3)] - \ln(x^2 + 3) \cdot \frac{d}{dx}[x - 1]}{(x - 1)^2} \). Simplify to find critical points.
Find the second derivative \( f''(x) \) to determine concavity and points of inflection. Use the derivative of \( f'(x) \) and analyze the sign of \( f''(x) \) to determine where the function is concave up or down.

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.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the function f(x) = ln(x² + 3) / (x - 1), the domain excludes x = 1, where the denominator becomes zero, causing a division by zero error. Additionally, the expression inside the logarithm, x² + 3, must be positive, which it always is for real numbers.
Video consigliato:
Percorso guidato
5:10
Finding the Domain and Range of a Graph

Asymptotes

Asymptotes are lines that a graph approaches but never touches. For f(x) = ln(x² + 3) / (x - 1), there is a vertical asymptote at x = 1 due to the division by zero. Horizontal asymptotes can be determined by analyzing the behavior of the function as x approaches infinity or negative infinity, which involves understanding the limits of the function.
Video consigliato:
Percorso guidato
5:37
Introduction to Cotangent Graph

Critical Points and Derivatives

Critical points occur where the derivative of a function is zero or undefined, indicating potential maxima, minima, or inflection points. To find these for f(x) = ln(x² + 3) / (x - 1), compute the first derivative and solve for x where the derivative equals zero or is undefined. This helps in understanding the function's increasing or decreasing behavior and concavity.
Video consigliato:
04:50
Critical Points
Pratica correlata
Domanda del libro di testo

Explain the Mean Value Theorem with a sketch.

280
views
Domanda del libro di testo

Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).

ƒ(x) = sin 2x + 3 on [-π , π]

211
views
Domanda del libro di testo

Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).

ƒ(x) = x³ - 6x² on [-1, 5]

264
views
Domanda del libro di testo

Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.


f(x) = x ln x - 2x + 3 on (0,∞)

244
views
Domanda del libro di testo

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 

lim_x→∞ ln ((x +1) / (x-1))

193
views
Domanda del libro di testo

Maximum printable area A rectangular page in a text (with width x and length y) has an area of 98 in² , top and bottom margins set at 1 in, and left and right margins set at 1/2 in. The printable area of the page is the rectangle that lies within the margins. What are the dimensions of the page that maximize the printable area?

227
views