Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

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

Briggs 3rd Edition

Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions

Problema 7.1.1

Capitolo 7, Problema 7.1.1

What are the domain and range of ln x?

Guida verificata passo dopo passo

Understand the function: The natural logarithm function, ln(x), is defined as the logarithm to the base e, where e is approximately 2.718. It is only defined for positive values of x.

Determine the domain: The domain of ln(x) consists of all x-values for which the function is defined. Since ln(x) is undefined for x ≤ 0, the domain is (0, ∞).

Analyze the behavior of ln(x): As x approaches 0 from the right (x → 0⁺), ln(x) decreases without bound, approaching negative infinity. As x increases (x → ∞), ln(x) increases without bound.

Determine the range: Based on the behavior of ln(x), the range includes all real numbers because the function can output any value from negative infinity to positive infinity. Thus, the range is (-∞, ∞).

Summarize the domain and range: The domain of ln(x) is (0, ∞), and the range is (-∞, ∞). These properties are fundamental to understanding the natural logarithm 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:

Concetti chiave

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

Natural Logarithm Function

The natural logarithm function, denoted as ln(x), is the logarithm to the base e, where e is approximately 2.71828. It is defined for positive real numbers and is the inverse of the exponential function e^x. Understanding this function is crucial for determining its domain and range.

Video consigliato:

05:18

Derivative of the Natural Logarithmic Function

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For the natural logarithm function ln(x), the domain is limited to positive real numbers, meaning x must be greater than zero, as ln(x) is undefined for x ≤ 0.

Video consigliato:

Percorso guidato

5:10

Finding the Domain and Range of a Graph

Range of a Function

The range of a function is the set of all possible output values (y-values) that the function can produce. For the natural logarithm function ln(x), the range is all real numbers, as ln(x) can take any value from negative infinity to positive infinity as x approaches zero from the right and increases without bound.

Video consigliato:

Percorso guidato

5:10

Finding the Domain and Range of a Graph

Pratica correlata

Domanda del libro di testo

Critical points Find the critical points of the function ƒ(x) = sinh² x cosh x.

views

Domanda del libro di testo

22–36. Derivatives Find the derivatives of the following functions.

f(x) = csch⁻¹(2/x)

views

Domanda del libro di testo

Sketch the graphs of y = cosh x, y = sinh x, and y = tanh x (include asymptotes), and state whether each function is even, odd, or neither.

views

Domanda del libro di testo

22–36. Derivatives Find the derivatives of the following functions.

f(x) = sinh 4x

views

Domanda del libro di testo

After the introduction of foxes on an island, the number of rabbits on the island decreases by 4.5% per month. If y(t) equals the number of rabbits on the island t months after foxes were introduced, find the rate constant k for the exponential decay function y(t) = y₀eᵏᵗ.

views

Domanda del libro di testo

15–20. Designing exponential growth functions Complete the following steps for the given situation.

a. Find the rate constant k and use it to devise an exponential growth function that fits the given data.

b. Answer the accompanying question.

Rising costs Between 2010 and 2016, the average rate of inflation was about 1.6%/yr. If a cart of groceries cost \$100 in 2010, what will it cost in 2025, assuming the rate of inflation remains constant at 1.6%?

views