Domanda del libro di testo
Visual approximation
a. Use a graphing utility to sketch the graph of y = coth x and then explain why ∫₅¹⁰ coth x dx ≈ 5.
42
views
Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testo
Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblema 7.3.78a
Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblema 7.3.78aCapitolo 7, Problema 7.3.78a
Evaluating hyperbolic functions Use a calculator to evaluate each expression or state that the value does not exist. Report answers accurate to four decimal places to the right of the decimal point.
a. coth 4
Guida verificata passo dopo passo1
Recall the definition of the hyperbolic cotangent function: \(\text{coth}(x) = \frac{\cosh(x)}{\sinh(x)}\).
Express \(\cosh(x)\) and \(\sinh(x)\) in terms of exponential functions: \(\cosh(x) = \frac{e^{x} + e^{-x}}{2}\) and \(\sinh(x) = \frac{e^{x} - e^{-x}}{2}\).
Substitute \(x = 4\) into these expressions to find \(\cosh(4)\) and \(\sinh(4)\).
Calculate the values of \(\cosh(4)\) and \(\sinh(4)\) using a calculator or computational tool.
Divide \(\cosh(4)\) by \(\sinh(4)\) to find \(\text{coth}(4)\) and round the result to four decimal places.
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:
2mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Hyperbolic Functions
Hyperbolic functions, such as sinh, cosh, and coth, are analogs of trigonometric functions but based on hyperbolas. The function coth(x) is defined as cosh(x) divided by sinh(x), and it is important to understand their definitions to evaluate expressions correctly.
Video consigliato:
Percorso guidato
5:50
Asymptotes of Hyperbolas
Domain and Existence of Hyperbolic Functions
The domain of hyperbolic functions like coth(x) excludes values where the denominator is zero. Since sinh(x) = 0 at x = 0, coth(x) is undefined there. For x = 4, sinh(4) ≠ 0, so coth(4) exists and can be evaluated.
Video consigliato:
Percorso guidato
5:10Finding the Domain and Range of a Graph
Using Calculators for Hyperbolic Functions
Modern calculators often have built-in hyperbolic function keys (sinh, cosh, tanh). To find coth(x), compute cosh(x) and sinh(x) separately and divide. Reporting answers to four decimal places requires rounding the calculator output accordingly.
Video consigliato:
Percorso guidato
5:50Asymptotes of Hyperbolas
Pratica correlata
Domanda del libro di testo
Depreciation of equipment A large die-casting machine used to make automobile engine blocks is purchased for \$2.5 million. For tax purposes, the value of the machine can be depreciated by 6.8% of its current value each year.
a. What is the value of the machine after 10 years?
26
views
Domanda del libro di testo
ln x is unbounded Use the following argument to show that lim (x → ∞) ln x = ∞ and lim (x → 0⁺) ln x = −∞.
a. Make a sketch of the function f(x) = 1/x on the interval [1, 2]. Explain why the area of the region bounded by y = f(x) and the x-axis on [1, 2] is ln 2.
104
views
Domanda del libro di testo
Energy consumption On the first day of the year (t=0), a city uses electricity at a rate of 2000 MW. That rate is projected to increase at a rate of 1.3% per year.
a. Based on these figures, find an exponential growth function for the power (rate of electricity use) for the city.
45
views
Domanda del libro di testo
Projection sensitivity
According to the 2014 national population projections published by the U.S. Census Bureau, the U.S. population is projected to be 334.4 million in 2020 with an estimated growth rate of 0.79%/yr.
a. Based on these figures, find the doubling time and the projected population in 2050. Assume the growth rate remains constant.
48
views
Domanda del libro di testo
Falling body When an object falling from rest encounters air resistance proportional to the square of its velocity, the distance it falls (in meters) after t seconds is given by d(t) = (m/k) ln (cosh (√(kg/m) t)), where m is the mass of the object in kilograms, g = 9.8 m/s² is the acceleration due to gravity, and k is a physical constant.
a. A BASE jumper (m = 75 kg) leaps from a tall cliff and performs a ten-second delay (she free-falls for 10 s and then opens her chute). How far does she fall in 10 s? Assume k = 0.2.
54
views