Domanda del libro di testo
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
51. lim (θ → 0) (θ - sin θ cos θ) / (tan θ - θ)
42
views
Ch. 7 - Transcendental Functions
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 7, Problema 7.7.8
Rewrite the expressions in Exercises 5–10 in terms of exponentials and simplify the results as much as you can.
8. cosh(3x) - sinh(3x)
Guida verificata passo dopo passo1
Recall the definitions of hyperbolic cosine and hyperbolic sine in terms of exponentials:
\(\text{cosh}(x) = \frac{e^{x} + e^{-x}}{2}\) and \(\text{sinh}(x) = \frac{e^{x} - e^{-x}}{2}\).
Substitute \$3x$ into these definitions:
\(\text{cosh}(3x) = \frac{e^{3x} + e^{-3x}}{2}\) and \(\text{sinh}(3x) = \frac{e^{3x} - e^{-3x}}{2}\).
Write the expression \(\text{cosh}(3x) - \text{sinh}(3x)\) using the exponential forms:
\(\frac{e^{3x} + e^{-3x}}{2} - \frac{e^{3x} - e^{-3x}}{2}\).
Combine the fractions since they have the same denominator:
\(\frac{(e^{3x} + e^{-3x}) - (e^{3x} - e^{-3x})}{2}\).
Simplify the numerator by distributing the minus sign and combining like terms:
\(\frac{e^{3x} + e^{-3x} - e^{3x} + e^{-3x}}{2} = \frac{2e^{-3x}}{2}\), which simplifies further.
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 cosh(x) and sinh(x), are analogs of trigonometric functions but based on exponential functions. They are defined as cosh(x) = (e^x + e^(-x))/2 and sinh(x) = (e^x - e^(-x))/2, which allows rewriting expressions involving them in terms of exponentials.
Video consigliato:
Percorso guidato
5:50
Asymptotes of Hyperbolas
Exponential Function Properties
Exponential functions follow specific algebraic rules, such as e^(a+b) = e^a * e^b and e^(-x) = 1/e^x. Understanding these properties helps simplify expressions involving sums or differences of exponentials, which is essential when rewriting hyperbolic functions.
Video consigliato:
Percorso guidato
06:21Properties of Functions
Simplification of Expressions
Simplification involves combining like terms and reducing expressions to their simplest form. For hyperbolic functions expressed as exponentials, this often means canceling terms or factoring to reveal simpler equivalent expressions, making the result easier to interpret or use.
Video consigliato:
6:36Simplifying Trig Expressions
Pratica correlata
Domanda del libro di testo
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
29. y=arcsec(1/t), 0<t<1
28
views
Domanda del libro di testo
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
27. y=arccsc(x²+1)
16
views
Domanda del libro di testo
82. Find a curve through the point (1, 0) whose length from x=1 to x=2 is
L = ∫(from 1 to 2)√(1 + 1/x²)dx.
28
views
Domanda del libro di testo
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
27. y = θ(sin(lnθ) + cos(lnθ))
22
views
Domanda del libro di testo
Verify the integration formulas in Exercises 37–40.
39. ∫x coth⁻¹(x)dx = ((x²-1)/2)coth⁻¹(x) + x/2 + C
19
views