Textbook Question
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)
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Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- 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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 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)
Verified step by step guidance1
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.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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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.
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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.
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6:36Simplifying Trig Expressions
Related Practice
Textbook Question
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
29. y=arcsec(1/t), 0<t<1
Textbook Question
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
27. y=arccsc(x²+1)
Textbook Question
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.
Textbook Question
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))
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Textbook Question
Verify the integration formulas in Exercises 37–40.
39. ∫x coth⁻¹(x)dx = ((x²-1)/2)coth⁻¹(x) + x/2 + C
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