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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 7 - Transcendental Functions문제 7.7.8
7장, 문제 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)

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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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주요 개념

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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.
추천 영상:
가이드 코스
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.
추천 영상:
가이드 코스
06:21
Properties 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.
추천 영상:
6:36
Simplifying Trig Expressions
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