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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 7 - Transcendental FunctionsProblema 7.7.6
Capitolo 7, Problema 7.7.6

Rewrite the expressions in Exercises 5–10 in terms of exponentials and simplify the results as much as you can.
6. sinh(2ln x)

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Recall the definition of the hyperbolic sine function: \(\sinh(y) = \frac{e^{y} - e^{-y}}{2}\).
Substitute \(y = 2 \ln x\) into the definition: \(\sinh(2 \ln x) = \frac{e^{2 \ln x} - e^{-2 \ln x}}{2}\).
Use the property of exponentials and logarithms: \(e^{2 \ln x} = (e^{\ln x})^{2} = x^{2}\) and similarly \(e^{-2 \ln x} = (e^{\ln x})^{-2} = x^{-2}\).
Rewrite the expression using these simplifications: \(\sinh(2 \ln x) = \frac{x^{2} - x^{-2}}{2}\).
Express the final simplified form clearly: \(\sinh(2 \ln x) = \frac{x^{2} - \frac{1}{x^{2}}}{2}\).

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Hyperbolic Sine Function (sinh)

The hyperbolic sine function, sinh(x), is defined as (e^x - e^(-x))/2. It relates exponential functions to hyperbolic trigonometry and is essential for rewriting expressions involving sinh in terms of exponentials.
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Graph of Sine and Cosine Function

Properties of Logarithms

Logarithms, especially natural logs (ln), allow expressions like ln(x^a) = a ln(x). Understanding how to manipulate ln expressions helps simplify arguments inside functions before rewriting them in exponential form.
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Change of Base Property

Exponential Simplification

After rewriting expressions using exponentials, simplifying involves applying exponent rules such as e^(a+b) = e^a * e^b and e^(ln x) = x. This step reduces complex expressions to simpler forms.
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Exponential Functions
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