Domanda del libro di testo
Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
93. lim(x→0) (csc(x) - cot(x))
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Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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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.PE.81
In Exercises 79–84, solve for y.
81. 9e^(2y) = = x^2
Guida verificata passo dopo passo1
Start with the given equation: \(9e^{2y} = x^2\).
Isolate the exponential term by dividing both sides of the equation by 9: \(e^{2y} = \frac{x^2}{9}\).
To solve for \(y\), take the natural logarithm (ln) of both sides to undo the exponential: \(\ln\left(e^{2y}\right) = \ln\left(\frac{x^2}{9}\right)\).
Use the logarithm property \(\ln\left(e^a\right) = a\) to simplify the left side: \(2y = \ln\left(\frac{x^2}{9}\right)\).
Finally, solve for \(y\) by dividing both sides by 2: \(y = \frac{1}{2} \ln\left(\frac{x^2}{9}\right)\).
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Exponential Functions
An exponential function involves a variable in the exponent, such as e^(2y). Understanding how to manipulate and isolate the variable in the exponent is essential for solving equations involving exponential expressions.
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6:13
Exponential Functions
Logarithmic Functions and Properties
Logarithms are the inverse operations of exponentials. Applying logarithms allows us to solve for variables in the exponent by converting exponential equations into linear ones, making it easier to isolate and solve for y.
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Percorso guidato
06:21Properties of Functions
Algebraic Manipulation
Solving for y requires careful algebraic steps, including isolating terms, simplifying expressions, and applying inverse operations. Mastery of algebraic manipulation ensures accurate and efficient problem-solving.
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05:25Determine Continuity Algebraically
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