Textbook Question
Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
93. lim(x→0) (csc(x) - cot(x))
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 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
Chapter 7, Problem 7.PE.79
In Exercises 79–84, solve for y.
79. 3^y = 2^(y+1)
Verified step by step guidance1
Start with the given equation: \(3^{y} = 2^{y+1}\).
Rewrite the right side using exponent rules: \(2^{y+1} = 2^{y} \cdot 2^{1} = 2^{y} \cdot 2\).
Express the equation as \(3^{y} = 2 \cdot 2^{y}\).
Divide both sides by \$2^{y}\( to isolate terms involving \)y$: \(\frac{3^{y}}{2^{y}} = 2\).
Rewrite the left side as a single exponential: \(\left(\frac{3}{2}\right)^{y} = 2\). Then, take the natural logarithm of both sides to solve for \(y\): \(y \cdot \ln\left(\frac{3}{2}\right) = \ln(2)\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Exponential Equations
Exponential equations involve variables in the exponent position, such as 3^y or 2^(y+1). Solving these requires techniques to isolate the variable, often by rewriting the equation or applying logarithms.
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Properties of Logarithms
Logarithms are the inverse operations of exponentials and help solve equations where the variable is an exponent. Key properties include log(a^b) = b log(a), which allows bringing down exponents to solve for the variable.
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Equating and Simplifying Exponents
When bases differ and cannot be rewritten to a common base, taking logarithms on both sides allows comparison of exponents. Simplifying the resulting equation leads to isolating the variable and finding its value.
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Related Practice
Textbook Question
110. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.
f. f(x) = sech(x), g(x) = e^(-x)
Textbook Question
Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
85. lim(x→1) (x² + 3x - 4)/(x - 1)
Textbook Question
133. What is the age of a sample of charcoal in which 90% of the carbon-14 originally present has decayed?
Textbook Question
110. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.
c. f(x) = 10x^3 + 2x^2, g(x) = e^x
Textbook Question
113. The function f(x) = e^x + x, being differentiable and one-to-one, has a differentiable inverse f^(-1)(x). Find the value of df^(-1)/dx at the point f (ln 2).