Textbook Question
In Exercises 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.
3. y = 1/x ∫(from 1 to x) e^t/t dt, x²y' + xy = e^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 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 7.7.6
Rewrite the expressions in Exercises 5–10 in terms of exponentials and simplify the results as much as you can.
6. sinh(2ln x)
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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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05:36Change 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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6:13Exponential Functions
Related Practice
Textbook Question
Evaluate the integrals in Exercises 33–54.
51. ∫ from ln(π/6) to ln(π/2) 2e^v cos(e^v) dv
Textbook Question
In Exercises 13–24, find the derivative of y with respect to the appropriate variable.
21. y = ln(cosh v) - 1/2 tanh²v
Textbook Question
82. For what values of a and b is
lim(x→0)(tan(2x/x³) + a/x² + sin(bx)/x) = 0?
1
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Textbook Question
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
65. y = (cos θ)^(√2)
Textbook Question
44. Silver cooling in air The temperature of an ingot of silver is 60°C above room temperature right now. Twenty minutes ago, it was 70°C above room temperature. How far above room temperature will the silver be
b. 2 hours from now?