Textbook Question
In Exercises 1–6, use l’Hôpital’s Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter 2.
5. lim (x → 0) (1 - cos x) / x²
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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 7.4.15
Solve the differential equation in Exercises 9–22.
15. √x (dy/dx) = e^(y+√x), x > 0
Verified step by step guidance1
Rewrite the given differential equation to isolate \( \frac{dy}{dx} \): \[ \sqrt{x} \frac{dy}{dx} = e^{y + \sqrt{x}} \implies \frac{dy}{dx} = \frac{e^{y + \sqrt{x}}}{\sqrt{x}} \].
Express the right-hand side as a product of exponentials to separate variables more easily: \[ \frac{dy}{dx} = \frac{e^y \cdot e^{\sqrt{x}}}{\sqrt{x}}. \]
Rewrite the equation in differential form to separate variables \( y \) and \( x \): \[ \frac{dy}{e^y} = \frac{e^{\sqrt{x}}}{\sqrt{x}} dx. \]
Integrate both sides: \[ \int \frac{1}{e^y} dy = \int \frac{e^{\sqrt{x}}}{\sqrt{x}} dx. \] The left integral simplifies to \( \int e^{-y} dy \).
For the right integral, use substitution: let \( t = \sqrt{x} \), so \( x = t^2 \) and \( dx = 2t dt \). Substitute and simplify the integral accordingly before integrating.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Separable Differential Equations
A separable differential equation can be written so that all terms involving y are on one side and all terms involving x are on the other. This allows integration of each side independently. Recognizing separability is key to solving equations like √x (dy/dx) = e^(y+√x).
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06:06
Solving Separable Differential Equations
Integration Techniques
Solving separable equations requires integrating functions of x and y separately. Familiarity with integrating exponential functions and substitutions, such as u = √x, helps simplify the integrals and find the general solution.
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06:18Integration by Parts for Definite Integrals
Implicit and Explicit Solutions
After integration, solutions may be implicit or explicit. Understanding how to manipulate and interpret these forms is important, especially when the solution involves expressions like e^(y+√x), which may require logarithmic manipulation to isolate y.
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05:14Finding The Implicit Derivative
Related Practice
Textbook Question
Each of Exercises 19–24 gives a formula for a function y=f(x) and shows the graphs of f and f^(-1). Find a formula for f^(-1) in each case.
f(x)=x³-1
Textbook Question
Solve the differential equation in Exercises 9–22.
11. (dy/dx) = e^(x-y)
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Textbook Question
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
59. y = 2^x
Textbook Question
133. Find the absolute maximum value of
f(x) = x^2 * ln(1/x)
and say where it is assumed.
Textbook Question
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
69. y = 2^(sin 3t)