Textbook Question
In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.
__________
y = √ 3 + 2𝓍 ―𝓍²
Ch. 4 - Applications of Derivatives
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 4, Problem 4.7.91
Initial Value Problems
Find the curve y = f(x) in the xy-plane that passes through the point (9,4) and whose slope at each point is 3√x.
Verified step by step guidance1
Identify the given differential equation from the problem: the slope of the curve at each point is given by \(\frac{dy}{dx} = 3\sqrt{x}\).
Rewrite the slope expression in terms of a power of \(x\) to make integration easier: \(\frac{dy}{dx} = 3x^{\frac{1}{2}}\).
Integrate both sides with respect to \(x\) to find the general form of \(y\): \(y = \int 3x^{\frac{1}{2}} \, dx\).
Perform the integration using the power rule for integrals: \(\int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration.
Use the initial condition \(y(9) = 4\) to solve for the constant \(C\) by substituting \(x=9\) and \(y=4\) into the integrated function.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Initial Value Problems
An initial value problem involves finding a function that satisfies a given differential equation and passes through a specific point, called the initial condition. This ensures a unique solution by fixing the constant of integration after solving the differential equation.
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Initial Value Problems
Differential Equations and Slope Fields
A differential equation relates a function to its derivatives. Here, the slope of the curve y = f(x) is given as a function of x, meaning dy/dx = 3√x. Understanding how to interpret and solve such equations is key to finding the original function.
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05:45Understanding Slope Fields
Integration to Find the Original Function
To find y = f(x) from its derivative dy/dx, integrate the given slope function with respect to x. After integration, use the initial condition to solve for the constant of integration, yielding the specific curve passing through the given point.
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05:11Integrals of General Exponential Functions
Related Practice
Textbook Question
Each of Exercises 67–88 gives the first derivative of a continuous function y=f(x). Find y'' and then use Steps 2–4 of the graphing procedure described in this section to sketch the general shape of the graph of f.
69. y' = x(x - 3)²
Textbook Question
6. You are planning to close off a corner of the first quadrant with a line segment 20 units long running from (a, 0) to (0,b). Show that the area of the triangle enclosed by the segment is largest when a = b.
Textbook Question
Theory and Examples
[Technology Exercise] Graph the functions in Exercises 63–66. Then find the extreme values of the function on the interval and say where they occur.
f(x) = |x − 2| + |x + 3|, −5 ≤ x ≤ 5
Textbook Question
Checking Antiderivative Formulas
Right, or wrong? Give a brief reason why.
∫−15(x + 3)² / (x − 2)⁴ dx = ((x + 3)/(x − 2))³ + C
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Textbook Question
Initial Value Problems
Solve the initial value problems in Exercises 71–90.
dy/dx = 1/x² + x, x > 0; y(2) = 1