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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 4.7.73
Chapter 4, Problem 4.7.73

Initial Value Problems


Solve the initial value problems in Exercises 71–90.


dy/dx = 1/x² + x, x > 0; y(2) = 1

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Identify the given differential equation and initial condition: \( \frac{dy}{dx} = \frac{1}{x^{2}} + x \), with \( y(2) = 1 \).
Rewrite the differential equation to express \( dy \) in terms of \( dx \): \( dy = \left( \frac{1}{x^{2}} + x \right) dx \).
Integrate both sides with respect to \( x \) to find the general solution: \( y = \int \left( \frac{1}{x^{2}} + x \right) dx + C \).
Compute the integral by splitting it into two parts: \( \int \frac{1}{x^{2}} dx \) and \( \int x \, dx \), then combine the results.
Use the initial condition \( y(2) = 1 \) to solve for the constant of integration \( C \) by substituting \( x = 2 \) and \( y = 1 \) into the general solution.

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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 as a product of a function of x and a function of y, allowing the variables to be separated on opposite sides of the equation. This technique simplifies solving by integrating each side independently.
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Solving Separable Differential Equations

Integration of Functions

Solving dy/dx = f(x) involves integrating the right-hand side with respect to x to find the general solution y(x). Understanding how to integrate functions like 1/x² and x is essential for finding the antiderivative.
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Integrals of General Exponential Functions

Initial Conditions and Particular Solutions

An initial value problem includes a condition like y(2) = 1, which allows determination of the constant of integration after finding the general solution. This yields a unique particular solution satisfying the given initial condition.
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Initial Value Problems
Related Practice
Textbook Question

Identifying Extrema


In Exercises 19–40:


a. Find the open intervals on which the function is increasing and those on which it is decreasing.


b. Identify the function’s local extreme values, if any, saying where they occur.


f(x) = x¹ᐟ³(x + 8)

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

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

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

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.

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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