Textbook Question
Finding Critical Points
In Exercises 41–50, determine all critical points and all domain endpoints for each function.
g(x) = √(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 4.7.29
Finding Indefinite Integrals
In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.
∫(8y − 2 / y¹ᐟ⁴) dy
Verified step by step guidance1
Identify the integral to solve: \(\int \left(8y - \frac{2}{y^{1/4}}\right) \, dy\).
Rewrite the integral by expressing the terms with exponents: \(\int \left(8y - 2y^{-1/4}\right) \, dy\).
Split the integral into two separate integrals: \(\int 8y \, dy - \int 2y^{-1/4} \, dy\).
Apply the power rule for integration to each term separately. Recall that \(\int y^n \, dy = \frac{y^{n+1}}{n+1} + C\) for \(n \neq -1\).
Integrate each term: For \$8y\(, treat it as \)8y^1\(, so the integral is \(8 \times \frac{y^{1+1}}{1+1}\). For \(2y^{-1/4}\), the integral is \(2 \times \frac{y^{-1/4 + 1}}{-1/4 + 1}\). Don't forget to add the constant of integration \)+ C$ at the end.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Indefinite Integral and Antiderivative
An indefinite integral represents the most general antiderivative of a function, including a constant of integration. It reverses differentiation, finding a function whose derivative matches the integrand. For example, ∫f(y) dy gives a family of functions F(y) + C where F'(y) = f(y).
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Introduction to Indefinite Integrals
Power Rule for Integration
The power rule states that ∫y^n dy = (y^(n+1)) / (n+1) + C for any real number n ≠ -1. This rule is essential for integrating terms like y raised to a power, including fractional and negative exponents, by increasing the exponent by one and dividing by the new exponent.
Recommended video:
04:04Power Rule for Indefinite Integrals
Verification by Differentiation
After finding an antiderivative, differentiating it should return the original integrand. This step confirms the correctness of the integral solution. Checking by differentiation helps identify and correct errors in the integration process.
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05:53Finding Differentials
Related Practice
Textbook Question
Finding Extreme Values
In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.
y = 𝓍³ + 𝓍² ― 8𝓍 + 5
Textbook Question
119. Find the values of constants a, b, and c such that the graph of y = ax^3 + bx^2 + cx has a
local maximum at x = 3, local minimum at x =- 1, and inflection point at (1, 11).
Textbook Question
Sketch the graphs of the rational functions in Exercises 53–60.
𝓍⁴ ― 1
y = ------------------
𝓍²
Textbook Question
Initial Value Problems
Solve the initial value problems in Exercises 71–90.
ds/dt = 1 + cos t, s(0) = 4
Textbook Question
Theory and Examples
In Exercises 53 and 54, show that the function has neither an absolute minimum nor an absolute maximum on its natural domain.
y = 3x + tan x