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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- 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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2mKey Concepts
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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.
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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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Related Practice
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
Textbook Question
"Roots (Zeros) Show that the functions in Exercises 19–26 have exactly one zero