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 ― 2𝓍 + 4
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 4.7.27
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.
∫(√x + ³√x) dx
Verified step by step guidance1
Rewrite the integral in terms of exponents to make it easier to integrate. Recall that \( \sqrt{x} = x^{1/2} \) and \( \sqrt[3]{x} = x^{1/3} \). So the integral becomes \( \int \left( x^{1/2} + x^{1/3} \right) \, dx \).
Use the power rule for integration, which states that for any real number \( n \neq -1 \), \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration.
Apply the power rule separately to each term inside the integral: \( \int x^{1/2} \, dx = \frac{x^{1/2 + 1}}{1/2 + 1} + C_1 = \frac{x^{3/2}}{3/2} + C_1 \) and \( \int x^{1/3} \, dx = \frac{x^{1/3 + 1}}{1/3 + 1} + C_2 = \frac{x^{4/3}}{4/3} + C_2 \).
Simplify the fractions in the denominators by multiplying numerator and denominator appropriately: \( \frac{x^{3/2}}{3/2} = \frac{2}{3} x^{3/2} \) and \( \frac{x^{4/3}}{4/3} = \frac{3}{4} x^{4/3} \).
Combine the results and write the most general antiderivative as \( \frac{2}{3} x^{3/2} + \frac{3}{4} x^{4/3} + C \), where \( C \) is the constant of integration. To verify, differentiate this expression and check if you get back the original integrand.
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Indefinite Integral (Antiderivative)
An indefinite integral represents the most general form of the antiderivative of a function, including a constant of integration. It reverses differentiation, finding a function whose derivative matches the integrand. For example, ∫f(x) dx = F(x) + C, where F'(x) = f(x).
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Introduction to Indefinite Integrals
Power Rule for Integration
The power rule states that ∫x^n dx = (x^(n+1))/(n+1) + C for any real number n ≠ -1. This rule is essential for integrating terms like √x (x^(1/2)) and ³√x (x^(1/3)) by rewriting roots as fractional exponents and applying the formula.
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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. It ensures no algebraic errors occurred during integration and that the constant of integration is properly accounted for.
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Related Practice
Textbook Question
In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.
53. y = x * √(8 - x²)
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 = √𝓍² ― 1
Textbook Question
Initial Value Problems
Solve the initial value problems in Exercises 71–90.
d²r/dt² = 2/t³; dr/dt|ₜ ₌ ₁ =1, r(1) = 1
Textbook Question
54. Fermat’s principle in optics Light from a source A is reflected by a plane mirror to a receiver at point B, as shown in the accompanying figure. Show that for the light to obey Fermat’s principle, the angle of incidence must equal the angle of reflection, both measured from the line normal to the reflecting surface. (This result can also be derived without calculus. There is a purely geometric argument, which you may prefer.)
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Textbook Question
37. What value of a makes f(x) = x^2 +(a/x) have
a. a local minimum at x = 2?
b. a point of inflection at x = 1?