Textbook Question
Root Finding
5. Use Newton's method to find the positive fourth root of 2 by solving the equation x^4 -2 = 0. Start with x_0 = 1 and find x_2.
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 4.7.54
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.
∫(1 − cot²x) dx
Verified step by step guidance1
Recall the trigonometric identity involving cotangent and cosecant: \(\cot^{2}x + 1 = \csc^{2}x\). From this, express \(1 - \cot^{2}x\) in terms of \(\csc^{2}x\).
Rewrite the integrand using the identity: \(1 - \cot^{2}x = \csc^{2}x\).
Recognize that the integral now becomes \(\int \csc^{2}x \, dx\), which is a standard integral in calculus.
Recall the antiderivative of \(\csc^{2}x\) is \(-\cot x + C\), where \(C\) is the constant of integration.
Write the general solution as \(-\cot x + C\) and verify by differentiating \(-\cot x\) to ensure it returns the original integrand \(1 - \cot^{2}x\).
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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 an arbitrary constant C. It reverses differentiation, finding a function whose derivative matches the integrand. Understanding this helps in solving integrals without specified limits.
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Introduction to Indefinite Integrals
Trigonometric Identities
Trigonometric identities, such as 1 + cot²x = csc²x, simplify integrals involving trigonometric functions. Recognizing and applying these identities transforms complex expressions into integrable forms, making the integration process more straightforward.
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7:17Verifying Trig Equations as Identities
Verification by Differentiation
After finding an antiderivative, differentiating it should return the original integrand. This verification step ensures the correctness of the solution and helps identify any errors in the integration process.
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05:53Finding Differentials
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.
71. y' = x(x² - 12)
Textbook Question
Finding Extrema from Graphs
In Exercises 1–6, determine from the graph whether the function has any absolute extreme values on [a, b]. Then explain how your answer is consistent with Theorem 1.
Textbook Question
Root Finding
2. Use Newton's method to estimate the one real solution of x^3 +3x + 1 = 0. Start with x_0 = 0 and then find x_2.
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.
82. y' = sin t, for 0 ≤ t ≤ 2π
Textbook Question
Initial Value Problems
Solve the initial value problems in Exercises 71–90.
d³y/dx³ = 6; y″(0) = −8, y′(0) = 0, y(0) = 5