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Roots (Zeros)
Show that the functions in Exercises 19–26 have exactly one zero in the given interval.
f(x) = x³ + 4x² + 7, (−∞, 0)
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Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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
Capitolo 4, Problema 4.7.41
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.
∫(cscθ cotθ) / 2 dθ
Guida verificata passo dopo passo1
Recognize that the integral is \( \int \frac{\csc \theta \cot \theta}{2} \, d\theta \). Since the constant \( \frac{1}{2} \) can be factored out, rewrite the integral as \( \frac{1}{2} \int \csc \theta \cot \theta \, d\theta \).
Recall the derivative of \( \csc \theta \) is \( -\csc \theta \cot \theta \). This suggests that \( \csc \theta \cot \theta \) is closely related to the derivative of \( \csc \theta \).
Use this relationship to guess that the antiderivative of \( \csc \theta \cot \theta \) is \( -\csc \theta \), because differentiating \( -\csc \theta \) gives \( \csc \theta \cot \theta \).
Therefore, the integral becomes \( \frac{1}{2} \times (-\csc \theta) + C \), where \( C \) is the constant of integration.
Finally, verify your result by differentiating \( -\frac{1}{2} \csc \theta + C \) to ensure it matches the original integrand \( \frac{\csc \theta \cot \theta}{2} \).
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Indefinite Integral and Antiderivative
An indefinite integral represents the most general form of an antiderivative of a function, including a constant of integration. It reverses differentiation, finding a function whose derivative matches the integrand. Understanding this helps in solving integrals without specified limits.
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05:04
Introduction to Indefinite Integrals
Trigonometric Functions and Identities
Knowledge of trigonometric functions like cosecant (csc) and cotangent (cot), and their relationships, is essential. Recognizing identities such as the derivative of cscθ being -cscθ cotθ aids in simplifying and integrating expressions involving these functions.
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6:04Introduction to Trigonometric Functions
Verification by Differentiation
After finding an antiderivative, differentiating it confirms the correctness of the integral. This step ensures the solution is accurate and helps identify any errors in the integration process, reinforcing understanding of the fundamental theorem of calculus.
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05:53Finding Differentials
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Checking Antiderivative Formulas
Verify the formulas in Exercises 57–62 by differentiation.
∫(3x + 5)⁻² dx = −(3x + 5)⁻¹/3 + C
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Domanda del libro di testo
A cone is formed from a circular piece of material of radius 1 meter by removing a section of angle θ and then joining the two straight edges. Determine the largest possible volume for the cone.
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Domanda del libro di testo
Roots (Zeros)
Show that the functions in Exercises 19–26 have exactly one zero in the given interval.
f(x) = x⁴ + 3x + 1, [−2, −1]
243
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Domanda del libro di testo
Absolute Extrema on Finite Closed Intervals
In Exercises 21–36, find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
h(x) = ³√x, −1 ≤ x ≤ 8
204
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Domanda del libro di testo
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.
∫csc θ/(csc θ − sin θ) dθ
30
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