Textbook Question
Finding Extrema from Graphs
In Exercises 7–10, find the absolute extreme values and where they occur.
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 4.7.56
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
Verified step by step guidance1
Start by rewriting the integrand to express everything in terms of sine and cosine functions. Recall that \(\csc \theta = \frac{1}{\sin \theta}\). So rewrite the integral as \(\int \frac{\frac{1}{\sin \theta}}{\frac{1}{\sin \theta} - \sin \theta} \, d\theta\).
Simplify the denominator by finding a common denominator inside it: \(\frac{1}{\sin \theta} - \sin \theta = \frac{1 - \sin^2 \theta}{\sin \theta}\). Use the Pythagorean identity \(1 - \sin^2 \theta = \cos^2 \theta\) to rewrite the denominator as \(\frac{\cos^2 \theta}{\sin \theta}\).
Now the integrand becomes \(\frac{\frac{1}{\sin \theta}}{\frac{\cos^2 \theta}{\sin \theta}} = \frac{1}{\sin \theta} \times \frac{\sin \theta}{\cos^2 \theta} = \frac{1}{\cos^2 \theta} = \sec^2 \theta\).
Recognize that the integral simplifies to \(\int \sec^2 \theta \, d\theta\). Recall the antiderivative of \(\sec^2 \theta\) is \(\tan \theta + C\).
Write the most general antiderivative as \(\tan \theta + C\), where \(C\) is the constant of integration. To verify, differentiate \(\tan \theta + C\) and confirm you get back the original integrand.
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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 of integration. It reverses differentiation, finding a function whose derivative matches the integrand. Understanding this concept is essential for solving integrals without specified limits.
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Introduction to Indefinite Integrals
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values in their domains. Simplifying integrals often requires rewriting expressions using identities like csc θ = 1/sin θ or combining terms to facilitate integration. Mastery of these identities helps transform complex integrands into manageable forms.
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7:17Verifying Trig Equations as Identities
Verification by Differentiation
After finding an antiderivative, verifying the solution by differentiating it ensures correctness. Differentiation should return the original integrand, confirming the integral was computed properly. This step is crucial for validating answers in indefinite integral problems.
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05:53Finding Differentials
Related Practice
Textbook Question
Motion with constant acceleration The standard equation for the position s of a body moving with a constant acceleration a along a coordinate line is s = (a/2)t² + v₀t + s₀, where v₀ and s₀ are the body’s velocity and position at time t = 0. Derive this equation by solving the initial value problem
Differential equation: d²s/dt² = a
Initial conditions: ds/dt = v₀ and s = s₀ when t=0.
Textbook Question
Checking the Mean Value Theorem
Find the value or values of c that satisfy the equation (f(b) − f(a)) / (b − a) = f′(c) in the conclusion of the Mean Value Theorem for the functions and intervals in Exercises 1–6.
f(x) =√(x − 1), [1, 3]
Textbook Question
Applications
A marathoner ran the 26.2-mi New York City Marathon in 2.2 hours. Show that at least twice the marathoner was running at exactly 11 mph, assuming the initial and final speeds are zero.
Textbook Question
The 8-ft wall shown here stands 27 ft from the building. Find the length of the shortest straight beam that will reach to the side of the building from the ground outside the wall.
Textbook Question
Checking Antiderivative Formulas
Verify the formulas in Exercises 57–62 by differentiation.
∫csc²((x − 1)/3)dx = −3cot((x − 1)/3) + C