Textbook Question
Initial Value Problems
Solve the initial value problems in Exercises 71–90.
dy/dx = 2x − 7, y(2) = 0
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 4.7.41
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
Verified step by step guidance1
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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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 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.
Recommended video:
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.
Recommended video:
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.
Recommended video:
05:53Finding Differentials
Related Practice
Textbook Question
Theory and Examples
Maximum height of a vertically moving body The height of a body moving vertically is given by s = −12gt² + υ₀t + s₀, g > 0, with s in meters and t in seconds. Find the body’s maximum height.
Textbook Question
Finding Position from Velocity or Acceleration
Exercises 45–48 give the acceleration a=d²s/dt², initial velocity, and initial position of an object moving on a coordinate line. Find the object’s position at time t.
a = 9.8, v(0) = −3, s(0) = 0
Textbook Question
[Technology Exercises] When solving Exercises 14–30, you may need to use appropriate technology (such as a calculator or a computer).
26. Factoring a quartic Find the approximate values of r_1 through r_4 in the factorization
8x^4-14x^3-9x^2+11x-1=8(x-r_1)(x-r_2)(x-r_3)(x-r_4)
Textbook Question
Finding Critical Points
In Exercises 41–50, determine all critical points and all domain endpoints for each function.
y = x − 3x²ᐟ³
Textbook Question
Identifying Extrema
In Exercises 15–18:
a. Find the open intervals on which the function is increasing and those on which it is decreasing.
b. Identify the function’s local and absolute extreme values, if any, saying where they occur.