Textbook Question
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
-(1/3)x⁻⁴ᐟ³
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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 4.7.15c
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
−π csc (πx/2) cot (πx/2)
Verified step by step guidance1
Recognize that the given function is \(-\pi \csc\left(\frac{\pi x}{2}\right) \cot\left(\frac{\pi x}{2}\right)\), which involves trigonometric functions cosecant and cotangent with an inner function \(\frac{\pi x}{2}\).
Recall the derivative formula: \(\frac{d}{dx}[\csc(u)] = -\csc(u) \cot(u) \cdot u'\) where \(u\) is a function of \(x\).
Identify the inner function \(u = \frac{\pi x}{2}\), so its derivative is \(u' = \frac{\pi}{2}\).
Notice that the given function resembles the derivative of \(\csc\left(\frac{\pi x}{2}\right)\) multiplied by a constant. Use this to set up the antiderivative as a constant multiple of \(\csc\left(\frac{\pi x}{2}\right)\).
Adjust the constant factor to account for the difference between the given function and the derivative formula, then write the antiderivative as \(F(x) = A \csc\left(\frac{\pi x}{2}\right) + C\), where \(A\) is the constant to be determined and \(C\) is the constant of integration.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Antiderivatives (Indefinite Integrals)
An antiderivative of a function is another function whose derivative equals the original function. Finding antiderivatives involves reversing differentiation, often represented as an indefinite integral with a constant of integration. This process helps solve problems involving accumulation and area under curves.
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05:04
Introduction to Indefinite Integrals
Trigonometric Functions and Their Derivatives
Understanding the derivatives of trigonometric functions like sine, cosine, cosecant, and cotangent is essential. For example, the derivative of cotangent is -csc²(x), and the derivative of cosecant is -csc(x)cot(x). Recognizing these relationships aids in identifying antiderivatives involving trigonometric expressions.
Recommended video:
06:35Derivatives of Other Inverse Trigonometric Functions
Chain Rule and Substitution Method
The chain rule helps differentiate composite functions, and its reverse is used in integration by substitution. When the integrand involves a function and its derivative, substitution simplifies finding antiderivatives. This technique is crucial for handling functions like csc(πx/2) cot(πx/2) where the argument is a linear function of x.
Recommended video:
05:02Intro to the Chain Rule
Related Practice
Textbook Question
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
√x + 1/√x
Textbook Question
Theory and Examples
In Exercises 51 and 52, give reasons for your answers.
Let f(x) = |x³ − 9x|.
b. Does f'(-3) exist?
Textbook Question
Analyzing Functions from Derivatives
Answer the following questions about the functions whose derivatives are given in Exercises 1–14:
c. At what points, if any, does f assume local maximum or minimum values?
f′(x) = (x − 1)²(x + 2)
Textbook Question
Finding displacement from an antiderivative of velocity
a. Suppose that the velocity of a body moving along the s-axis is
ds/dt = v = 9.8t − 3.
iii. Now find the body’s displacement from t = 1 to t = 3 given that s = s₀ when t = 0.
Textbook Question
Analyzing Functions from Derivatives
Answer the following questions about the functions whose derivatives are given in Exercises 1–14:
c. At what points, if any, does f assume local maximum or minimum values?
f′(x) = 1− 4/x², x ≠ 0