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.
−3x⁻⁴
1
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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 4.3.59a
Identifying Extrema
In Exercises 53–60:
a. Find the local extrema of each function on the given interval, and say where they occur.
f(x) = csc²x − 2cot x, 0 < x < π
Verified step by step guidance1
To find the local extrema of the function \( f(x) = \csc^2 x - 2 \cot x \) on the interval \( 0 < x < \pi \), we first need to find the derivative \( f'(x) \).
Recall that \( \csc x = \frac{1}{\sin x} \) and \( \cot x = \frac{\cos x}{\sin x} \). Use the chain rule and quotient rule to differentiate \( \csc^2 x \) and \( \cot x \).
The derivative of \( \csc^2 x \) is \( -2 \csc^2 x \cot x \), and the derivative of \( -2 \cot x \) is \( 2 \csc^2 x \). Therefore, \( f'(x) = -2 \csc^2 x \cot x + 2 \csc^2 x \).
Set \( f'(x) = 0 \) to find critical points: \( -2 \csc^2 x \cot x + 2 \csc^2 x = 0 \). Simplify to find \( \cot x = 1 \).
Solve \( \cot x = 1 \) within the interval \( 0 < x < \pi \) to find the critical points. Evaluate \( f(x) \) at these critical points to determine the local extrema.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Functions
Understanding the trigonometric functions involved, such as cosecant (csc) and cotangent (cot), is crucial. The cosecant function is the reciprocal of sine, and cotangent is the reciprocal of tangent. These functions have specific properties and behaviors, especially within the interval 0 < x < π, which affect the function's behavior and the location of extrema.
Recommended video:
6:04
Introduction to Trigonometric Functions
Derivatives and Critical Points
To find local extrema, we need to calculate the derivative of the function and identify critical points where the derivative is zero or undefined. These points are potential candidates for local maxima or minima. Analyzing the derivative helps determine the function's increasing or decreasing behavior, which is essential for locating extrema.
Recommended video:
04:50Critical Points
Interval Analysis
The given interval, 0 < x < π, restricts the domain of the function, affecting where extrema can occur. It's important to consider the behavior of the function at the boundaries and within this interval. Since trigonometric functions can have discontinuities or undefined points, careful analysis within the specified range is necessary to accurately identify local extrema.
Recommended video:
02:59Finding Area Between Curves that Cross on the Interval
Related Practice
Textbook Question
52. Two masses hanging side by side from springs have positions s_1 = 2 sin t and s_2 = sin 2t,
respectively.
a. At what times in the interval 0 < t do the masses pass each other? (Hint: sin 2t = 2 sint cost.)
Textbook Question
Identifying Extrema
In Exercises 53–60:
a. Find the local extrema of each function on the given interval, and say where they occur.
f(x) = x / 2 − 2sin (x/2), 0 ≤ x ≤ 2π
Textbook Question
Identifying Extrema
In Exercises 53–60:
a. Find the local extrema of each function on the given interval, and say where they occur.
f(x) = sin 2x, 0 ≤ x ≤ π
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 / x²
1
views
Textbook Question
Identifying Extrema
In Exercises 41–52:
a. Identify the function’s local extreme values in the given domain, and say where they occur.
f(t) = 12t − t³, −3 ≤ t < ∞