Textbook Question
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
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 4.1.50
Finding Critical Points
In Exercises 41–50, determine all critical points and all domain endpoints for each function.
y = x − 3x²ᐟ³
Verified step by step guidance1
Step 1: Understand the concept of critical points. Critical points occur where the derivative of a function is zero or undefined. These points are potential locations for local maxima, minima, or points of inflection.
Step 2: Find the derivative of the function y = x - 3x^(2/3). Use the power rule and the constant multiple rule to differentiate each term separately.
Step 3: Set the derivative equal to zero to find where the slope of the tangent is horizontal. Solve the equation for x to find potential critical points.
Step 4: Determine where the derivative is undefined. This occurs when the denominator of any fraction in the derivative is zero or when the derivative involves a root that results in an undefined expression.
Step 5: Identify the domain of the original function y = x - 3x^(2/3). Consider any restrictions on x that might affect the existence of critical points, such as values that make the function undefined.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Critical Points
Critical points of a function occur where its derivative is zero or undefined. These points are essential for identifying local maxima, minima, or points of inflection. To find critical points, compute the derivative of the function and solve for values of x where the derivative equals zero or does not exist.
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04:50
Critical Points
Derivative
The derivative of a function represents the rate of change of the function with respect to its variable. It is a fundamental tool in calculus used to find slopes of tangent lines, velocities, and optimize functions. For the function y = x − 3x²ᐟ³, the derivative helps identify critical points by setting it to zero or finding where it is undefined.
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05:44Derivatives
Domain Endpoints
Domain endpoints are the boundary values of the domain of a function, where the function is defined. These endpoints are crucial when analyzing the behavior of a function over its entire domain, especially when determining absolute extrema. For y = x − 3x²ᐟ³, consider the domain of x and identify any endpoints that might affect the function's critical points.
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5:10Finding the Domain and Range of a Graph
Related Practice
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
Roots (Zeros)
Show that the functions in Exercises 19–26 have exactly one zero in the given interval.
f(x) = x³ + 4x² + 7, (−∞, 0)
Textbook Question
Finding Critical Points
In Exercises 41–50, determine all critical points and all domain endpoints for each function.
y = x² + 2/x
Textbook Question
Initial Value Problems
Solve the initial value problems in Exercises 71–90.
d²r/dt² = 2/t³; dr/dt|ₜ ₌ ₁ =1, r(1) = 1
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.