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.
(2/3)x⁻¹ᐟ³
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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 4.3.50a
Identifying Extrema
In Exercises 41–52:
a. Identify the function’s local extreme values in the given domain, and say where they occur.
f(x) = √(x² − 2x − 3), 3 ≤ x < ∞
Verified step by step guidance1
Step 1: Begin by understanding the function f(x) = √(x² − 2x − 3). This is a square root function, and we need to ensure the expression inside the square root is non-negative for the function to be defined.
Step 2: Set the expression inside the square root greater than or equal to zero: x² − 2x − 3 ≥ 0. Solve this inequality to find the domain where the function is defined.
Step 3: Factor the quadratic expression: x² − 2x − 3 = (x - 3)(x + 1). Use this factorization to solve the inequality (x - 3)(x + 1) ≥ 0. Determine the intervals where the product is non-negative.
Step 4: Analyze the critical points and endpoints within the domain 3 ≤ x < ∞. Critical points occur where the derivative is zero or undefined. Find the derivative of f(x) and solve for x where f'(x) = 0 or is undefined.
Step 5: Evaluate the function at the critical points and endpoints to identify local extrema. Compare the values to determine the local maximum or minimum values within the given domain.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Local Extrema
Local extrema refer to the points in a function where it reaches a local maximum or minimum value. These points occur where the derivative of the function is zero or undefined, indicating a change in the direction of the curve. Identifying local extrema involves finding these critical points and using tests like the first or second derivative test to determine their nature.
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05:58
Finding Extrema Graphically
Domain of a Function
The domain of a function is the set of all possible input values (x-values) for which the function is defined. In this problem, the domain is specified as 3 ≤ x < ∞, meaning we only consider x-values starting from 3 and extending to infinity. Understanding the domain is crucial for correctly identifying where extrema can occur within the given constraints.
Recommended video:
5:10Finding the Domain and Range of a Graph
Derivative and Critical Points
The derivative of a function provides the rate of change of the function with respect to its variable. Critical points occur where the derivative is zero or undefined, indicating potential local maxima or minima. To find these points, differentiate the function and solve for x where the derivative equals zero or does not exist, considering the given domain.
Recommended video:
04:50Critical Points
Related Practice
Textbook Question
51. Frictionless cart A small frictionless cart, attached to the wall by a spring, is pulled 10 cm from its rest position and released at time t = 0 to roll back and forth for 4 sec. Its position at time t is s = 10 cos πt.
a. What is the cart's maximum speed? When is the cart moving that fast? Where is it then? What is the magnitude of the acceleration then?
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Textbook Question
Identifying Extrema
In Exercises 19–40:
a. Find the open intervals on which the function is increasing and those on which it is decreasing.
f(r) = 3r³ + 16r
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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) = csc²x − 2cot x, 0 < x < π
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Textbook Question
25. Paper folding A rectangular sheet of 8.5-in.-by-11-in. paper is placed on a flat surface. One of the corners is placed on the opposite longer edge, as shown in the figure, and held there as the paper is smoothed flat. The problem is to make the length of the crease as small as possible. Call the length L. Try it with paper.
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a. Show that L^2=2x^3/(2x-8.5).
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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.
g(x) = x² − 4x + 4, 1 ≤ x < ∞
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