Textbook Question
Checking Antiderivative Formulas
Right, or wrong? Say which for each formula and give a brief reason for each answer.
∫√(2x + 1) dx = √(x² + x +C)
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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.
Recommended video:
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
Theory and Examples
In Exercises 51 and 52, give reasons for your answers.
Let f(x) = |x³ − 9x|.
a. Does f'(0) exist?
Textbook Question
Theory and Examples
Sketch the graph of a differentiable function y = f(x) that has a local minimum at (1, 1) and a local maximum at (3, 3).
Textbook Question
Checking Antiderivative Formulas
Right, or wrong? Say which for each formula and give a brief reason for each answer.
∫tanθ sec²θ dθ = sec³θ / 3 + C
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Textbook Question
Identifying Extrema
In Exercises 63 and 64, the graph of f' is given. Assume that f has domain (-2, 2).
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a. Either use the graph to determine which intervals f is increasing on and which intervals f is decreasing on, or explain why this information cannot be determined from the graph.
Textbook Question
20.The U.S. Postal Service will accept a box for domestic shipment only if the sum of its length and girth (distance around) does not exceed 108 in. a.What dimensions will give a box with a square end the largest possible volume?
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