Textbook Question
Finding Extrema from Graphs
In Exercises 7–10, find the absolute extreme values and where they occur.
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 4.4.77
Each of Exercises 67–88 gives the first derivative of a continuous function y=f(x). Find y'' and then use Steps 2–4 of the graphing procedure described in this section to sketch the general shape of the graph of f.
77. y' = cot(θ/2), for 0 < θ < 2π
Verified step by step guidance1
Step 1: To find the second derivative y'', start by differentiating the given first derivative y' = cot(θ/2). Use the chain rule and the derivative of cot(x), which is -csc²(x). Specifically, differentiate cot(θ/2) with respect to θ.
Step 2: Apply the chain rule to account for the inner function θ/2. The derivative of θ/2 with respect to θ is 1/2. Combine this with the derivative of cot(θ/2) to find y''.
Step 3: Simplify the expression for y'' to obtain y'' = -(1/2)csc²(θ/2). This represents the second derivative of the function.
Step 4: Analyze the behavior of y'' to determine concavity. Recall that if y'' > 0, the graph of f is concave up, and if y'' < 0, the graph of f is concave down. Use the interval 0 < θ < 2π to identify regions of concavity based on the sign of y''.
Step 5: Combine the information from y' and y'' to sketch the general shape of the graph of f. Use the critical points (where y' = 0), inflection points (where y'' changes sign), and the concavity analysis to create a rough sketch of the graph.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
First Derivative
The first derivative of a function, denoted as y' or f'(x), represents the rate of change of the function with respect to its variable. It provides information about the slope of the tangent line to the graph of the function at any given point. Understanding the first derivative is crucial for determining critical points, where the function may have local maxima or minima.
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The First Derivative Test: Finding Local Extrema
Second Derivative
The second derivative, denoted as y'' or f''(x), is the derivative of the first derivative. It indicates the rate of change of the slope of the function, providing insights into the concavity of the graph. A positive second derivative suggests the graph is concave up, while a negative second derivative indicates concave down, which is essential for sketching the function's general shape.
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06:02The Second Derivative Test: Finding Local Extrema
Graphing Procedure
The graphing procedure involves analyzing the first and second derivatives to sketch the function's graph accurately. Steps typically include identifying critical points, determining intervals of increase or decrease, and assessing concavity. This systematic approach helps in visualizing the behavior of the function, including its turning points and inflection points, leading to a more complete understanding of its overall shape.
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06:15Graphing The Derivative
Related Practice
Textbook Question
Identifying Extrema
In Exercises 61 and 62, the graph of f' is given. Assume that f is continuous, and determine the x-values corresponding to local minima and local maxima.
Textbook Question
Identifying Extrema
In Exercises 61 and 62, the graph of f' is given. Assume that f is continuous, and determine the x-values corresponding to local minima and local maxima.
Textbook Question
Absolute Extrema on Finite Closed Intervals
In Exercises 21–36, find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
f(x) = (2/3)x − 5, −2 ≤ x ≤ 3
Textbook Question
Identify the inflection points and local maxima and minima of the functions graphed in Exercises 1–8. Identify the open intervals on which the functions are differentiable and the graphs are concave up and concave down.
2. y=x^4/4-2x^2+4
Textbook Question
Absolute Extrema on Finite Closed Intervals
In Exercises 21–36, find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
f(t) = 2 − |t|, −1 ≤ t ≤ 3