Textbook Question
A cone is formed from a circular piece of material of radius 1 meter by removing a section of angle θ and then joining the two straight edges. Determine the largest possible volume for the cone.
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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 4.1.9
Finding Extrema from Graphs
In Exercises 7–10, find the absolute extreme values and where they occur.
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Identify the endpoints of the graph. The graph starts at the point (0, 5) and ends at the point (2, 0).
Determine the type of extrema by analyzing the graph. The graph is a straight line decreasing from (0, 5) to (2, 0).
The absolute maximum value occurs at the highest point on the graph, which is at (0, 5).
The absolute minimum value occurs at the lowest point on the graph, which is at (2, 0).
Verify that there are no other points on the graph that exceed the y-values at the endpoints, confirming the absolute extrema.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Absolute Extrema
Absolute extrema refer to the highest or lowest points on a graph over a given interval. The absolute maximum is the highest point, while the absolute minimum is the lowest. These points can occur at critical points or endpoints of the interval. In the context of the graph, identifying these points involves examining the y-values at the endpoints and any critical points within the interval.
Recommended video:
05:58
Finding Extrema Graphically
Endpoints of a Graph
Endpoints are the points at the boundaries of a graph's domain. They are crucial when determining absolute extrema, as extrema can occur at these points. In the given graph, the endpoints are at (2, 5) and (2, 0). Evaluating the function's value at these points helps in identifying the absolute maximum and minimum values.
Recommended video:
06:15Graphing The Derivative
Critical Points
Critical points are where the derivative of a function is zero or undefined, indicating potential local maxima or minima. However, in a linear segment like the one shown, there are no critical points within the interval, as the slope is constant. Thus, for this graph, the focus is on the endpoints to find the absolute extrema.
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04:50Critical Points
Related Practice
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.
∫x⁻¹ᐟ³ dx
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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.
∫(8y − 2 / y¹ᐟ⁴) dy
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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.
∫(1 + tan²θ)dθ (Hint:1 + tan²θ = sec²θ)
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Textbook Question
Roots (Zeros)
Show that the functions in Exercises 19–26 have exactly one zero in the given interval.
f(x) = x⁴ + 3x + 1, [−2, −1]
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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.
h(x) = ³√x, −1 ≤ x ≤ 8
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