Ch. 4 - Applications of Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 4 - Applications of Derivatives

Problem 4.1.7

Chapter 4, Problem 4.1.7

Finding Extrema from Graphs

In Exercises 7–10, find the absolute extreme values and where they occur.

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First, identify the endpoints of the graph. The graph has endpoints at (-1, 1) and (1, -1). These points are important because absolute extrema can occur at endpoints.

Next, observe the graph to identify any local extrema. Local extrema are points where the graph changes direction. In this graph, there is a local maximum at (0, 0) because the graph changes from increasing to decreasing.

Determine the y-values at the endpoints and the local extrema. At (-1, 1), the y-value is 1; at (0, 0), the y-value is 0; and at (1, -1), the y-value is -1.

Compare the y-values to find the absolute maximum and minimum. The absolute maximum is the highest y-value, which is 1 at (-1, 1). The absolute minimum is the lowest y-value, which is -1 at (1, -1).

Conclude that the absolute maximum value is 1 occurring at x = -1, and the absolute minimum value is -1 occurring at x = 1.

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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. Identifying these points involves evaluating the function at critical points and endpoints to determine the largest and smallest values.

Critical Points

Critical points are where the derivative of a function is zero or undefined, indicating potential local maxima, minima, or points of inflection. To find these points, take the derivative of the function and solve for where it equals zero or is undefined. These points are essential in determining where the function's slope changes, which helps in identifying extrema.

Graph Analysis

Graph analysis involves examining a graph to identify key features such as intercepts, slopes, and extrema. By analyzing the graph, one can visually determine where the function reaches its highest or lowest values. This process includes observing the behavior of the graph at endpoints and critical points, which is crucial for finding absolute extrema.

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Related Practice

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.

8. y = 2cosx - √2x, -π≤x≤3π/2

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Textbook Question

Finding Extrema from Graphs

In Exercises 11–14, match the table with a graph.

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Textbook Question

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.

82. y' = sin t, for 0 ≤ t ≤ 2π

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Textbook Question

99. In Exercises 99 and 100, the graph of f' is given. Determine x-values corresponding to inflection points for the graph of f.

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Textbook Question

Initial Value Problems

Solve the initial value problems in Exercises 71–90.

dr/dθ = −π sin (πθ), r(0) = 0

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