Ch. 4 - Applications of Derivatives

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

Hass 15th Edition

Ch. 4 - Applications of Derivatives

Problem 4.4.99

Chapter 4, Problem 4.4.99

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

Verified step by step guidance

Step 1: Recall that inflection points occur where the second derivative of a function changes sign. This means we need to analyze the graph of f' to determine where the slope of f' (which represents f'') changes from positive to negative or vice versa.

Step 2: Observe the graph of f'. The slope of f' changes sign at points where the graph of f' transitions from increasing to decreasing or decreasing to increasing. These are the x-values where the graph of f' has local maxima or minima.

Step 3: Identify the x-values of local maxima and minima on the graph of f'. From the graph, it appears that there are local maxima at x ≈ -2 and local minima at x ≈ 2.

Step 4: Verify that these points correspond to changes in the sign of the slope of f'. At x ≈ -2, the slope of f' changes from positive to negative, and at x ≈ 2, the slope of f' changes from negative to positive.

Step 5: Conclude that the x-values corresponding to inflection points for the graph of f are approximately x = -2 and x = 2.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivative and Its Graph

The derivative of a function, denoted as f', represents the rate of change of the function f. The graph of f' provides insights into the behavior of f, such as where it is increasing or decreasing. Points where f' changes from positive to negative indicate local maxima, while points where it changes from negative to positive indicate local minima.

Inflection Points

Inflection points occur where the concavity of a function changes, which can be identified by analyzing the second derivative, f''. However, when given the graph of f', inflection points correspond to where f' changes its increasing or decreasing behavior. This means that the x-values where f' has local extrema (maxima or minima) are potential inflection points for f.

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Related Practice

Textbook Question

113. If b, c, and d are constants, for what value of b will the curve y = x^3 + bx^2 + cx + d have a

point of inflection at x = 1? Give reasons for your answer.

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Textbook Question

Checking the Mean Value Theorem

Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.

f(x) = {2x − 3, 0 ≤ x ≤ 2

6x − x² − 7, 2 < x ≤ 3

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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

Finding Extrema from Graphs

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

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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

Initial Value Problems

Solve the initial value problems in Exercises 71–90.

dr/dθ = −π sin (πθ), r(0) = 0

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