Ch. 4 - Applications of Derivatives

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

Hass 15th Edition

Ch. 4 - Applications of Derivatives

Problem 4.1.29

Chapter 4, Problem 4.1.29

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.

g(x) = √(4 − x²), −2 ≤ x ≤ 1

Verified step by step guidance

First, understand that the function g(x) = √(4 − x²) is defined for the interval −2 ≤ x ≤ 1. This is because the expression under the square root, 4 − x², must be non-negative.

Next, identify the critical points of the function within the interval. To do this, find the derivative of g(x) with respect to x. The derivative is g'(x) = -x / √(4 − x²). Set g'(x) = 0 to find critical points, which gives x = 0.

Evaluate the function g(x) at the critical point and at the endpoints of the interval. Calculate g(−2), g(0), and g(1) to determine the function values at these points.

Compare the values obtained in the previous step to identify the absolute maximum and minimum values of the function on the interval. The largest value is the absolute maximum, and the smallest value is the absolute minimum.

Finally, graph the function g(x) = √(4 − x²) over the interval −2 ≤ x ≤ 1. Mark the points where the absolute extrema occur on the graph, and label these points with their coordinates.

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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 and lowest values a function attains on a given interval. To find these, evaluate the function at critical points and endpoints of the interval. The largest value is the absolute maximum, and the smallest is the absolute minimum.

Critical Points

Critical points occur where the derivative of a function is zero or undefined. These points are potential locations for local extrema. For the function g(x) = √(4 − x²), find the derivative, set it to zero, and solve for x to identify critical points within the interval.

Graphing Functions

Graphing a function involves plotting its values over a specified interval to visually identify features like extrema. For g(x) = √(4 − x²), plot points within the interval [-2, 1] and mark the coordinates of the absolute maximum and minimum to illustrate where these extrema occur.

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Related Practice

Textbook Question

110. Suppose the derivative of the function y = f(x) is

y'=(x-1)^22(x-2)(x-4).

At what points, if any, does the graph of f have a local minimum, local maximum, or

point of inflection?

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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) = {x² − x, −2 ≤ x ≤−1

2x² − 3x − 3, −1 < x ≤ 0

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Textbook Question

56. Airplane landing path An airplane is flying at altitude H when it begins its descent to an airport runway that is at horizontal ground distance L from the airplane, as shown in the accompanying figure. Assume that the landing path of the airplane is the graph of a cubic polynomial function y = ax^3+bx^2+cx+d, where y(-L)= H and y(0)=0.

a. What is dy/dx at x = 0?

b. What is dy/dx at x = -L?

c. Use the values for dy/dx at x = 0 and x =- L together with y(0) = 0 and y(-L) = H to show that y(x)=H[2(x/L)^3+3(x/L)^2]

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Textbook Question

Identifying Extrema

In Exercises 19–40:

a. Find the open intervals on which the function is increasing and those on which it is decreasing.

b. Identify the function’s local extreme values, if any, saying where they occur.

f(x) = x − 6√(x − 1)

Textbook Question

Roots (Zeros)

Show that the functions in Exercises 19–26 have exactly one zero in the given interval.

g(t) = √t + √(1 + t) − 4, (0, ∞)

Textbook Question

Identifying Extrema

In Exercises 19–40:

a. Find the open intervals on which the function is increasing and those on which it is decreasing.

b. Identify the function’s local extreme values, if any, saying where they occur.

g(x) = x√8 − x²