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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

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.

Recommended video:

5:53

Graph of Sine and Cosine Function

Related Practice

Textbook Question

Business and Economics

62. Production level Suppose that c(x)=x^3-20x^2 + 20,000x is the cost of manufacturing x items. Find a production level that will minimize the average cost of making x items.

182

views

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

218

views

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]

158

views

Textbook Question

The 8-ft wall shown here stands 27 ft from the building. Find the length of the shortest straight beam that will reach to the side of the building from the ground outside the wall.

211

views

Textbook Question

Checking Antiderivative Formulas

Verify the formulas in Exercises 57–62 by differentiation.

∫csc²((x − 1)/3)dx = −3cot((x − 1)/3) + C

views

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.

∫2x(1 − x⁻³) dx

views