Ch. 4 - Applications of Derivatives

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

Hass 15th Edition

Ch. 4 - Applications of Derivatives

Problem 4.3.60a

Chapter 4, Problem 4.3.60a

Identifying Extrema

In Exercises 53–60:

a. Find the local extrema of each function on the given interval, and say where they occur.

f(x) = sec²x − 2tan x, −π/2 < x < π/2

Verified step by step guidance

First, understand that local extrema occur where the derivative of the function is zero or undefined. Begin by finding the derivative of the function f(x) = sec²x − 2tan x.

To find the derivative, use the chain rule and trigonometric identities. The derivative of sec²x is 2sec²x tan x, and the derivative of -2tan x is -2sec²x. Therefore, f'(x) = 2sec²x tan x - 2sec²x.

Set the derivative f'(x) = 0 to find critical points. This gives the equation 2sec²x tan x - 2sec²x = 0. Simplify this to find tan x = 1.

Solve tan x = 1 within the interval -π/2 < x < π/2. The solution is x = π/4, as tan(π/4) = 1.

Evaluate the function f(x) at x = π/4 and check the endpoints of the interval to determine the local extrema. Compare these values to identify the local maximum and minimum.

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Key Concepts

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

Local Extrema

Local extrema refer to the points in a function where it reaches a local maximum or minimum. These are points where the function changes direction, and they can be found by analyzing the first derivative. A local maximum is where the function changes from increasing to decreasing, and a local minimum is where it changes from decreasing to increasing.

First Derivative Test

The first derivative test is a method used to identify local extrema of a function. By taking the derivative of the function and finding its critical points (where the derivative is zero or undefined), we can determine where the function's slope changes sign. This change in sign indicates a local maximum or minimum, depending on the direction of the change.

Trigonometric Derivatives

Understanding the derivatives of trigonometric functions is crucial for solving problems involving functions like sec²x and tan x. The derivative of sec²x is 2sec²x tan x, and the derivative of tan x is sec²x. These derivatives help in finding critical points and analyzing the behavior of the function within the given interval.

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Derivatives of Other Inverse Trigonometric Functions

Related Practice

Textbook Question

Identifying Extrema

In Exercises 53–60:

a. Find the local extrema of each function on the given interval, and say where they occur.

f(x) = √3cos x + sin x, 0 ≤ x ≤ 2π

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Textbook Question

Identifying Extrema

In Exercises 53–60:

a. Find the local extrema of each function on the given interval, and say where they occur.

f(x) = sin x − cos x,0 ≤ x ≤ 2π

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Textbook Question

20.The U.S. Postal Service will accept a box for domestic shipment only if the sum of its length and girth (distance around) does not exceed 108 in. a.What dimensions will give a box with a square end the largest possible volume?

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Textbook Question

Identifying Extrema

In Exercises 41–52:

a. Identify the function’s local extreme values in the given domain, and say where they occur.

f(x) = (x + 1)², −∞ < x ≤ 0

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Textbook Question

Identifying Extrema