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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 4 - Applications of Derivatives문제 4.3.60a
4장, 문제 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

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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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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.
추천 영상:
05:58
Finding Extrema Graphically

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.
추천 영상:
07:09
The First Derivative Test: Finding Local Extrema

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.
추천 영상:
06:35
Derivatives of Other Inverse Trigonometric Functions
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교과서 질문

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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교과서 질문

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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교과서 질문

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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교과서 질문

Identifying Extrema


In Exercises 19–40:


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


f(r) = 3r³ + 16r

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교과서 질문

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) = csc²x − 2cot x, 0 < x < π

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