교과서 질문
Theory and Examples
In Exercises 51 and 52, give reasons for your answers.
Let f(x) = (x − 2)²ᐟ³.
a. Does f′(2) exist?
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Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
4장, 문제 4.3.53a
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 2x, 0 ≤ x ≤ π
검증된 단계별 안내1
To find the local extrema of the function f(x) = sin(2x) on the interval [0, π], we first need to find the critical points. This involves taking the derivative of the function and setting it equal to zero.
The derivative of f(x) = sin(2x) is f'(x) = 2cos(2x). Set this equal to zero to find the critical points: 2cos(2x) = 0.
Solve the equation cos(2x) = 0. The solutions to this equation are 2x = π/2 + kπ, where k is an integer. Divide by 2 to solve for x: x = π/4 + kπ/2.
Determine which of these critical points lie within the interval [0, π]. For k = 0, x = π/4; for k = 1, x = 3π/4. Both are within the interval.
Evaluate the function f(x) at the critical points and the endpoints of the interval, x = 0 and x = π, to determine the local extrema. Compare these values to identify the local maxima and minima.
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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Local Extrema
Local extrema refer to the points in a function where it reaches a local maximum or minimum value. These points occur where the derivative of the function is zero or undefined, indicating a change in the direction of the function's slope. Identifying local extrema involves finding these critical points and evaluating the function's behavior around them.
추천 영상:
05:58
Finding Extrema Graphically
Derivative and Critical Points
The derivative of a function provides the rate of change or slope of the function at any given point. Critical points occur where the derivative is zero or undefined, which are potential locations for local extrema. To find these points, differentiate the function and solve for the values of x that satisfy these conditions within the given interval.
추천 영상:
04:50Critical Points
Trigonometric Functions and Their Derivatives
Understanding the behavior of trigonometric functions like sine is crucial for solving problems involving them. The derivative of sin(2x) is 2cos(2x), which helps in finding critical points. Analyzing the sine and cosine functions over the interval [0, π] allows us to determine where the function increases or decreases, aiding in identifying local extrema.
추천 영상:
가이드 코스
6:04Introduction to Trigonometric Functions
관련 실천
교과서 질문
52. Two masses hanging side by side from springs have positions s_1 = 2 sin t and s_2 = sin 2t,
respectively.
a. At what times in the interval 0 < t do the masses pass each other? (Hint: sin 2t = 2 sint cost.)
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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) = x / 2 − 2sin (x/2), 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.
k(x) = x³ + 3x² + 3x + 1, −∞ < x ≤ 0
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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
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
1 / x²
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