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 4.3.48a
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
검증된 단계별 안내1
To find the local extrema of the function k(x) = x³ + 3x² + 3x + 1 on the domain (-∞, 0], we first need to find the critical points. This involves taking the derivative of k(x) and setting it equal to zero.
Calculate the derivative of k(x): k'(x) = 3x² + 6x + 3. This derivative will help us find the critical points where the slope of the tangent is zero or undefined.
Set the derivative equal to zero to find the critical points: 3x² + 6x + 3 = 0. Solve this quadratic equation for x to find the critical points within the given domain.
Use the quadratic formula x = (-b ± √(b² - 4ac)) / 2a, where a = 3, b = 6, and c = 3, to solve for x. This will give you the potential critical points.
Evaluate the function k(x) at the critical points and at the endpoint x = 0 to determine the local extrema. Compare these values to identify the local maximum and minimum values within the domain (-∞, 0].
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
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Critical Points
Critical points of a function occur where its derivative is zero or undefined. These points are potential locations for local extrema (maximum or minimum values). For the function k(x) = x³ + 3x² + 3x + 1, finding the derivative and solving k'(x) = 0 will help identify critical points within the given domain.
추천 영상:
04:50
Critical Points
First Derivative Test
The First Derivative Test helps determine whether a critical point is a local maximum, minimum, or neither. By analyzing the sign changes of the derivative around the critical points, we can infer the behavior of the function. If the derivative changes from positive to negative, the point is a local maximum; if it changes from negative to positive, it's a local minimum.
추천 영상:
07:09The First Derivative Test: Finding Local Extrema
Domain Consideration
Understanding the domain is crucial when identifying extrema, as it limits where extrema can occur. For k(x) = x³ + 3x² + 3x + 1, the domain is −∞ < x ≤ 0, meaning we only consider critical points and endpoints within this range. The endpoint x = 0 must also be evaluated to determine if it is an extremum.
추천 영상:
가이드 코스
5:10Finding the Domain and Range of a Graph
관련 실천
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 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
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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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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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