교과서 질문
Use formal definitions to prove the limit statements in Exercises 93–96.
lim x → 0 (−1 / x²) = −∞
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Ch. 2 - Limits and Continuity
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- 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
2장, 문제 2.5.69
Use the Intermediate Value Theorem in Exercises 69–74 to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations.
x³ − 3x − 1 = 0
검증된 단계별 안내1
Identify the function f(x) = x³ − 3x − 1 and note that it is a polynomial, which is continuous everywhere.
Choose an interval [a, b] where the function changes sign, indicating a root exists. For example, test f(0) and f(2).
Calculate f(0) = 0³ − 3(0) − 1 = -1 and f(2) = 2³ − 3(2) − 1 = 3. Since f(0) < 0 and f(2) > 0, there is a sign change.
Apply the Intermediate Value Theorem: Since f(x) is continuous on [0, 2] and f(0) < 0 < f(2), there exists at least one c in (0, 2) such that f(c) = 0.
Use a graphing calculator or computer grapher to approximate the root of the equation x³ − 3x − 1 = 0 within the interval [0, 2].
비슷한 문제에 대한 검증된 영상 답변:
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.
Intermediate Value Theorem
The Intermediate Value Theorem states that if a continuous function f(x) takes on values f(a) and f(b) at two points a and b, and if f(a) and f(b) have opposite signs, then there exists at least one c in the interval (a, b) such that f(c) = 0. This theorem is crucial for proving the existence of a solution to the equation x³ − 3x − 1 = 0.
추천 영상:
06:11
Fundamental Theorem of Calculus Part 1
Continuous Functions
A function is continuous if there are no breaks, jumps, or holes in its graph over its domain. For the Intermediate Value Theorem to apply, the function x³ − 3x − 1 must be continuous over the interval being considered. Polynomial functions, like this one, are continuous everywhere, which allows us to use the theorem effectively.
추천 영상:
05:34Intro to Continuity
Graphical Analysis
Graphical analysis involves using a graphing calculator or computer software to visualize the function and identify where it crosses the x-axis, indicating a root. By graphing x³ − 3x − 1, we can visually confirm the existence of a solution and approximate its value, complementing the theoretical proof provided by the Intermediate Value Theorem.
추천 영상:
05:02Determining Differentiability Graphically
관련 실천
교과서 질문
Domains and Asymptotes
Determine the domain of each function in Exercises 69–72. Then use various limits to find the asymptotes.
y = (√(x² + 4)) / x
288
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교과서 질문
Infinite Limits
Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.
lim x→0 (−1) / (x² (x + 1))
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교과서 질문
Slope of a Curve at a Point
In Exercises 7–18, use the method in Example 3 to find (a) the slope of the curve at the given point P, and (b) an equation of the tangent line at P.
y=x³−3x²+4, P(2,0)
562
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교과서 질문
Limits of Rational Functions
In Exercises 13–22, find the limit of each rational function (a) as x → ∞ and (b) as x → −∞. Write ∞ or −∞ where appropriate.
f(x) = (2x + 3)/(5x + 7)
414
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교과서 질문
Removable discontinuity Give an example of a function f (x) that is continuous for all values of x except x = 2, where it has a removable discontinuity. Explain how you know that f is discontinuous at x = 2, and how you know the discontinuity is removable.
499
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