교과서 질문
Using the Formal Definition
Prove the limit statements in Exercises 37–50.
limx →4 (9 − x) = 5
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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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
2장, 문제 2.5.72
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 − 15x + 1 = 0 (three roots)
검증된 단계별 안내1
Understand the Intermediate Value Theorem (IVT): It states that if a continuous function f(x) takes on values of opposite signs at two points a and b, then there exists at least one c in the interval (a, b) such that f(c) = 0.
Identify the function: Here, the function is f(x) = x³ − 15x + 1. This is a polynomial function, which is continuous everywhere.
Choose appropriate intervals: To apply the IVT, we need to find intervals where the function changes sign. Start by evaluating f(x) at various points to find intervals where f(x) changes from positive to negative or vice versa.
Evaluate the function at chosen points: Calculate f(x) at several points, such as f(-3), f(0), f(3), etc., to find intervals where the sign changes. For example, if f(-3) < 0 and f(0) > 0, then there is a root in the interval (-3, 0). Repeat this process to find all intervals where the function changes sign.
Use a graphing calculator or computer grapher: Once you have identified the intervals using the IVT, use a graphing tool to visualize the function and find the approximate values of the roots within those intervals.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
2m도움이 되었나요?
주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Intermediate Value Theorem
The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b] and takes on different signs at the endpoints (f(a) and f(b)), then there exists at least one c in (a, b) such that f(c) = 0. This theorem is fundamental for proving the existence of roots in continuous functions.
추천 영상:
06:11
Fundamental Theorem of Calculus Part 1
Continuous Functions
A continuous function is one where small changes in the input result in small changes in the output. Formally, a function f is continuous at a point x = c if the limit of f as x approaches c equals f(c). Understanding continuity is crucial for applying the Intermediate Value Theorem effectively.
추천 영상:
05:34Intro to Continuity
Graphing Techniques
Graphing techniques involve using tools like graphing calculators or software to visualize functions. By plotting the function, one can identify where it crosses the x-axis, indicating the roots of the equation. This visual approach complements analytical methods like the Intermediate Value Theorem in finding solutions.
추천 영상:
06:15Graphing The Derivative
관련 실천
교과서 질문
Limits as x → ∞ or x → −∞
The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x. Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36. Write ∞ or −∞ where appropriate.
lim x→∞ (x − 3) / √(4x² + 25)
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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) = (x + 1)/(x² + 3)
365
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교과서 질문
Finding Limits of Differences When x → ±∞
Find the limits in Exercises 84–90. (Hint: Try multiplying and dividing by the conjugate.)
lim x → ∞ (√(x + 9) − √(x + 4))
325
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교과서 질문
Using the Formal Definition
Prove the limit statements in Exercises 37–50.
limx→1 f(x) = 1 if f(x) = {x², x ≠ 1
2, x = 1
252
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교과서 질문
Limits of Average Rates of Change
Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form limh→0 (f(x+h) − f(x)) / h occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function f.
f(x) = x², x = 1
333
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