교과서 질문
Find the limits in Exercises 49–52. Write ∞ or −∞ where appropriate.
lim x→(−π/2)⁺ sec x
226
views
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 55
[Technology Exercise] Grinding engine cylinders Before contracting to grind engine cylinders to a cross-sectional area of 9in², you need to know how much deviation from the ideal cylinder diameter of c = 3.385in. you can allow and still have the area come within 0.01in² of the required 9in². To find out, you let A=π(x/2)² and look for the largest interval in which you must hold x to make |A − 9| ≤ 0.01. What interval do you find?
검증된 단계별 안내1
Start by understanding the problem: You need to find the interval for the diameter x such that the area A of the cylinder's cross-section is within 0.01 in² of 9 in². The formula for the area A is given by A = π(x/2)².
Set up the inequality |A - 9| ≤ 0.01. Substitute the expression for A into the inequality: |π(x/2)² - 9| ≤ 0.01.
Solve the inequality: First, express the inequality without the absolute value by considering two cases: π(x/2)² - 9 ≤ 0.01 and π(x/2)² - 9 ≥ -0.01.
For the first case, π(x/2)² - 9 ≤ 0.01, solve for x by isolating x: π(x/2)² ≤ 9.01. Then, divide both sides by π and take the square root to find x.
For the second case, π(x/2)² - 9 ≥ -0.01, solve for x similarly: π(x/2)² ≥ 8.99. Divide both sides by π and take the square root to find x. Combine the results from both cases to determine the interval for x.
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.
Area of a Circle
The area of a circle is calculated using the formula A = π(r²), where r is the radius. In this context, the radius is half of the diameter (x/2). Understanding this formula is crucial for determining how changes in the diameter affect the area, which is central to solving the problem.
추천 영상:
05:06
Finding Area When Bounds Are Not Given
Deviation and Tolerance
Deviation refers to the difference between a measured value and a standard or ideal value. In this problem, the tolerance is set at 0.01 in², meaning the area must remain within 0.01 in² of the target area of 9 in². This concept is essential for establishing the acceptable range of values for the diameter.
Inequalities in Calculus
Inequalities are mathematical expressions that show the relationship between two values when they are not equal. In this case, the inequality |A - 9| ≤ 0.01 is used to find the range of diameters that keep the area within the specified limits. Understanding how to manipulate and solve inequalities is key to finding the required interval for x.
추천 영상:
06:11Fundamental Theorem of Calculus Part 1
관련 실천
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Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.
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