“The Simpson’s Rule states that ∫ a b f ( x ) d x ≈ _ _ _ _ _ _ where Δ x = b − a n and n is ______. The error E n in approximating ∫ a b f ( x ) d x by the Simpson’s Rule satisfies | E n | ≤ _ _ _ _ ”.

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Answer 1

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Chapter 7, Problem 4CRQ

To determine

To fill: “The Simpson’s Rule states that ∫abf(x)dx≈ where Δx=b−an and n is . The error E_n in approximating ∫abf(x)dx by the Simpson’s Rule satisfies |En|≤____ ”.

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1. Abel's Theorem. The goal in this problem is to prove Abel's theorem by following a series of steps (each step must be justified). Theorem 0.1 (Abel's Theorem). If y1 and y2 are solutions of the differential equation y" + p(t) y′ + q(t) y = 0, where p and q are continuous on an open interval, then the Wronskian is given by W (¥1, v2)(t) = c exp(− [p(t) dt), where C is a constant that does not depend on t. Moreover, either W (y1, y2)(t) = 0 for every t in I or W (y1, y2)(t) = 0 for every t in I. 1. (a) From the two equations (which follow from the hypotheses), show that y" + p(t) y₁ + q(t) y₁ = 0 and y½ + p(t) y2 + q(t) y2 = 0, 2. (b) Observe that Hence, conclude that (YY2 - Y1 y2) + P(t) (y₁ Y2 - Y1 Y2) = 0. W'(y1, y2)(t) = yY2 - Y1 y2- W' + p(t) W = 0. 3. (c) Use the result from the previous step to complete the proof of the theorem.

2. Observations on the Wronskian. Suppose the functions y₁ and y2 are solutions to the differential equation p(x)y" + q(x)y' + r(x) y = 0 on an open interval I. 1. (a) Prove that if y₁ and y2 both vanish at the same point in I, then y₁ and y2 cannot form a fundamental set of solutions. 2. (b) Prove that if y₁ and y2 both attain a maximum or minimum at the same point in I, then y₁ and Y2 cannot form a fundamental set of solutions. 3. (c) show that the functions & and t² are linearly independent on the interval (−1, 1). Verify that both are solutions to the differential equation t² y″ – 2ty' + 2y = 0. Then justify why this does not contradict Abel's theorem. 4. (d) What can you conclude about the possibility that t and t² are solutions to the differential equation y" + q(x) y′ + r(x)y = 0?

Question 4 Find an equation of (a) The plane through the point (2, 0, 1) and perpendicular to the line x = y=2-t, z=3+4t. 3t, (b) The plane through the point (3, −2, 8) and parallel to the plane z = x+y. (c) The plane that contains the line x = 1+t, y = 2 − t, z = 4 - 3t and is parallel to the plane 5x + 2y + z = 1. (d) The plane that passes through the point (1,2,3) and contains the line x = 3t, y = 1+t, and z = 2-t. (e) The plane that contains the lines L₁: x = 1 + t, y = 1 − t, z = 2t and L2 : x = 2 − s, y = s, z = 2.

Answer 2

Question

Chapter 7, Problem 4CRQ

To determine

To fill: “The Simpson’s Rule states that ∫abf(x)dx≈ where Δx=b−an and n is . The error E_n in approximating ∫abf(x)dx by the Simpson’s Rule satisfies |En|≤____ ”.

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Answer 3

Question

Chapter 7, Problem 4CRQ

To determine

To fill: “The Simpson’s Rule states that ∫abf(x)dx≈ where Δx=b−an and n is . The error E_n in approximating ∫abf(x)dx by the Simpson’s Rule satisfies |En|≤____ ”.

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Answer 4

Question

Chapter 7, Problem 4CRQ

To determine

To fill: “The Simpson’s Rule states that ∫abf(x)dx≈ where Δx=b−an and n is . The error E_n in approximating ∫abf(x)dx by the Simpson’s Rule satisfies |En|≤____ ”.

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Answer 5

Chapter 7, Problem 4CRQ

To determine

To fill: “The Simpson’s Rule states that ∫abf(x)dx≈ where Δx=b−an and n is . The error E_n in approximating ∫abf(x)dx by the Simpson’s Rule satisfies |En|≤____ ”.

Answer 6

Chapter 7, Problem 4CRQ

To determine

To fill: “The Simpson’s Rule states that ∫abf(x)dx≈ where Δx=b−an and n is . The error E_n in approximating ∫abf(x)dx by the Simpson’s Rule satisfies |En|≤____ ”.

Answer 7

Chapter 7, Problem 4CRQ

To determine

To fill: “The Simpson’s Rule states that ∫abf(x)dx≈ where Δx=b−an and n is . The error E_n in approximating ∫abf(x)dx by the Simpson’s Rule satisfies |En|≤____ ”.

Answer 8

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Answer 9

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Answer 10

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Answer 11

Ch. 7.1 - Write the formula for integration by parts.Ch. 7.1 - Prob. 2CQ Ch. 7.1 - Prob. 1E Ch. 7.1 - Prob. 2E Ch. 7.1 - In Exercises 1-26, find each indefinite integral....Ch. 7.1 - Prob. 4E Ch. 7.1 - In Exercises 1-26, find each indefinite integral....Ch. 7.1 - Prob. 6E Ch. 7.1 - In Exercises 1-26, find each indefinite integral....Ch. 7.1 - Prob. 8E

“The Simpson’s Rule states that ∫ a b f ( x ) d x ≈ _ _ _ _ _ _ where Δ x = b − a n and n is ______. The error E n in approximating ∫ a b f ( x ) d x by the Simpson’s Rule satisfies | E n | ≤ _ _ _ _ ”.

Solution Summary: The author explains how the Simpson's Rule satisfies left|E_nright|

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To fill: “The Simpson’s Rule states that ∫abf(x)dx≈ where Δx=b−an and n is . The error E_n in approximating ∫abf(x)dx by the Simpson’s Rule satisfies |En|≤____ ”.

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Chapter 7 Solutions

“The Simpson’s Rule states that ∫ a b f ( x ) d x ≈ _ _ _ _ _ _ where Δ x = b − a n and n is ______. The error E n in approximating ∫ a b f ( x ) d x by the Simpson’s Rule satisfies | E n | ≤ _ _ _ _ ”.

Solution Summary: The author explains how the Simpson's Rule satisfies left|E_nright|

Concept explainers

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To fill: “The Simpson’s Rule states that ∫abf(x)dx≈______ where Δx=b−an and n is ______. The error En in approximating ∫abf(x)dx by the Simpson’s Rule satisfies |En|≤____ ”.

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Chapter 7 Solutions

To fill: “The Simpson’s Rule states that ∫abf(x)dx≈ where Δx=b−an and n is . The error E_n in approximating ∫abf(x)dx by the Simpson’s Rule satisfies |En|≤____ ”.