Calculus
Calculus
5th Edition
ISBN: 9781429241861
Author: Laura Taalman, Peter Kohn
Publisher: W. H. Freeman
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Chapter 1, Problem 30LRIF
To determine

To identify:The limit form is indeterminate or not indeterminate, and describe the behavior of a limit of not indeterminant form.

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4. Which substitution would you use to simplify the following integrand? S a) x = sin b) x = 2 tan 0 c) x = 2 sec 3√√3 3 x3 5. After making the substitution x = = tan 0, the definite integral 2 2 3 a) ៖ ស្លឺ sin s π - dᎾ 16 0 cos20 b) 2/4 10 cos 20 π sin30 6 - dᎾ c) Π 1 cos³0 3 · de 16 0 sin20 1 x²√x²+4 3 (4x²+9)2 π d) cos²8 16 0 sin³0 dx d) x = tan 0 dx simplifies to: de 6. In order to evaluate (tan 5xsec7xdx, which would be the most appropriate strategy? a) Separate a sec²x factor b) Separate a tan²x factor c) Separate a tan xsecx factor 7. Evaluate 3x x+4 - dx 1 a) 3x+41nx + 4 + C b) 31n|x + 4 + C c) 3 ln x + 4+ C d) 3x - 12 In|x + 4| + C x+4
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?

Chapter 1 Solutions

Calculus

Ch. 1.1 - Prob. 8CCh. 1.1 - Prob. 9CCh. 1.1 - Prob. 10CCh. 1.1 - Prob. 11CCh. 1.1 - Prob. 12CCh. 1.1 - Prob. 13CCh. 1.1 - Prob. 14CCh. 1.1 - Prob. 15CCh. 1.1 - Prob. 16CCh. 1.1 - Prob. 17CCh. 1.1 - Prob. 18CCh. 1.1 - Prob. 19CCh. 1.1 - Prob. 20CCh. 1.1 - Prob. 21SCh. 1.1 - Prob. 22SCh. 1.1 - Prob. 23SCh. 1.1 - Prob. 24SCh. 1.1 - Prob. 25SCh. 1.1 - Prob. 26SCh. 1.1 - Prob. 27SCh. 1.1 - Prob. 28SCh. 1.1 - Prob. 29SCh. 1.1 - Prob. 30SCh. 1.1 - Prob. 31SCh. 1.1 - Prob. 32SCh. 1.1 - Prob. 33SCh. 1.1 - Prob. 34SCh. 1.1 - Prob. 35SCh. 1.1 - Prob. 36SCh. 1.1 - Prob. 37SCh. 1.1 - Prob. 38SCh. 1.1 - Prob. 39SCh. 1.1 - Prob. 40SCh. 1.1 - Prob. 41SCh. 1.1 - Prob. 42SCh. 1.1 - Prob. 43SCh. 1.1 - Prob. 44SCh. 1.1 - Prob. 45SCh. 1.1 - Prob. 46SCh. 1.1 - Prob. 47SCh. 1.1 - Prob. 48SCh. 1.1 - Prob. 49SCh. 1.1 - Prob. 50SCh. 1.1 - Prob. 51SCh. 1.1 - Prob. 52SCh. 1.1 - Prob. 53SCh. 1.1 - Prob. 54SCh. 1.1 - Prob. 55SCh. 1.1 - Prob. 56SCh. 1.1 - Prob. 57SCh. 1.1 - Prob. 58SCh. 1.1 - Prob. 59SCh. 1.1 - Prob. 60SCh. 1.1 - 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Prob. 34SCh. 1.4 - Prob. 35SCh. 1.4 - Prob. 36SCh. 1.4 - Prob. 37SCh. 1.4 - Prob. 38SCh. 1.4 - Prob. 39SCh. 1.4 - Prob. 40SCh. 1.4 - Prob. 41SCh. 1.4 - Prob. 42SCh. 1.4 - Prob. 43SCh. 1.4 - Prob. 44SCh. 1.4 - Prob. 45SCh. 1.4 - Prob. 46SCh. 1.4 - Prob. 47SCh. 1.4 - Prob. 48SCh. 1.4 - Prob. 49SCh. 1.4 - Prob. 50SCh. 1.4 - Prob. 51SCh. 1.4 - Prob. 52SCh. 1.4 - Prob. 53SCh. 1.4 - Prob. 54SCh. 1.4 - Prob. 55SCh. 1.4 - Prob. 56SCh. 1.4 - Prob. 57SCh. 1.4 - Prob. 58SCh. 1.4 - Prob. 59SCh. 1.4 - Prob. 60SCh. 1.4 - Prob. 61SCh. 1.4 - Prob. 62SCh. 1.4 - Prob. 63SCh. 1.4 - Prob. 64SCh. 1.4 - Prob. 65SCh. 1.4 - Prob. 66SCh. 1.4 - Prob. 67SCh. 1.4 - Prob. 68SCh. 1.4 - Prob. 69SCh. 1.4 - Prob. 70SCh. 1.4 - Prob. 71SCh. 1.4 - Prob. 72SCh. 1.4 - Prob. 73SCh. 1.4 - Prob. 74SCh. 1.4 - Prob. 75ACh. 1.4 - Prob. 76ACh. 1.4 - Prob. 77ACh. 1.4 - Prob. 78ACh. 1.4 - Prob. 79ACh. 1.4 - Prob. 80ACh. 1.4 - Prob. 81PCh. 1.4 - Prob. 82PCh. 1.4 - Prob. 83PCh. 1.4 - Prob. 84PCh. 1.4 - Prob. 85PCh. 1.4 - Prob. 86PCh. 1.4 - Prob. 87PCh. 1.4 - Prob. 88PCh. 1.4 - Prob. 89PCh. 1.4 - Prob. 90PCh. 1.4 - Prob. 91PCh. 1.4 - Prob. 92PCh. 1.4 - Prob. 93PCh. 1.4 - Prob. 1TFCh. 1.4 - Prob. 2TFCh. 1.4 - Prob. 3TFCh. 1.4 - Prob. 4TFCh. 1.4 - Prob. 5TFCh. 1.5 - Prob. 1TBCh. 1.5 - Prob. 2TBCh. 1.5 - Prob. 0CCh. 1.5 - Prob. 1CCh. 1.5 - Prob. 2CCh. 1.5 - Prob. 3CCh. 1.5 - Prob. 4CCh. 1.5 - Prob. 5CCh. 1.5 - Prob. 6CCh. 1.5 - Prob. 7CCh. 1.5 - Prob. 8CCh. 1.5 - Prob. 9CCh. 1.5 - Prob. 10CCh. 1.5 - Prob. 11CCh. 1.5 - Prob. 12CCh. 1.5 - Prob. 13CCh. 1.5 - Prob. 14CCh. 1.5 - Prob. 15CCh. 1.5 - Prob. 16CCh. 1.5 - Prob. 17CCh. 1.5 - Prob. 18CCh. 1.5 - Prob. 19CCh. 1.5 - Prob. 20CCh. 1.5 - Prob. 21CCh. 1.5 - Prob. 22CCh. 1.5 - Prob. 23CCh. 1.5 - Prob. 24CCh. 1.5 - Prob. 25SCh. 1.5 - Prob. 26SCh. 1.5 - Prob. 27SCh. 1.5 - Prob. 28SCh. 1.5 - Prob. 29SCh. 1.5 - Prob. 30SCh. 1.5 - Prob. 31SCh. 1.5 - Prob. 32SCh. 1.5 - Prob. 33SCh. 1.5 - Prob. 34SCh. 1.5 - Prob. 35SCh. 1.5 - Prob. 36SCh. 1.5 - Prob. 37SCh. 1.5 - Prob. 38SCh. 1.5 - Prob. 39SCh. 1.5 - Prob. 40SCh. 1.5 - Prob. 41SCh. 1.5 - Prob. 42SCh. 1.5 - Prob. 43SCh. 1.5 - Prob. 44SCh. 1.5 - Prob. 45SCh. 1.5 - Prob. 46SCh. 1.5 - Prob. 47SCh. 1.5 - Prob. 48SCh. 1.5 - Prob. 49SCh. 1.5 - Prob. 50SCh. 1.5 - Prob. 51SCh. 1.5 - Prob. 52SCh. 1.5 - Prob. 53SCh. 1.5 - Prob. 54SCh. 1.5 - Prob. 55SCh. 1.5 - Prob. 56SCh. 1.5 - Prob. 57SCh. 1.5 - Prob. 58SCh. 1.5 - Prob. 59SCh. 1.5 - Prob. 60SCh. 1.5 - Prob. 61SCh. 1.5 - Prob. 62SCh. 1.5 - Prob. 63SCh. 1.5 - Prob. 64SCh. 1.5 - Prob. 65SCh. 1.5 - Prob. 66SCh. 1.5 - Prob. 67SCh. 1.5 - Prob. 68SCh. 1.5 - Prob. 69SCh. 1.5 - Prob. 70SCh. 1.5 - Prob. 71SCh. 1.5 - Prob. 72SCh. 1.5 - Prob. 73SCh. 1.5 - Prob. 74SCh. 1.5 - Prob. 75SCh. 1.5 - Prob. 76SCh. 1.5 - Prob. 77SCh. 1.5 - Prob. 78SCh. 1.5 - Prob. 79SCh. 1.5 - Prob. 80SCh. 1.5 - Prob. 81SCh. 1.5 - Prob. 82SCh. 1.5 - Prob. 83SCh. 1.5 - Prob. 84SCh. 1.5 - Prob. 85SCh. 1.5 - Prob. 86SCh. 1.5 - Prob. 87ACh. 1.5 - Prob. 88ACh. 1.5 - Prob. 89PCh. 1.5 - Prob. 90PCh. 1.5 - Prob. 91PCh. 1.5 - Prob. 92PCh. 1.5 - 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ACPCh. 1 - Prob. BCPCh. 1 - Prob. CCPCh. 1 - Prob. DCP
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