(a) To prove: S ( t ) = ∫ 0 t ‖ r ' ( u ) ‖ d u Is equal to S ( t ) = ln ( cosh ( t ) ) .

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Answer 1

Question

Chapter 14.3, Problem 43E

To determine

(a)

To prove:

$S (t) = \int_{0}^{t} ‖ r' (u) ‖ d u$ Is equal to $S (t) = \ln (\cosh (t))$ .

To determine

(b)

To prove:

$t = g (s) = \ln (e^{x} + \sqrt{e^{2 s} - 1})$ is an inverse of $S (t)$ , along with verification that

$r_{1} (s) = 〈 \tan h^{- 1} (\sqrt{1 - e^{- 2 s}}) - \sqrt{1 - e^{- 2 s}}, e^{- s} 〉$ is an arc length parameterization of the tractrix?

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9. a) Determie values of a and b so that the function is continuous. ax - 2b f(x) 2 x≤-2 -2x+a, x ≥2 \-ax² - bx + 1, −2 < x < 2) 9b) Consider f(x): = 2x²+x-3 x-b and determine all the values of b such that f(x) does not have a vertical asymptote. Show work.

Answer 2

Question

Chapter 14.3, Problem 43E

To determine

(a)

To prove:

$S (t) = \int_{0}^{t} ‖ r' (u) ‖ d u$ Is equal to $S (t) = \ln (\cosh (t))$ .

To determine

(b)

To prove:

$t = g (s) = \ln (e^{x} + \sqrt{e^{2 s} - 1})$ is an inverse of $S (t)$ , along with verification that

$r_{1} (s) = 〈 \tan h^{- 1} (\sqrt{1 - e^{- 2 s}}) - \sqrt{1 - e^{- 2 s}}, e^{- s} 〉$ is an arc length parameterization of the tractrix?

Expert Solution & Answer

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See solution

Answer 3

Question

Chapter 14.3, Problem 43E

To determine

(a)

To prove:

$S (t) = \int_{0}^{t} ‖ r' (u) ‖ d u$ Is equal to $S (t) = \ln (\cosh (t))$ .

To determine

(b)

To prove:

$t = g (s) = \ln (e^{x} + \sqrt{e^{2 s} - 1})$ is an inverse of $S (t)$ , along with verification that

$r_{1} (s) = 〈 \tan h^{- 1} (\sqrt{1 - e^{- 2 s}}) - \sqrt{1 - e^{- 2 s}}, e^{- s} 〉$ is an arc length parameterization of the tractrix?

Expert Solution & Answer

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Answer 4

Question

Chapter 14.3, Problem 43E

To determine

(a)

To prove:

$S (t) = \int_{0}^{t} ‖ r' (u) ‖ d u$ Is equal to $S (t) = \ln (\cosh (t))$ .

To determine

(b)

To prove:

$t = g (s) = \ln (e^{x} + \sqrt{e^{2 s} - 1})$ is an inverse of $S (t)$ , along with verification that

$r_{1} (s) = 〈 \tan h^{- 1} (\sqrt{1 - e^{- 2 s}}) - \sqrt{1 - e^{- 2 s}}, e^{- s} 〉$ is an arc length parameterization of the tractrix?

Expert Solution & Answer

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Answer 5

Chapter 14.3, Problem 43E

To determine

(a)

To prove:

$S (t) = \int_{0}^{t} ‖ r' (u) ‖ d u$ Is equal to $S (t) = \ln (\cosh (t))$ .

To determine

(b)

To prove:

$t = g (s) = \ln (e^{x} + \sqrt{e^{2 s} - 1})$ is an inverse of $S (t)$ , along with verification that

$r_{1} (s) = 〈 \tan h^{- 1} (\sqrt{1 - e^{- 2 s}}) - \sqrt{1 - e^{- 2 s}}, e^{- s} 〉$ is an arc length parameterization of the tractrix?

Answer 6

Chapter 14.3, Problem 43E

To determine

(a)

To prove:

$S (t) = \int_{0}^{t} ‖ r' (u) ‖ d u$ Is equal to $S (t) = \ln (\cosh (t))$ .

Answer 7

Chapter 14.3, Problem 43E

To determine

(a)

To prove:

$S (t) = \int_{0}^{t} ‖ r' (u) ‖ d u$ Is equal to $S (t) = \ln (\cosh (t))$ .

Answer 8

To determine

(b)

To prove:

$t = g (s) = \ln (e^{x} + \sqrt{e^{2 s} - 1})$ is an inverse of $S (t)$ , along with verification that

$r_{1} (s) = 〈 \tan h^{- 1} (\sqrt{1 - e^{- 2 s}}) - \sqrt{1 - e^{- 2 s}}, e^{- s} 〉$ is an arc length parameterization of the tractrix?

Answer 9

To determine

(b)

To prove:

$t = g (s) = \ln (e^{x} + \sqrt{e^{2 s} - 1})$ is an inverse of $S (t)$ , along with verification that

$r_{1} (s) = 〈 \tan h^{- 1} (\sqrt{1 - e^{- 2 s}}) - \sqrt{1 - e^{- 2 s}}, e^{- s} 〉$ is an arc length parameterization of the tractrix?

Answer 10

Expert Solution & Answer

Answer 11

Expert Solution & Answer

Answer 12

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Answer 13

Ch. 14.1 - Prob. 1PQ Ch. 14.1 - Prob. 2PQ Ch. 14.1 - Prob. 3PQ Ch. 14.1 - Prob. 4PQ Ch. 14.1 - Prob. 5PQ Ch. 14.1 - Prob. 6PQ Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E

(a) To prove: S ( t ) = ∫ 0 t ‖ r ' ( u ) ‖ d u Is equal to S ( t ) = ln ( cosh ( t ) ) .

Solution Summary: The author explains that r (t ) = t, sech, and tractrix, respectively, are equal to ln ( cosh).

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(a)

To prove:

$S (t) = \int_{0}^{t} ‖ r' (u) ‖ d u$ Is equal to $S (t) = \ln (\cosh (t))$ .

(b)

To prove:

$t = g (s) = \ln (e^{x} + \sqrt{e^{2 s} - 1})$ is an inverse of $S (t)$ , along with verification that

$r_{1} (s) = 〈 \tan h^{- 1} (\sqrt{1 - e^{- 2 s}}) - \sqrt{1 - e^{- 2 s}}, e^{- s} 〉$ is an arc length parameterization of the tractrix?

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Chapter 14 Solutions

(a) To prove: S ( t ) = ∫ 0 t ‖ r ' ( u ) ‖ d u Is equal to S ( t ) = ln ( cosh ( t ) ) .

Solution Summary: The author explains that r (t ) = t, sech, and tractrix, respectively, are equal to ln ( cosh).

Videos