Explanation Given: F ( θ ) = f m g cos θ + f sin θ Where, θ is the angle between the rope and the horizontal. f is the co-efficient of friction g = 9.8 m / s 2 f = 0.4 Calculation: F ( θ ) = f m g cos θ + f sin θ If the denominator in the above equation is maximum then we obtain the minimum value of F ( θ ) . Let, g ( θ ) = cos θ + f sin θ Differentiating the above equation and equating to zero, we get g ' ( θ ) = − sin θ + f cos θ − sin θ + f cos θ = 0 From the above, we obtain f = tan θ From the above relation, we can find that the angle θ lies between 0 and π 2 . Consider the triangle as shown in Figure. Let, the base be 1 and the height be f ...

Applied Calculus (with Infotrac) 3rd Edition By Waner, Stefan; Costenoble, Steven Published By Brooks Cole Hardcover

3rd Edition

ISBN: 9781319048518

Author: Colin Adams Jon Rogawski

Publisher: Macmillan Higher Education

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Question

Chapter 4.7, Problem 66E

To determine

Part (a)

To show that the minimum force required is proportional to $\frac{1}{\sqrt{1 + f^{2}}}$ .

Expert Solution & Answer

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Chapter 4.7, Problem 65E

Chapter 4.7, Problem 67E

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The OU process studied in the previous problem is a common model for interest rates. Another common model is the CIR model, which solves the SDE: dX₁ = (a = X₁) dt + σ √X+dWt, - under the condition Xoxo. We cannot solve this SDE explicitly. = (a) Use the Brownian trajectory simulated in part (a) of Problem 1, and the Euler scheme to simulate a trajectory of the CIR process. On a graph, represent both the trajectory of the OU process and the trajectory of the CIR process for the same Brownian path. (b) Repeat the simulation of the CIR process above M times (M large), for a large value of T, and use the result to estimate the long-term expectation and variance of the CIR process. How do they compare to the ones of the OU process? Numerical application: T = 10, N = 500, a = 0.04, x0 = 0.05, σ = 0.01, M = 1000. 1 (c) If you use larger values than above for the parameters, such as the ones in Problem 1, you may encounter errors when implementing the Euler scheme for CIR. Explain why.

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Chapter 4 Solutions

Applied Calculus (with Infotrac) 3rd Edition By Waner, Stefan; Costenoble, Steven Published By Brooks Cole Hardcover

Ch. 4.1 - Prob. 1PQ Ch. 4.1 - Prob. 2PQ Ch. 4.1 - Prob. 3PQ Ch. 4.1 - Prob. 4PQ Ch. 4.1 - Prob. 1E Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E Ch. 4.1 - Prob. 6E

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Knowledge Booster

Part (a) To show that the minimum force required is proportional to 1 1 + f 2 .

Part (a) To show that the minimum force required is proportional to 11+f2.

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Chapter 4 Solutions

Part (a)

To show that the minimum force required is proportional to $\frac{1}{\sqrt{1 + f^{2}}}$ .