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### hw2

Course: MATH 471, Winter 2008
School: Michigan
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Word Count: 420

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2 Math Homework 471, Fall 2007 Assigned: Friday, September 14, 2007 Due: Friday, September 21, 2007 Include a cover page Clearly label all plots using title, xlabel, ylabel, legend Use the subplot command to compare multiple plots Include printouts of all Matlab code, labeled with your name, date, section, etc. (1) (Rootfinding and Optimization) (a) Suppose f (x) is differentiable on [a, b]. Discuss how you...

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2 Math Homework 471, Fall 2007 Assigned: Friday, September 14, 2007 Due: Friday, September 21, 2007 Include a cover page Clearly label all plots using title, xlabel, ylabel, legend Use the subplot command to compare multiple plots Include printouts of all Matlab code, labeled with your name, date, section, etc. (1) (Rootfinding and Optimization) (a) Suppose f (x) is differentiable on [a, b]. Discuss how you might use a rootfinding method to identify a local extremum of f (x) inside [a, b]. (b) Let f (x) = log x - sin x. Prove that f (x) has a unique maximum in the interval [4, 6]. (Note that log means natural logarithm.) (c) Approximate this local maximum using six iterations of the enclosure methods (Bisection and False Position) with starting interval [4, 6]. (d) Approximate this local maximum using six iterations of the two fixed-point methods (Secant and Newton). For Newton's Method, use p0 = 4. For the Secant Method, use p0 = 6 and p1 = 4. (e) What is your best estimate for p, the location of the maximum? (f) Compare the four algorithms using the following two tables. Table 1: Approximation pn versus iteration number n Iteration n Bisection False Position Secant Newton Table 2: Absolute error |pn - p| versus iteration number n Iteration n Bisection Position False Secant Newton (g) Plot the log of the absolute error for all four methods on the same graph. (h) What happens if you attempt to approximate the maximum by starting Newton's Method with p0 = 6? (i) Don't forget to turn in your code! Warning: Do not hardwire the function into the algorithms; you should pass the function to them as an argument. (2) (Order of Convergence) Apply the theorems on pp. 90-91 to the following problems: (a) Section 2.3 #8: Verify that x = g(x) = a is a fixed point of the function 1 a x+ . 2 x Use the techniques of this section to determine the order of convergence and the as ymptotic error constant of the sequence pn = g(pn-1 ) toward x = a. 1 2 (b) Section 2.3 #9: Verify that x = a is a fixed point of the function x3 + 3ax . 3x2 + a Use the techniques of this section to determine the order of convergence and the as ymptotic error constant of the sequence pn = g(pn-1 ) toward x = a. g(x) = (3) (Linearly Convergent Fixed-Point Schemes) Section 2.3 #1: Suppose the sequence {pn } is generated by the fixed point iteration scheme pn = g(pn-1 . Further, suppose that te sequence converges lineraly to the fixed point p. (a) Show that pn - pn-1 . g (p) pn-1 - pn-2 (b) Show that g (p) |en | |pn - pn-1 |. g (p) - 1
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Michigan - MATH - 471
Homework 2 SolutionsMath 471, Fall 2006 Assigned: Friday, September 15, 2006 Due: Friday, September 22, 2006 Include a cover page Clearly label all plots using title, xlabel, ylabel, legend Use the subplot command to compare multiple plots Inclu
Michigan - MATH - 471
Homework 3Math 471, Fall 2007 Assigned: Monday, September 24, 2007 Due: Monday, October 1, 2007 Include a cover page Clearly label all plots using title, xlabel, ylabel, legend Use the subplot command to compare multiple plots Include printouts
Michigan - MATH - 471
Homework 3, Solution SketchesMath 471, Fall 2007 Assigned: Friday, September 24, 2007 Due: Friday, October 1, 2007 (1) (Newton versus Secant) Bradie, p. 113, #12. The sequence of iterates generated by the secant method follows. == n p(n) |e(n)| = 0
Michigan - MATH - 471
Homework 4Math 471, Fall 2007 Assigned: Friday, September 28, 2007 Due: Friday, October 5, 2007 Include a cover page Clearly label all plots using title, xlabel, ylabel, legend Use the subplot command to compare multiple plots Include printouts
Michigan - MATH - 471
Homework 4 SolutionsMath 471, Fall 2007 Assigned: Friday, September 28, 2007 Due: Friday, October 5, 2007 Include a cover page Clearly label all plots using title, xlabel, ylabel, legend Use the subplot command to compare multiple plots Include
Michigan - MATH - 471
Homework 5Math 471, Fall 2007 Assigned: Friday, October 5, 2006 Due: Friday, October 12, 2006 Include a cover page Clearly label all plots using title, xlabel, ylabel, legend Use the subplot command to compare multiple plots Include printouts of
Michigan - MATH - 471
Homework 5Math 471, Fall 2007 Assigned: Friday, October 5, 2007 Due: Friday, October 12, 2007 Include a cover page Clearly label all plots using title, xlabel, ylabel, legend Use the subplot command to compare multiple plots Include printouts of
Michigan - MATH - 471
Homework 6Math 471, Fall 2007 Assigned: Friday, October 19, 2006 Due: Friday, October 26, 2006 Include a cover page Clearly label all plots using title, xlabel, ylabel, legend Use the subplot command to compare multiple plots Include printouts o
Michigan - MATH - 471
Homework 6 SolutionsMath 471, Fall 2006 Assigned: Friday, October 20, 2006 Due: Friday, October 27, 2006 Include a cover page Clearly label all plots using title, xlabel, ylabel, legend Use the subplot command to compare multiple plots Include p
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Cal Poly - PHYS - 131-133
7.1. Solve: (a) From t = 0 s to t = 1 s the particle rotates clockwise from the angular position +4 rad to -2 rad. Therefore, = -2 - ( +4 ) = -6 rad in one sec, or = -6 rad s . From t = 1 s to t = 2 s, = 0 rad/s. From t = 2 s to t = 4 s the partic
Cal Poly - PHYS - 131-133
8.1. Visualize:Solve: Figure (i) shows a weightlifter (WL) holding a heavy barbell (BB) across his shoulders. He is standing on a rough surface (S) that is a part of the earth (E). We distinguish between the surface (S), which exerts a contact forc
Cal Poly - PHYS - 131-133
Solve: (a) The momentum p = mv = (1500 kg)(10 m /s) = 1.5 10 4 kg m /s . (b) The momentum p = mv = (0.2 kg)( 40 m /s) = 8.0 kg m /s .9.1. Model: Model the car and the baseball as particles.9.2. Model: Model the bicycle and its rider as a particl
Cal Poly - PHYS - 131-133
10.1. Model: We will use the particle model for the bullet (B) and the bowling ball (BB).Visualize:Solve:For the bullet,KB =For the bowling ball,1 1 2 mB vB = (0.01 kg)(500 m /s) 2 = 1250 J 2 2 1 1 2 mBB vBB = (10 kg)(10 m / s) 2 = 500 J 2
Cal Poly - PHYS - 131-133
11.1. Visualize:r Please refer to Figure Ex11.1. rSolve: (b) (c)(a) A B = AB cos = ( 4)(5)cos 40 = 15.3. r r C D = CD cos = (2)( 4)cos120 = -4.0. r r E F = EF cos = (3)( 4)cos 90 = 0.11.2. Visualize:r Please refer to Figure Ex11.2. rSolve
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12.1.Solve: (b)Model: Model the sun (s), the earth (e), and the moon (m) as spherical. (a)Fs on e =Gms me (6.67 10 -11 N m 2 / kg 2 )(1.99 10 30 kg)(5.98 10 24 kg) = 3.53 10 22 N = (1.50 1011 m ) 2 rs2 e -Fm on e =GMm Me (6.67 10 -1
Cal Poly - PHYS - 131-133
13.1. Model: The crankshaft is a rotating rigid body.Solve: The crankshaft at t = 0 s has an angular velocity of 250 rad/s. It gradually slows down to 50 rad/s in 2 s, maintains a constant angular velocity for 2 s until t = 4 s, and then speeds up
Cal Poly - PHYS - 131-133
14.1. Solve: The frequency generated by a guitar string is 440 Hz. The period is the inverse of the frequency, henceT= 1 1 = = 2.27 10 -3 s = 2.27 ms f 440 Hz14.2. Solve: Your pulse or heart beat is 75 beats per minute. The frequency of your hear
Cal Poly - PHYS - 131-133
15.1. Solve: The density of the liquid is=m 0.120 kg 0.120 kg = = = 1200 kg m 3 V 100 mL 100 10 -3 10 -3 m 3Assess: The liquid's density is more than that of water (1000 kg/m3) and is a reasonable number.15.2. Solve: The volume of the helium
Cal Poly - PHYS - 131-133
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Cal Poly - PHYS - 131-133
18.1. Solve: We can use the ideal-gas law in the form pV = NkBT to determine the Loschmidt number (N/V):1.013 10 5 Pa N p = 2.69 10 25 m -3 = = V kB T (1.38 10 -23 J K )(273 K )()18.2. Solve: Nitrogen is a diatomic molecule, so r 1.0 10-1
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19.1. Model: The heat engine follows a closed cycle, starting and ending in the original state. The cycleconsists of three individual processes. Visualize: Please refer to Figure Ex19.1. Solve: (a) The work done by the heat engine per cycle is the a
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Berkeley - MCB - 130
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Berkeley - MCB - 130
Berkeley - MCB - 130
Berkeley - MCB - 130
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