Numerical Analysis
3rd Edition
ISBN: 9780134696454
Author: Sauer, Tim
Publisher: Pearson,
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Chapter 9.1, Problem 3E
Approximate the area under the curve
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What is a solution to a differential equation? We said that a differential equation is an equation that
describes the derivative, or derivatives, of a function that is unknown to us. By a solution to a differential
equation, we mean simply a function that satisfies this description.
2. Here is a differential equation which describes an unknown position function s(t):
ds
dt
318
4t+1,
ds
(a) To check that s(t) = 2t2 + t is a solution to this differential equation, calculate
you really do get 4t +1.
and check that
dt'
(b) Is s(t) = 2t2 +++ 4 also a solution to this differential equation?
(c) Is s(t)=2t2 + 3t also a solution to this differential equation?
ds
1
dt
(d) To find all possible solutions, start with the differential equation = 4t + 1, then move dt to the
right side of the equation by multiplying, and then integrate both sides. What do you get?
(e) Does this differential equation have a unique solution, or an infinite family of solutions?
these are solutions to a tutorial that was done and im a little lost. can someone please explain to me how these iterations function, for example i Do not know how each set of matrices produces a number if someine could explain how its done and provide steps it would be greatly appreciated thanks.
Chapter 9 Solutions
Numerical Analysis
Ch. 9.1 - Find the period of the linear congruential...Ch. 9.1 - Find the period of the LCG defined by a=4,b=0,m=9...Ch. 9.1 - Approximate the area under the curve y=x2 for 0x1,...Ch. 9.1 - Approximate the area under the curve y=1x for 0x1,...Ch. 9.1 - Prob. 5ECh. 9.1 - Prove that u1=x21+x22 in the Box-Muller Rejection...Ch. 9.1 - Implement the Minimal Standard random number...Ch. 9.1 - Implement randu and find the Monte Carlo...Ch. 9.1 - (a) Using calculus, find the area bounded by the...Ch. 9.1 - Carry out the steps of Computer Problem 3 for the...
Ch. 9.1 - Use n=104 pseudo-random points to estimate the...Ch. 9.1 - Use n=104 pseudo-random points to estimate the...Ch. 9.1 - (a) Use calculus to evaluate the integral 01x2x,...Ch. 9.1 - Prob. 8CPCh. 9.1 - Prob. 9CPCh. 9.1 - Devise a Monte Carlo approximation problem that...Ch. 9.2 - Prob. 1CPCh. 9.2 - Prob. 2CPCh. 9.2 - Prob. 3CPCh. 9.2 - Prob. 4CPCh. 9.2 - Prob. 5CPCh. 9.2 - One of the best-known Monte Carlo problems is the...Ch. 9.2 - Prob. 7CPCh. 9.2 - Prob. 8CPCh. 9.2 - Prob. 9CPCh. 9.3 - Design a Monte Carlo simulation to estimate the...Ch. 9.3 - Calculate the mean escape time for the random...Ch. 9.3 - In a biased random walk, the probability of going...Ch. 9.3 - Prob. 4CPCh. 9.3 - Design a Monte Carlo simulation to estimate the...Ch. 9.3 - Calculate the mean escape time for Brownian motion...Ch. 9.3 - Prob. 7CPCh. 9.4 - Use Itos formula to show that the solutions of the...Ch. 9.4 - Use Itos formula to show that the solutions of the...Ch. 9.4 - Use Itos formula to show that the solutions of the...Ch. 9.4 - Prob. 4ECh. 9.4 - Prob. 5ECh. 9.4 - Prob. 6ECh. 9.4 - Use the Euler-Maruyama Method to find approximate...Ch. 9.4 - Use the Euler-Maruyama Method to find approximate...Ch. 9.4 - Apply the Euler-Maruyama Method with step size...Ch. 9.4 - Prob. 4CPCh. 9.4 - Prob. 5CPCh. 9.4 - Prob. 6CPCh. 9.4 - Use the Milstein Method to find approximate...Ch. 9.4 - Prob. 8CPCh. 9.4 - Prob. 9CPCh. 9.4 - Prob. 10CPCh. 9.4 - Prob. 11CPCh. 9.4 - Prob. 12CPCh. 9.4 - Prob. 1SACh. 9.4 - Prob. 2SACh. 9.4 - Prob. 3SACh. 9.4 - Prob. 4SACh. 9.4 - Compare your approximation in step 4 with the...Ch. 9.4 - Prob. 6SA
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- Q1) Classify the following statements as a true or false statements a. Any ring with identity is a finitely generated right R module.- b. An ideal 22 is small ideal in Z c. A nontrivial direct summand of a module cannot be large or small submodule d. The sum of a finite family of small submodules of a module M is small in M A module M 0 is called directly indecomposable if and only if 0 and M are the only direct summands of M f. A monomorphism a: M-N is said to split if and only if Ker(a) is a direct- summand in M & Z₂ contains no minimal submodules h. Qz is a finitely generated module i. Every divisible Z-module is injective j. Every free module is a projective module Q4) Give an example and explain your claim in each case a) A module M which has two composition senes 7 b) A free subset of a modale c) A free module 24 d) A module contains a direct summand submodule 7, e) A short exact sequence of modules 74.arrow_forward************* ********************************* Q.1) Classify the following statements as a true or false statements: a. If M is a module, then every proper submodule of M is contained in a maximal submodule of M. b. The sum of a finite family of small submodules of a module M is small in M. c. Zz is directly indecomposable. d. An epimorphism a: M→ N is called solit iff Ker(a) is a direct summand in M. e. The Z-module has two composition series. Z 6Z f. Zz does not have a composition series. g. Any finitely generated module is a free module. h. If O→A MW→ 0 is short exact sequence then f is epimorphism. i. If f is a homomorphism then f-1 is also a homomorphism. Maximal C≤A if and only if is simple. Sup Q.4) Give an example and explain your claim in each case: Monomorphism not split. b) A finite free module. c) Semisimple module. d) A small submodule A of a module N and a homomorphism op: MN, but (A) is not small in M.arrow_forwardProve that Σ prime p≤x p=3 (mod 10) 1 Ρ = for some constant A. log log x + A+O 1 log x "arrow_forward
- Prove that, for x ≥ 2, d(n) n2 log x = B ― +0 X (금) n≤x where B is a constant that you should determine.arrow_forwardProve that, for x ≥ 2, > narrow_forwardI need diagram with solutionsarrow_forwardT. Determine the least common denominator and the domain for the 2x-3 10 problem: + x²+6x+8 x²+x-12 3 2x 2. Add: + Simplify and 5x+10 x²-2x-8 state the domain. 7 3. Add/Subtract: x+2 1 + x+6 2x+2 4 Simplify and state the domain. x+1 4 4. Subtract: - Simplify 3x-3 x²-3x+2 and state the domain. 1 15 3x-5 5. Add/Subtract: + 2 2x-14 x²-7x Simplify and state the domain.arrow_forwardQ.1) Classify the following statements as a true or false statements: Q a. A simple ring R is simple as a right R-module. b. Every ideal of ZZ is small ideal. very den to is lovaginz c. A nontrivial direct summand of a module cannot be large or small submodule. d. The sum of a finite family of small submodules of a module M is small in M. e. The direct product of a finite family of projective modules is projective f. The sum of a finite family of large submodules of a module M is large in M. g. Zz contains no minimal submodules. h. Qz has no minimal and no maximal submodules. i. Every divisible Z-module is injective. j. Every projective module is a free module. a homomorp cements Q.4) Give an example and explain your claim in each case: a) A module M which has a largest proper submodule, is directly indecomposable. b) A free subset of a module. c) A finite free module. d) A module contains no a direct summand. e) A short split exact sequence of modules.arrow_forward1 2 21. For the matrix A = 3 4 find AT (the transpose of A). 22. Determine whether the vector @ 1 3 2 is perpendicular to -6 3 2 23. If v1 = (2) 3 and v2 = compute V1 V2 (dot product). .arrow_forward7. Find the eigenvalues of the matrix (69) 8. Determine whether the vector (£) 23 is in the span of the vectors -0-0 and 2 2arrow_forward1. Solve for x: 2. Simplify: 2x+5=15. (x+3)² − (x − 2)². - b 3. If a = 3 and 6 = 4, find (a + b)² − (a² + b²). 4. Solve for x in 3x² - 12 = 0. -arrow_forward5. Find the derivative of f(x) = 6. Evaluate the integral: 3x3 2x²+x— 5. - [dz. x² dx.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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