DISCRETE MATHEMATICS+ITS APPL. (LL)-W/A
8th Edition
ISBN: 9781260521337
Author: ROSEN
Publisher: MCG
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Chapter 13, Problem 3CP
To determine
To find out the all strings using twenty or fewer application with help of Backus-Naur form (BNF)a type 2 grammar.
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Consider the initial value problem
mx" + cx' + kx = F(t),
x(0) = 0, x'(0) = 0
modeling the motion of a damped mass-spring system initially at rest and subjected to an
applied force F(t), where the unit of force is the Newton (N). Assume that m = = 2
kilograms, c = 8 kilograms per second, k 80 Newtons per meter, and F(t) = 20e¯*
=
Newtons.
Solve the initial value problem.
x(t) =
=
help (formulas)
Determine the long-term behavior of the system (steady periodic solution). Is lim x(t) = 0
t→∞
? If it is, enter zero. If not, enter a function that approximates x(t) for very large positive
values of t.
For very large positive values of t,
x(t) ≈ x sp(t)
=
help (formulas)
Book: Section 2.6 of Notes on Diffy Qs
Consider the initial value problem
mx" + cx' + kx = F(t), x(0) = 0, x'(0) = 0
modeling the motion of a damped mass-spring system initially at rest and subjected to an
applied force F(t), where the unit of force is the Newton (N). Assume that m = 2
kilograms, c = 8 kilograms per second, k = 80 Newtons per meter, and
F(t) = 100 cos(8t) Newtons.
Solve the initial value problem.
x(t) =
help (formulas)
Determine the long-term behavior of the system (steady periodic solution). Is lim x(t) = 0
t→∞
? If it is, enter zero. If not, enter a function that approximates x(t) for very large positive
values of t.
For very large positive values of t,
x(t)≈ x sp(t)
=
help (formulas)
Book: Section 2.6 of Notes on Diffy Qs
Chapter 13 Solutions
DISCRETE MATHEMATICS+ITS APPL. (LL)-W/A
Ch. 13.1 - Exercises 1-3 refer to the grammar with start...Ch. 13.1 - Exercises 1-3 refer to the grammar with start...Ch. 13.1 - Prob. 3ECh. 13.1 - Let G=(V,T,S,P) be the phrase-structure grammar...Ch. 13.1 - Prob. 5ECh. 13.1 - Prob. 6ECh. 13.1 - Prob. 7ECh. 13.1 - Show that the grammar given in Example 5 generates...Ch. 13.1 - Prob. 9ECh. 13.1 - Prob. 10E
Ch. 13.1 - Construct a derivation of 021222 in the grammar...Ch. 13.1 - Show that the grammar given in Example 7 generates...Ch. 13.1 - s13. Find a phrase-structure grammar for each of...Ch. 13.1 - Find a phrase-structure grammar for each of these...Ch. 13.1 - Find a phrase-structure grammar for each of these...Ch. 13.1 - Construct phrase-structure grammars to generate...Ch. 13.1 - Construct phrase-structure grammars to generate...Ch. 13.1 - Construct phrase-structure grammars to generate...Ch. 13.1 - Prob. 19ECh. 13.1 - A palindrome is a string that reads the same...Ch. 13.1 - Let G1 and G2 be context-free grammars, generating...Ch. 13.1 - Prob. 22ECh. 13.1 - Construct derivation trees for the sentences in...Ch. 13.1 - Let G be the grammar with V={a,b,c,S};T={a,b,c} ;...Ch. 13.1 - Prob. 25ECh. 13.1 - Prob. 26ECh. 13.1 - Prob. 27ECh. 13.1 - a) Explain what the productions are in a grammar...Ch. 13.1 - Prob. 29ECh. 13.1 - a) Construct a phrasestructure grammar for the set...Ch. 13.1 - Give production rules in Backus-Naur form for an...Ch. 13.1 - Prob. 32ECh. 13.1 - Prob. 33ECh. 13.1 - Prob. 34ECh. 13.1 - Prob. 35ECh. 13.1 - Prob. 36ECh. 13.1 - Prob. 37ECh. 13.1 - Prob. 38ECh. 13.1 - Prob. 39ECh. 13.1 - Prob. 40ECh. 13.1 - Prob. 41ECh. 13.1 - Let G be a grammar and let R be the relation...Ch. 13.2 - Draw the state diagrams for the finite-state...Ch. 13.2 - Give the state tables for the finite-state machine...Ch. 13.2 - Find the output generated from the input string...Ch. 13.2 - Find the output generated from the input string...Ch. 13.2 - Find the output for each of these input strings...Ch. 13.2 - Find the output for each of these input strings...Ch. 13.2 - Construct a finite-state machine that models an...Ch. 13.2 - Prob. 8ECh. 13.2 - Construct a finite-state machine that delays an...Ch. 13.2 - Construct a finite-state machine that changes...Ch. 13.2 - Construct a finite-state machine for the log-on...Ch. 13.2 - Construct a finite-state machine for lock that...Ch. 13.2 - Construct a finite-state machine for a toll...Ch. 13.2 - Construct a finite-state machine for entering a...Ch. 13.2 - Construct a finite-state machine for a restricted...Ch. 13.2 - Construct a finite-state machine that gives an...Ch. 13.2 - Prob. 17ECh. 13.2 - Construct a finite-state machine that determines...Ch. 13.2 - Construct a finite-state machine that determines...Ch. 13.2 - Prob. 20ECh. 13.2 - Prob. 21ECh. 13.2 - Find the output string generated by the Moore...Ch. 13.2 - Prob. 23ECh. 13.2 - Construct a Moore machine that gives an output of...Ch. 13.2 - Prob. 25ECh. 13.3 - Prob. 1ECh. 13.3 - 2. Show that if A is a set of strings, then.
Ch. 13.3 - Find all pairs of sets of strings A and B for...Ch. 13.3 - Show that these equalities hold. a) {}*={} b)...Ch. 13.3 - Prob. 5ECh. 13.3 - Prob. 6ECh. 13.3 - Prob. 7ECh. 13.3 - Prob. 8ECh. 13.3 - Prob. 9ECh. 13.3 - Determine whether the string 01001 is in each of...Ch. 13.3 - Determine whether each of these strings is...Ch. 13.3 - Determine whether each of these strings is...Ch. 13.3 - Determine whether all the strings in each of these...Ch. 13.3 - Show that if M=(S,I,f,so,F) is a deterministic...Ch. 13.3 - Given a finite-state automaton M=(S,I,f,so,F) ,...Ch. 13.3 - In Exercises 16—22 find the language recognized by...Ch. 13.3 - In Exercises 16—22 find the language recognized by...Ch. 13.3 - Prob. 18ECh. 13.3 - Prob. 19ECh. 13.3 - In Exercises 16—22 find the language recognized by...Ch. 13.3 - In Exercises 16—22 find the language recognized by...Ch. 13.3 - Prob. 22ECh. 13.3 - Construct a deterministic finite-state automaton...Ch. 13.3 - Construct a deterministic finite-state automaton...Ch. 13.3 - Construct a deterministic finite-state automaton...Ch. 13.3 - Construct a deterministic finite-state automaton...Ch. 13.3 - Prob. 27ECh. 13.3 - Construct a deterministic finite-state automaton...Ch. 13.3 - Prob. 29ECh. 13.3 - Construct a deterministic finite-state automaton...Ch. 13.3 - Construct a deterministic finite-state automaton...Ch. 13.3 - Construct a deterministic finite-state automaton...Ch. 13.3 - Prob. 33ECh. 13.3 - Prob. 34ECh. 13.3 - Prob. 35ECh. 13.3 - Prob. 36ECh. 13.3 - Prob. 37ECh. 13.3 - Prob. 38ECh. 13.3 - Prob. 39ECh. 13.3 - Use Exercise 39 finite-state automata constructed...Ch. 13.3 - Prob. 41ECh. 13.3 - Prob. 42ECh. 13.3 - Prob. 43ECh. 13.3 - Prob. 44ECh. 13.3 - Prob. 45ECh. 13.3 - In Exercises 43-49 find the language recognized by...Ch. 13.3 - Prob. 47ECh. 13.3 - In Exercises 43-49 find the language recognized by...Ch. 13.3 - Prob. 49ECh. 13.3 - Find a deterministic finite-state automaton that...Ch. 13.3 - Prob. 51ECh. 13.3 - Find a deterministic finite-state automaton that...Ch. 13.3 - Find a deterministic finite-state automaton that...Ch. 13.3 - Find a deterministic finite-state automaton that...Ch. 13.3 - Find a deterministic finite-state automaton that...Ch. 13.3 - Find a nondeterministic finite-state automaton...Ch. 13.3 - Prob. 57ECh. 13.3 - Prob. 58ECh. 13.3 - Prob. 59ECh. 13.3 - Prob. 60ECh. 13.3 - Prob. 61ECh. 13.3 - Prob. 62ECh. 13.4 - Describe in words the strings in each of these...Ch. 13.4 - Prob. 2ECh. 13.4 - Prob. 3ECh. 13.4 - Prob. 4ECh. 13.4 - Express each of these sets using a regular...Ch. 13.4 - Express each of these sets using a regular...Ch. 13.4 - Express each of these sets using a regular...Ch. 13.4 - Construct deterministic finite-state automata that...Ch. 13.4 - Construct nondeterministic finite-state automata...Ch. 13.4 - Construct nondeterministic finite-state automata...Ch. 13.4 - Show that if A is a regular set, then AR, the set...Ch. 13.4 - Using the construction described in the proof of...Ch. 13.4 - Using the construction described in the proof of...Ch. 13.4 - Construct a nondeterministic finite-state...Ch. 13.4 - In Exercises 15-17 conflict a regular grammar...Ch. 13.4 - In Exercises 15-17 conflict a regular grammar...Ch. 13.4 - In Exercises 15-17 conflict a regular grammar...Ch. 13.4 - Show that the finite-state automaton constructed...Ch. 13.4 - Show that the regular grammar constructed from a...Ch. 13.4 - Show that every nondeterministic finite-state...Ch. 13.4 - Let M=(S,I,f,s0,F) be a deterministic finite-state...Ch. 13.4 - One important technique used to prove that certain...Ch. 13.4 - Show that the set 02n1nn=0,1,2,... is not regular...Ch. 13.4 - Show that the set {1n2n=0,1,2,...} is not regular...Ch. 13.4 - Show that the set of palindromes over {0, 1} is...Ch. 13.4 - Prob. 26ECh. 13.4 - Prob. 27ECh. 13.4 - Prob. 28ECh. 13.4 - Prob. 29ECh. 13.4 - Prob. 30ECh. 13.4 - Use Exercise 29 to show that the language...Ch. 13.5 - Let T be the Turing machine defined by the...Ch. 13.5 - Let T be the Turing machine defined by the...Ch. 13.5 - What does the Turing machine defined by the...Ch. 13.5 - What does the Turing machine described by the...Ch. 13.5 - What does the Turing machine described by the...Ch. 13.5 - Construct a Turing machine with tape 0, 1, and B...Ch. 13.5 - Construct a Turning machine with tape symbols 0,...Ch. 13.5 - Construct a Turing machine with tape symbols 0, 1,...Ch. 13.5 - Construct a Turing machine with tape symbols 0, 1,...Ch. 13.5 - Construct a Turing machine with tape symbols 0, 1,...Ch. 13.5 - Construct a Turing machine that recognizes the set...Ch. 13.5 - Construct a Turing machine that recognizes the set...Ch. 13.5 - Construct a Turing machine that recognizes the set...Ch. 13.5 - Show at each step the contents of the tape of the...Ch. 13.5 - Explain why the Turing machine in Example 3...Ch. 13.5 - Construct a Turing machine that recognizes the set...Ch. 13.5 - Construct a Turing machine that recognizes the set...Ch. 13.5 - Construct a Turing machine that computes the...Ch. 13.5 - Construct a Turing machine that computes the...Ch. 13.5 - Construct a Turing machine that computes the...Ch. 13.5 - Construct a Turing machine that computes the...Ch. 13.5 - Construct a Turing machine that computes the...Ch. 13.5 - Construct a Turing machine that computes the...Ch. 13.5 - Construct a Turing machine that computes the...Ch. 13.5 - Construct a Turing machine that computes the...Ch. 13.5 - Construct a Turning machine that computes the...Ch. 13.5 - Prob. 27ECh. 13.5 - Prob. 28ECh. 13.5 - Which of the following problems is a decision...Ch. 13.5 - Which of the following problems is a decision...Ch. 13.5 - Prob. 31ECh. 13.5 - Show that the function B(n) cannot be computed by...Ch. 13 - a) Define a phrase-structure grammar. b) What does...Ch. 13 - a) What is the language generated by a...Ch. 13 - Prob. 3RQCh. 13 - Prob. 4RQCh. 13 - Prob. 5RQCh. 13 - a) What is a finite-state machine? b) Show how a...Ch. 13 - Prob. 7RQCh. 13 - Prob. 8RQCh. 13 - Prob. 9RQCh. 13 - Prob. 10RQCh. 13 - a) Define a nondeterministic finite-state...Ch. 13 - a) Define the set of regular expressions over a...Ch. 13 - Prob. 13RQCh. 13 - Prob. 14RQCh. 13 - Prob. 15RQCh. 13 - Prob. 16RQCh. 13 - Describe how Turing machines are used to recognize...Ch. 13 - Prob. 18RQCh. 13 - Prob. 19RQCh. 13 - Prob. 1SECh. 13 - Prob. 2SECh. 13 - Prob. 3SECh. 13 - Prob. 4SECh. 13 - Prob. 5SECh. 13 - Prob. 6SECh. 13 - Prob. 7SECh. 13 - Prob. 8SECh. 13 - Prob. 9SECh. 13 - Prob. 10SECh. 13 - Prob. 11SECh. 13 - Prob. 12SECh. 13 - Prob. 13SECh. 13 - Construct a finite-state machine with output that...Ch. 13 - Construct a finite-state machine with output that...Ch. 13 - Prob. 16SECh. 13 - Prob. 17SECh. 13 - Prob. 18SECh. 13 - Construct a deterministic finite-state automaton...Ch. 13 - Prob. 20SECh. 13 - Prob. 21SECh. 13 - Prob. 22SECh. 13 - Prob. 23SECh. 13 - Prob. 24SECh. 13 - Prob. 25SECh. 13 - Show that {02nnN} is not regular. You may use the...Ch. 13 - Prob. 27SECh. 13 - Prob. 28SECh. 13 - Construct a Turing machine that computes the...Ch. 13 - Prob. 30SECh. 13 - Prob. 1CPCh. 13 - Prob. 2CPCh. 13 - Prob. 3CPCh. 13 - Prob. 4CPCh. 13 - Given the state table of a Moore machine and an...Ch. 13 - Given the state table of a Mealy machine and an...Ch. 13 - Given the state table of a deterministic...Ch. 13 - Prob. 8CPCh. 13 - Prob. 9CPCh. 13 - Prob. 10CPCh. 13 - Given a regular grammar, construct a finite-state...Ch. 13 - Given a finite-state automaton, construct a...Ch. 13 - Prob. 13CPCh. 13 - Solve the busy beaver problem for two states by...Ch. 13 - Prob. 2CAECh. 13 - Prob. 3CAECh. 13 - Prob. 4CAECh. 13 - Prob. 5CAECh. 13 - Prob. 1WPCh. 13 - Describe the Backus-Naur form (and extended...Ch. 13 - Explain how finite-state machines are used by...Ch. 13 - Explain how finite-state machines are used in the...Ch. 13 - Explain how finite-state machines are used in...Ch. 13 - Compare the use of Moore machines versus Mealy...Ch. 13 - Explain the concept of minimizing finite-state...Ch. 13 - Give the definition of cellular automata, Explain...Ch. 13 - Define a pushdown automaton. Explain how pushdown...Ch. 13 - Define a linear-bounded automaton. Explain how...Ch. 13 - Prob. 11WPCh. 13 - Prob. 12WPCh. 13 - Prob. 13WPCh. 13 - Show that a Turing machine can simulate any action...Ch. 13 - Prob. 15WPCh. 13 - Describe the basic concepts of the lambda-calculus...Ch. 13 - Show that a Turing machine as defined in this...Ch. 13 - Prob. 18WP
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- Consider the initial value problem mx" cx' + kx F(t), x(0) = 0, x'(0) = 0 modeling the motion of a damped mass-spring system initially at rest and subjected to an applied force F(t), where the unit of force is the Newton (N). Assume that m = 2 80 Newtons per meter, and F(t) = 20 sin(6t) kilograms, c = 8 kilograms per second, k = Newtons. Solve the initial value problem. x(t) = help (formulas) Determine the long-term behavior of the system (steady periodic solution). Is lim x(t) = 0 0047 ? If it is, enter zero. If not, enter a function that approximates x(t) for very large positive values of t. For very large positive values of t, x(t) ≈ x sp(t) = ☐ help (formulas) Book: Section 2.6 of Notes on Diffy Qsarrow_forwardConsider the differential equation y' = - 4xy with initial condition y(0) = 1.9. Recall that Runge-Kutta method has the following formula for computing the next step, where h is the step size: f(xi, Yi) = fx i + (++) k1 = h k2 2 ¯‚ Yi + k₁ h h k3 = fxi 2 `, Yi + k₂· 2 k4 = f(xi+h, yikзh) i+1=i+h k12k22k3 + k4 Yi+1 Yi + h 6 Using Runge-Kutta step size h = 0.4: Estimate y(0.4) ≈ help (numbers) Estimate y(0.8) ≈ help (numbers) Book: Section 1.7 of Notes on Diffy Qsarrow_forwardDetermine which differential equation corresponds to each phase diagram. You should be able to state briefly how you know your choices are correct. х x 4 4 4 4 3 3 3 3 2 2 2 2 dx ? ✰ dt = 1. = x² - 3x 1 1 1 1 ? ◇ 2. dx dt = x(x − 2) - 0 0 0 0 ? ◇ 3. dx dt = x(2 − x)² -1 -1 -1 -1 Q -2 -2 -2 dx ? ◇ 4. ༤་ dt = = 3x - x² -3 -3 -3 -3 x³- 4x = x²|x − 2| ? ◇ 5. ம் dx dt བི་ dx ? ◇ 6. dt ཝེ་ dx ? 7. dt ཝེ་ dx ? ◇ 8. ཝེ་ dt -4 -4 -4 -4 A B 0 D = = 2x = x² * x * * x * K 4 4 4 4 = 4x - x³ 3 3 3 • 3 Book: Section 1.6 of Notes on Diffy Qs dit for this problem 2 2 2 2 1 1 1 1 0 0 0 8 -1 -1 -1 -1 N 心 -2 -2 -3 -3 -3 -4 -4 -4 -4 E FL G Harrow_forward
- Dear expert Chatgpt gives wrong answer Plz don't use chat gptarrow_forwardAn improved method that is similar to Euler's method is what is usually called the Improved Euler's method. It works like this: Consider an equation y' = f(x, y). From (xn, Yn), our approximation to the solution of the differential equation at the n-th stage, we find the next stage by computing the x-step Xn+1 = xn +h, and then k1, the slope at (xn, Yn). The predicted new value of the solution . İs Zn+1 = Yn + h · k₁. Then we find the slope at the predicted new point k₁ = f(xn+1, Zn+1) and get the corrected point by averaging slopes h Yn+1 = = Yn + 1½ ½ (k1 + k₂). Suppose that we use the Improved Euler's method to approximate the solution to the differential equation dy dx = x - 0.5y, y(0.5) = 9. We let xo = 0.5 and yo 9 and pick a step size h = 0.25. Complete the following table: n xn Yn k1 Zn+1 k₂ 0 0.59-48 -3.25 ♡ <+ help (numbers) The exact solution can also be found for the linear equation. Write the answer as a function of x. y(x) = = help (formulas) Thus the actual value of the…arrow_forwardAlready got wrong Chatgpt answer If ur also Chatgpt user leave itarrow_forward
- The graph of the function f(x) is 1,0 (the horizontal axis is x.) Consider the differential equation x' = f(x). List the constant (or equilibrium) solutions to this differential equation in increasing order and indicate whether or not these equalibria are stable, semi-stable (stable from one side, unstable from the other), or unstable. x = help (numbers) x = help (numbers) x = help (numbers) x = help (numbers) Book: Section 1.6 of Notes on Diffy Qsarrow_forward= A 10 kilogram object suspended from the end of a vertically hanging spring stretches the spring 9.8 centimeters. At time t = 0, the resulting mass-spring system is disturbed from its rest state by the force F(t) = 60 cos(8t). The force F(t) is expressed in Newtons and is positive in the downward direction, and time is measured in seconds. Determine the spring constant k. k = Newtons/meter help (numbers) Hint is to use earth gravity of 9.8 meters per second squared, and note that Newton is kg meter per second squared. Formulate the initial value problem for x(t), where x(t) is the displacement of the object from its equilibrium rest state, measured positive in the downward direction. Give your answer in terms of x, x',x",t. Differential equation: | help (equations) Initial conditions: x (0) = and '(0) = help (numbers) Solve the initial value problem for x(t). x(t) = ☐ help (formulas) Plot the solution and determine the maximum displacement from equilibrium made by the object on the…arrow_forwardSuppose f(x) is a continuous function that is zero when x is −1, 3, or 6 and nowhere else. Suppose we tested the function at a few points and found that ƒ(−2) 0, and f(7) < 0. Let x(t) be the solution to x' f(x) and x(0) = 1. Compute: lim x(t) help (numbers) t→∞ Book: Section 1.6 of Notes on Diffy Qsarrow_forward
- Consider the initial value problem У y' = sin(x) + y(-4) = 5 4 Use Euler's Method with five steps to approximate y(-2) to at least two decimal places (but do not round intermediate results). y(-2) ≈ help (numbers) Book: Section 1.7 of Notes on Diffy Qsarrow_forwardConsider the differential equation y' = 5y with initial condition y(0) : The actual solution is y(1) = 207.78 help (numbers) = 1.4. We wish to analyze what happens to the error when estimating y(1) via Euler's method. Start with step size h = 1 (1 step). Compute y(1) Error 8.4 help (numbers) 199.38 help (numbers) Note: Remember that the error is the absolute value! Let us half the step size to h = 0.5 (2 steps). Compute y(1) ≈ 17.15 help (numbers) Error = 190.63 help (numbers) The error went down by the factor: Error Previous error Let us half the step size to h = 0.25 (4 steps). Compute y(1) 35.88046875 help (numbers) Error = 171.90 help (numbers) help (numbers) The error went down by the factor: Error Previous error help (numbers) Euler's method is a first order method so we expect the error to go down by a factor of 0.5 each halving. Of course, that's only very approximate, so the numbers you get above are not exactly 0.5. Book: Section 1.7 of Notes on Diffy Qsarrow_forwardAnswer all the boxes and box the answers. Thank you write it downarrow_forward
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