MATH 1700 FINAL SPRING MOON

Transcription

1 MATH 700 FINAL SPRING 0 - MOON Write your answer neatly and show steps If there is no explanation of your answer, then you may not get the credit Except calculators, any electronic devices including laptops and cell phones are not allowed Do not use the graphing function on your calculator ) Consider the following infinite sum a) pts) Write the given sum by using sigma notation ) n 4 n0 Using wrong range of index: - pt b) 4 pts) Evaluate the sum ) n 4 ) n 4 n0 n0 4 By using the sum of infinite geometric series, obtaining 4 : pts Getting the correct answer : 4 pts Date: May 4, 0

2 MATH 700 Final ) pts) Compute 000 n n + ) + n n n + ) + n Finding the correct answer 000: pts Using a wrong initial or final term: - pts each Spring 0 - Moon ) pts) Choose one of the following terminologies and write its definition as precisely as you can Bifurcation of a fixed point Locally stable fixed point Dense subset of an interval For a one-parameter family x n+ f r x n ) of discrete dynamical systems, we say that a fixed point p r undergoes a bifurcation at r r 0 if it changes from stable to unstable, and either another stable fixed point or a stable -cycle emerges from it For a discrete dynamical system x n+ fx n ), a fixed point p is called locally stable if ) there is an interval I containing p such that for every point x 0 in I, x 0 p > x p > x p > and ) lim n x n p For an interval I and a subset J of I, we say that J is a dense subset of I if for each x in J and every ɛ > 0, there is y in I such that x y < ɛ If one states an equivalent condition instead of the definition, one can get pts If one misses one of conditions on the definition, - pts for each

3 MATH 700 Final Spring 0 - Moon 4) Suppose that a contaminated lake is being cleaned by a filtering process Each week this process is capable of filtering out 6% of all the pollutants present at that time, but another tons of pollutants seep in a) pts) At the beginning, 000 tons of pollutants were initially in that lake Find an iterative model describing the amount of pollutants P n in n-th week P n+ P n 06P n + 084P n +, P Missing the initial condition: - pt b) pts) Solve the model in a) and find the closed formula for P n P n 084 n P n 084 P n n n ) ) 084 n + 06 Unless you get the correct answer for b), you can t get any point for c) and d) c) 4 pts) When does the remaining amount of the pollutant become less than 0 tons? P n ) 084 n n log 084 log ) 084 n ) n n ) log 084 log Therefore it takes at least 4 weeks to become less than 0 tons Using a wrong unit: - pt d) pts) Describe the asymptotic behavior of P n Because lim 084 n 0, n lim 000 ) 084 n + n Therefore the remaining amount of the pollutants approaches 87 tons

4 MATH 700 Final ) Consider the logistic model P n+ P n P ) n 000 a) pts) Find all fixed points Let fx) x x ) x 000x 000 fx) x x 000x x 000x x 0 x000x ) 0 x 0, x Therefore there are two fixed points x 0, x 600 Spring 0 - Moon b) 4 pts) Determine their stability f x) 000x f 0) f 0) > Therefore 0 is unstable f 600) f 600) 0 < So 600 is stable c) pts) Determine any oscillation around them Because f 0) > 0, there is no oscillation around it On the other hand, f 600) 0 < 0, so there is oscillation around it 4

5 MATH 700 Final Spring 0 - Moon 6) Consider a dynamical system x n+ x n a) 4 pts) Find all -cycles Let fx) x fx) x x x x + x xx + ) 0 x 0 So there is only one fixed point x 0 ffx)) x f x ) x x 9 x x 9 x 0 xx 8 ) 0 xx 4 )x 4 + ) 0 xx )x + )x 4 + ) 0 xx )x + )x + )x 4 + ) 0 x 0,, x 0 is a fixed point Because f) and f ), {, } is a -cycle Finding the fixed point x 0: + pt Finding all solutions x 0,, of ffx)) x: + pts Getting the answer {, }: + pt b) 4 pts) Determine their stability f x) x f )f ) ) 9 > Therefore the -cycle is unstable Evaluating the derivative f x) x : pt Determining the stability: 4 pts

6 MATH 700 Final Spring 0 - Moon 7) In this problem, do not use your calculator Unless you show every computational step, you can t get any credit Let A and B a) pts) Compute A B 0 A B b) pts) Find AB ) 4 + ) ) AB ) ) ) c) pts) Write the definition of the inverse matrix For a matrix A, the inverse matrix of A is a matrix A such that AA I A A If one states only one of two equalities AA I and A A I: - pt d) pts) Compute A A ) )

7 MATH 700 Final Spring 0 - Moon 8) Consider the following homogeneous linear model with x 0, y 0 x n+ x n + y n y n+ x n + y n, a) pts) Write a matrix equation v n+ A v n of the model xn+ xn x0, y n+ y n y 0 b) 4 pts) Find the eigenvalues of A and corresponding eigenvectors of A r A ri r deta ri) r) r) r r + r r + 0 r )r ) 0 r, r For r, A ri A ri)v 0 has a nonzero solution v For r, A ri A ri)v 0 has a nonzero solution v So there are two eigenvalues r, and their corresponding eigenvectors are, respectively Finding the determinant r r + of A ri: pt Getting two eigenvalues, : pts Obtaining two corresponding eigenvectors, Finding one eigenvalue and one eigenvector: pts c) pts) Determine the stability of the origin Because < and >, it is a saddle : 4 pts 7

8 MATH 700 Final Spring 0 - Moon xn d) 6 pts) Compute the closed formula for x n and y n A P DP, where P 0, D 0 Then P ) A y n x0 P D n y n P 0 n 0 0 n 0 n 0 n 4 4 n n Therefore x n + 4 n n, y n + 4 n n n + 4 n n + 4 n Finding P and D so that A P DP : + pts Writing the formula P D n P : + pts Getting the answer x n + 4 n n, y n + 4 n n : + pts 9) In this problem, do not use your calculator Unless you show every computational step, you can t get any credit Let z + i, w + i a) pts) Compute z + w z + w + i) + + i) + ) + + )i 4 + 4i b) pts) Compute zw zw + i) + i) + ) i + i + i i + i 8 + i 8

9 MATH 700 Final Spring 0 - Moon c) pts) Evaluate z w z w + i + i + i + i i i + 8i 4 + i + i i i + i 9i i Having the idea to compute the conjugate i to the numerator and denominator: pt Getting the answer i: pts 4 d) pts) Find the polar coordinate r and θ of z r ) + + i cos θ + i sin θ cos θ, sin θ Therefore θ π 4 Getting r : pt Finding θ π 4 : pts e) pts) Compute z 0 z r cos θ + ir sin θ re iθ z 0 re iθ) 0 r 0 e i0θ ) 0 cos 0 π 4 + i sin 0 π 4 ) cos π + i sin π ) i Describing z in terms of a complex exponent re iθ : + pt Knowing Euler s identity: + pt By using the law of exponents, obtaining ) 0 cos 0 π 4 π ): + pts 4 Getting the answer i: + pts + i sin 0 9

10 MATH 700 Final Spring 0 - Moon 0) Consider the following prey-predator model: P n+ P n Q n + 00 Q n+ P n + Q n 00 a) pts) Find the fixed point P, Q) P P Q + 00 Q P + Q 00 P P Q + 00 Q Q 00 Q P + Q 00 P 00 0 P 0 P, Q) 0, 00) b) 4 pts) Determine the stability of P, Q) Set A A ri r r deta ri) r) ) r r + r r + 0 r ± ) 4 + i So the fixed point is a source ± 8 + ) > Finding two complex eigenvalues ± : pts Determining the stability: 4 pts ± i c) pts) Is there any periodic point of this model? Explain your answer For a linear dynamical system with complex eigenvalues, a periodic point occurs only if the modulus of a complex eigenvalue is Therefore there is no periodic points If there is no reasonable explanation of the reason, pt 0