dn (a) Show that L[i" f(t)] = (-1)" (t) for any positive integer n 2 1 dsn a d K(s, t)f(t) dt / ) est = tne-8t...
9. [10 points) Consider the following algorithm: procedure Algorithm(n: positive integer; ddd: distinet integers) for k:=1 to n-1 for 1-1 to n-k print(k, I, di,da...-1,dn) if ds dti then interchange dy and d (a) Assume that this algorithm receives as input the integer n 6 and the input sequence 하하하하하하, Miss ^-ruteae rehen i12|3141516 Fill out the table below: ds ds (b) Assume that the algorithm receives the same input values as in part a). Once the algorithm finishes, what...
17. Consider the following algorithm: procedure Algorithm(n: positive integer; di,d2.. ,dn: distinct integers) for 1 to n-1 for 1 to n-k if ddi+ then interchange di and di+ print(k, I, d,ddn-1, dn) (a) |3 points Assume that this algorithm receives as input the integer-6 and the corresponding input sequence 41 36 27 31 17 20 Fill out the table below ds (b) 1 point Assume that the algorithm receives the same input values as in part a). Once the algo-...
For s > 0 define the gamma function I (s) by T () = [co-dt. Show that I (8) extends to an analytic function in the half-plane 20 = {ZEC: Rez >0}, and that the above formula continues to hold there. Hint: Show that S T. (s) ds = 0 for every triangle T in C where I (8) = le-+48-1dt for S E C and 0 <€ < 1.
Use Theorem 7.4.1. THEOREM 7.4.1 Derivatives of Transforms If F(s) = ℒ{f(t)} and n = 1, 2, 3, . . . , thenℒ{tnf(t)} = (−1)n dn dsn F(s). Evaluate the given Laplace transform. (Write your answer as a function of s.) ℒ{t cos(7t)}
Elementary Laplace Transtorms Y(S) = {f} -L e-stf(t)dt fc = C-'{F(s)} F(s) = {f} f(t) =-'{F(s)) F(s) = {f} -CS 1. 1 1 12. uct) le S> 0 S> 0 . s S 2. eat 1 13. ucOf(t-c) e-csF(s) S> a S-a n! 3. t",n e Z 14. ectf(t) F( sc) S> 0 sh+1 4. tP, p>-1 (p+1) S> 0 SP+1 15. f(ct) F). c>0 16. SFt - 1)g(t)dt F(s)G(*) 5. sin at S> 0 16. cos at 17. 8(t...
Consider the sequence defined as a[1] = 2; and a[k] = a[k-1]+2*k-1 for all positive integer k >= 2; . Show that a[n] = 1+sum(2*i-1, i = 1 .. n); . Hint: Start with sum(2*i-1, i = 1 .. n);and use the recursive definition of the sequence.
Let n be a non-negative integer. Letf() be such that f(x), f'(x).f"(x).,fn+exist, and are continuous, on an interval containing a. In this assignment, you will prove by induction on n that for any r in that interval f'(c) f"(c) fm (c) (t) (x -t)" dt. 7n n! 1. (a) Explain why the claim given above is true for n-0 (b) Use the fact that the claim is true for n-0 to explain why the claim is true for n =...
Prove: Let k be a positive integer, and set n :=2k-1(2k – 1). Then (2k+1 – 1)2 = 8n +1 Prove: Let n be a positive integer, and let s and t be integers. Show that Hire (st) = n(s) in (t) mod n.
1. a) Deduce from the relation Fk+1F-1-F2 = (-1)k, that for any positive integer k > 2 Fk b) Deduce from part a) that for any positive integer n, Fn+1 = Σ 1)k+1 1. a) Deduce from the relation Fk+1F-1-F2 = (-1)k, that for any positive integer k > 2 Fk b) Deduce from part a) that for any positive integer n, Fn+1 = Σ 1)k+1
Prove that the solution of the recurrence T(n) = T(n/2) +6(logk n) with T(1-6(1), for any integer k 2 0, is T(n) = Θ(logk+1 n) (Hint: the upper bound T(n) = O(logk+1 n) is easy; the lower bound T(n) = Ω(logk +1 n) is harder.) Prove that the solution of the recurrence T(n) = T(n/2) +6(logk n) with T(1-6(1), for any integer k 2 0, is T(n) = Θ(logk+1 n) (Hint: the upper bound T(n) = O(logk+1 n) is easy;...