Say that a < 0) and k is a positive integer. Find a constant c such...
question 5 5. (a) Informally find a positive integer k for which the following is true: 3n + 1 < n2 for all integers n > k-4 (b) Use induction to prove that 3n +1 < n2 for all integers n 2 k. 6. Consider the following interval sets in R: B-4.7, E = (1,5), G = (5,9), M-[3,6]. (a) Find (E × B) U (M × G) and sketch this set in the-y plane. (b) Find (EUM) x (BUG)...
For all n E N prove that 0 <e- > < 2 k!“ (n + 1)! k=0 Hint: Think about Taylor approximations of the function e".
Prove that is an integer for all n > 0.
4. (10 points) Find the solution to the wave problem Ut = c+421 +COSI, <0, t>0, with initial conditions u(1,0) = sin r, 4(1,0) = 1+I.
Find the Laplace transform of f(0) = 1, for 0 <t<1 5, for 1<t<2. e-l for t > 2
Find the Laplace Transform (d) f(t) = te, 0<t<1, et, t > 1. l
Problem 5. Let a < b and c > 0 and let f be integrable on [ca, cb]. Show that f c Ca where g(a) f(ex)
Find the length of spiral curve T() = ----- 0 < > < 2”
Find the Laplace Transform of f(t)=0 if t< 1; f(t) = t if 1sts 2; f(t)=0 if t> 2.
Problem 5.1.3. Prove by induction on n that (1+ n < n for every integer n > 3.