[RBH 13.14] Problem 2: Prove the equality a2 dw 4a4 w 1 sin2(at)dt -2at TJO 0
1. Prove“inclusion-exclusion,”thatP(A∪B)=P(A)+P(B)−P(A∩B). 2. Prove the “unionbound, ”thatP(A1∪A2)≤P(A1)+P(A2). Under what conditions does the equality hold? 3. Provethat, for A1 andA2 disjoint, P(A1∪A2|B)=P(A1|B)+P(A2|B). 4. A and B are independent events with nonzero probability. Prove whether or not A and Bc are independent.
Solve the PDE using laplace: (dw/dx)+(x*(dw/x)) =xu(t-1) w(x,0)=0 if x>=0 and w(0,t)=0 if t<=0
For natural number n, an = 1+1+3+--+--log n . x dt By use log x = hen x > 0,· w 1 t Prove that the series is convergence and for any n 2 1, - 0<an n+12n(n+1) For natural number n, an = 1+1+3+--+--log n . x dt By use log x = hen x > 0,· w 1 t Prove that the series is convergence and for any n 2 1, - 0
0=1/2at^2+vt solve for t , two answers.
6. Prove by induction that forn > 1 72 a2 0 0 b
matlab code Given the boundary value problem: daw = W 3 dw dc2 de w(x = 1) = 5 w(x = 10) = 5 Find the minimum value of wover this region using the shooting method. Round your answer to TWO decimal points rounding up for 5 or more. 0.55 0 -15.33 -0.55 10.25
(8) Prove that dt= 1-t n=1 for x e [-a, a],0< a< 1 and deduce from there a power series expansion for -In(1-x) (8) Prove that dt= 1-t n=1 for x e [-a, a],0
9. (a) Let Ao(x) = / (1-t*)dt, Ai(z) = / (1-t2) dt, and A2(z) = / (1-t2)dt. Compute these explicitly in terms ofェusing Part 2 of the Fundamental Theorem of Calculus. b) Over the interval [0,2], use your answers in part (a) to sketch the graphs of y Ao(x), y A1(x), and y A2(x) on the same set of axes. (c) How are the three graphs in part (a) related to each other? In particular, what does Part 1 of...
8, Prove that if R is an integral domain and a E R such that a2 + 2a + 1-0 then a =-1. Then give an example of a ring that is not an integral domain for which a2a 1-0 but a f -1. 8, Prove that if R is an integral domain and a E R such that a2 + 2a + 1-0 then a =-1. Then give an example of a ring that is not an integral domain...
(1 point) Solve the boundary value problem by using the Laplace transform 22 w ²w + sin(6ax) sin(16t) = 0 < x < 1, t> 0 дх2 dt2 w(0,t) = 0, w(1,t) = 0, t> 0, w(x,0) = 0, dw -(x,0) = 0, 0 < x < 1. dt First take the Laplace transform of the partial differential equation. Let W be the Laplace transform of w. Then W satisfies the ordinary differential equation W" = subject to W(0) =...