3. Show that if A-1 exists, then f eAT dr = A-(eAt – I) = (e4t – I)A-1.

Q: 12. Define def M e-$dWg• s=0 For each t > 0, compute (a) E[M¡]. (b) E[M?]

A: Given  Mt=∫s=0te-s dWs      , where W is a Brownian motion.

Q: Find the general solution to the yiven Euler x* j -3xy = 2.

A: Given differential equation is x^2y" - 3xy' + 9y = 0

Q: A. let w-e te Find W2

A: We can solve this using u/v rule

Q: If z + In(e 2 – e" + e*) = 0 -2x az find at the point (0, 0) O-3 O 3 O-2 O 2

A:

Q: 6. Prove that for x > -1 then Vn E N, (1 + x)" >1+ nx .

A:

Q: A large but sparsely populated county has two small hospitals, one at the south end of the county…

A: "Since you have posted a question with multiple subparts, we will solve first 3 sub-parts for you.…

Q: 8. Given that (2x-x*); 0<x<2, S(x) = ; elsewhere. a) Calculate E(X), E(X²) and E 2- X

A:

Q: If f(z, y) – e . In(9y) , then fz is given by: O - 9e In(9y) o - 27 e In(9y) O 9 e In(9y) O 27 e…

A:

Q: 2. If w = f(x + ct) + g(x- ct) show that c²wxx = Wtt C= constant.

A:

Q: If u = f(x), then ſ dx x++2x²+1 k + C for some constant k . Compute kf(1). x2+1

A: We have to find kf(1) Given that: u=f(x),∫xx4+2x2+1dx=kx2+1+C  for some constant k

Q: 1. Show (in terms of e – 6) that a function f : [2, 7] → R be defined by f(x) = Vx² + 1 is uniformly…

A:

Q: Let a e R" and let A = aaT. Show that xT Ax = (a" x)².

A:

Q: Consider the econometric model y = Bo + B1x +u and the OLS estimator B, for B1. We know that: O…

A:

Q: Solve the following Cauchy-Euler D.E: x²y" – xy' +y = 2x2 + In x %3D

A:

Q: 1. Find the following. Iy(-1, –1) if g(x, y) = x³ – xy² + 3

A: Given g(x,y)=x3-xy2+3 Differentiating g partially with respect to y, gyx,y=∂g∂y=0-x(2y)+0…

Q: Let SINGLEDFA = { | M is a DFA and for all s e L(M),|s| = 1}. Show that SINGLEDFA is decidable.

A: Detailed explanation mentioned below.

Q: 3. If f(x, y, z)= In xyz te(3.0²), find f,,f, and f-· 4

A:

Q: -2 Let f(t,y) = y + 3t?e'. Then f satisfies a Lipschitz condition in y with L = 2, given that 1<t<2.…

A:

Q: 1 1 Prove that: L' a exp(a't) erfe( avt), where erfc is complementary a+v error function.

A: Formula Used:- L1t=πsLeat=1s-aLerf(at)=as(s+a)Leatf(t)=F(s-a)                   where…

Q: 9. Verify that p(t) = te/50 satisfies the condition p(t) dt = 1

A:

Q: Let a be a constant. Given that 3x In(ax)dr = e18 ( 36 -89 , find In a. 3

A:

Q: If f:(0,∞)→Rf:(0,∞)→R is defined by f(x)=xln(2x)f(x)=xln⁡(2x) for every x∈(0,∞),x∈(0,∞),  then…

A:

Q: 12. Solve the problem: Utt = c*upx + Ax, 0 0, A= constant, и (х,0) — 0, и (0, t) — 0, Ut (x, 0) =…

A:

Q: 2. and XR %3D C. Two tails with n = 34 , x =

A: Solution: From the given information, level of significance is 0.01 and n=34.

Q: = 1 – 3, h(x) = VT, then If f(x) = x* + 6, g(x) fogo h(z)-

A: Find your answer below

Q: 1. Differentiate the following with respect to x: a. y= 5x - + 5Vx + 3p ex + In(x) - 7 (9)

A:

Q: 4 16 Define T: R? - R° by T(x) = Ax, where A = 1 3 -

A:

Q: Q4 Prove that E(",") = ("::)- () .n 2mzr. %3D +1 +1

A: we have to prove that                                                   ∑k=0mn-kr=n+1r+1-n-mr+1 now,…

Q: 2. A firm produces two goods in pure ompetition &amp; has the pllowing TR &amp; TC functions R= 15Q…

A: In this question, concept of optimization is applied. Optimization Mathematical optimization is the…

Q: d [e'- In (21+1)]dt, find g(x) . dx -2 A ex-In(2x+1) 2 B ex_ (2x + 1)2 4 C (2x + 1)² 4 D (2x + 1)²…

A:

Q: If 8(x) = [e²-m (21+ - dx S[e²-In (21+1) ]dt, find g(x). -2 A e-In(2x+1) 4 B (2x + 1)² 2 Ⓒ (2x + 1)²…

A:

Q: Show that E(SSE) = (n - k-1), as in Theorem 7.3f, using the following approach. Show that SSE = y'y…

A: Please find the explanation below. Thank you

Q: If f(t)dt = 1, S g(t)dt = 3, evaluate (3f(t) + 2g(t) – 5)dt.

A:

Q: Verify that u' (t – a) = 8(t – a) by solving the problem x' = 8(t – a); x (0) = 0 to obtain x (1) =…

A: What is Laplace Transformation: Laplace transformation is crucial for control system design. Laplace…

Q: An M.V.U. is unique in the sense that if T, 1 M.V.U. estimators for y (0), then T, = T, almost

A: A sample statistics t is said to be unbiased estimator of population parameterγ(θ) if expected value…

Q: In order to obtain a good linear approximation for In (7.4) what would a particularly useful nearby…

A:

Q: 2.4.10 Show that det(e4) = etr(A) for all A E Mn(C). D %3D

A: The given problem is to show the given identity. To show: det(eA)=etr(A) for all A belongs to Mn(C).

Q: 10. Show that S t2 if 0 1, 2(1 - e-) then Ldf %3D

A: Laplace of differential equation.

Q: 2. Let P:C0, 1] → C|0, 1] be given by Pp = " – e°. Compute dP(y). Be sure to justify your…

A: We have Pφ=φ''-eφ We need to find dPφ If we have  y=fx⇒dydx=f'x⇒dy=f'x dx Similarly, we find dPφ for…

Q: c). Invest diFrgate whether the given en equation is homogene eous dy 2. oxy

A: A first order differential equation is homogeneous when it can be written as a function of y/x, that…

Q: d If g(x)=- S* [e²-In (21+1) ]dt, find g(x). -2 2 (2x + 1)2 4 B (2x + 1)² Ⓒe-In(2x+1) 4 (2x + 1)² di…

A: A function is given by, gx=ddx∫-2xet-ln2t+1dt. The objective is to find the second derivative g''x…

Q: 18. (2) = t"(-t)" dt, show that F(m). m! +1, n+1 %3D m and n are positive integers.

A:

Q: Theorem 8.3.3 If X ~ t(k), then E(X2r + 1) = 0 and µz, = k"- (2r – 1)(2r – 3) ... .1_ (k– 2)(k – 4).…

A: Answer: For the given theorem, The proof is:

Q: 4. Now, determine the value of c in (0,1) such that f'(c) = 2 fx) =4x

A:

Q: Find fCx) if flo)=0 and the tanGat (k,F CX) has ine af Slore x(x+2) a

A: Given for the function F(x), F(0)=0 and tangent line to F(x) at x,F(x) has slope xx2+23.

Q: If z3 + In(e 2 -2 e+e) = 0 az find at the point (0,0) O -3 0 3 О-2 O 2

A:

Q: 2/ Compute fc E.dr and Where F Gxry2=(3+ 12x²,3xy²2 is a line segment from Co,01 to (4,2) %D Fdr =

A:

Q: 2) Evaluate each of the following. Show your work. pIn(3) e2t +5 dt et a)

A: Given:        (a): ∫0ln(3)e2t+5etdt

Q: 13. a) Show that EX=o (-1)*(")=0, and then verify that ) +C) + C) + - - G) +G) + () +- ... = b)…

A: We will use of binomial theorem to solve it

Q: Prove that (A')' = A for each A E Mmxn(F).

A: