(5) Use the recurrence for the Catalan numbers Cn+1 ating function Ek=o CkCn-k to prove that the gener- f (x) = Cna" %3D n=0 satisfies the equation 1- f(x) + xf(x)² = 0. (This implies that f(x) = 1-V-4=). Hint: The coefficient of x" in (Eo Chak )² is E-o CkCn-k· –4x k=0

Q: Prove that Ją (2) = =- J¼ (2) – J_4 (=)

A:

Q: Find L[et tsin(2t) - e-2t+3 +e-2t cos 3t]

A: BY USING THE LAPLACE TRANSFORMATION RULES  WE GET OPTION (D) IS CORRECT OPTION.

Q: Let {(=) = In ="(z +2)*(z² + 9)] f'(x) =

A:

Q: -35 Find s?(s°+4)

A: We will find out the required value.

Q: dH f(5) = 8, f' (5) = 4 f(26) = 8, f'(26) = 6, g(5) = 1, g'(5) = -9, and H(x) = f(r² + g(x)). Find…

A:

Q: (b). Use the definition of the derivative to evaluate f (x)=x". [Hint: Use the Binomial Theorem:…

A: Given: i. fx=xnii. a+bn=an+nan-1b+nn-12!an-2b2+·····+nabn-1+bn. To Find: Use definition of…

Q: (4) VL1, L2 ((L1 UL2)* = L₁* U L₂¹).

A: This is always True

Q: Find Cn in terms of C0 when: 1. Cn = Cn-1/(3n2 + 4n): (Note that C1 = C0/7, C2 = C0/140, C3 =…

A:

Q: 2(д-1)(1-и) 9-1)(2-и)и+1 log((q-1)(2—и)и+1) u? > 0 for u > Show that u E - 0, 1], q > 3

A:

Q: f(x) = ln(3x + 5). Show with induction that the n-derivative of f is given with "the function from…

A:

Q: f: ℤ675,683 → ℤ821 × ℤ823 satisfies f(x) = (mod821(x), mod823 (x)). You may use: 1=411⋅821− 410⋅823…

A:

Q: a/2 2. Evaluate f[(e)i-2sin(t) j+3cos(1)k]dt

A:

Q: Suppose f(x, y, z) = xyz + 2xy + 2² · Compute the following values -(2,8₁16) - (-2,-8, 16) (1,-1,0)…

A: We have to find functional value.

Q: Show that /(-1)-7(1) but there is Let f(x) 1 Show that /(-1) (1) but there is no number e in (-1, 1)…

A:

Q: Verify that det(AB) = det(A) det(B) for the following:

A: Given: The given matrices are A=1-23-231010 and B=1023-25213. First, find the determinant of A and…

Q: Suppose that an = 4" a₁x² + 2a₂x³ + 3a3x4 + 4a4x³. as 00 (i) f(n)x+2 n=0 00 (ii) Σ g(n)x¹+1 n=1 00…

A:

Q: a) x³ - x f (x) = 2x² + x − 1 - b) f(x) = ax + b cx - d

A: Here we have to evaluate for a) and b) the value of f'(x).

Q: 17. Let P1.Pn be positive real numbers such that P1+ + Pn=1. (a) Prove the following: @P² ≥ (ii)…

A:

Q: 1 -dx x(x² + 1)

A:

Q: hd the invense transformof eack of t FX) 6x-5 FCx)=

A: Since you have posted a question with multiple sub-parts, we will solve the first three sub-parts…

Q: Suppose P(Xi=1)=pi=1-P(Xi=0),  i=1,2 if p1> p2 show that X1>( greater than equal to (st)) X2

A:

Q: 2020-21 Algeb.► Briana Berisha - Topics8H Q4-2 - 20-21.pdf Simplify 2x 4 19. 2x + 8 a. ズー2 C. X + 4…

A: To simplify the given expression.

Q: LET • (CX)= In [x²(x+5) ³ (x² + 4)⁰7 f'(x) = LET F(x) = \n [X²(X + 3) ³ (X² + 2)²] Cico

A:

Q: 3x'e* (x* +1)* 3e a) -+C 2(x² +1) 3e* b) 2(x² +1) +C x? +3 x +1 3 c) +C .2

A:

Q: RECURSION RELATIONS: L Jp(x}] = ° Jp=1(x), d.r Jp-1(x) + Jp+1(x) =2J,(x). Jp-1(x) – Jp+1(1) =…

A: As per our company guidelines, we are supposed to solve only the first three subparts. Kindly re…

Q: Iff

A:

Q: A particular finite difference formula for the first derivative of a funtion is: f '(xi) =…

A: Given: f'(xi)=-fxi+3+9fxi+1-8fxi6h, where xi, xi+1, xi+2 and xi+3 are equally spaced with step size…

Q: Let h(x) = x³e" k6*. Find dr (h(x)) wherek is a real number (i.e. k is a constant).

A: Given: h(x)=x3ex3-k6x To find: ddxh(x)

Q: Let f (x) = -2r, de fined on -1, 1], then L (P ) = 2 (n– 1) 2- 4 (n+1) 4 (n – 1) 4 (п-1) 2n2

A: We need to find the lower Riemann sum

Q: _dp„dp (27)? - exp (-it [b² – 2p*p¯] +[-ip &* (1 – e2nQL,") – ip*8¯ (1 – e-nQt.m) + 27Qt,ns]) . (1)…

A: Consider the given equation.