(4) Determine the TAYLOR'S EXPANSION of exactly one of the following expres- sions: (4.1) 2+1 2...
(3) Determine the TAYLOR'S EXPANSION of exactly one of the following expres- sions: (3.1) cosh(z) 22 (3.2) COS (+)
Question 4 4.1 Express the limit as a definite integral on the given integral: 1-x} Ax , [2,6] lim Σ=1 a. (2 Marks) n->00 4+x} lim (?-1 - Ax ,[1,3] (xi) - 4 b. (2 Marks) n->00 4.2 Evaluate the following expressions. Show your calculations. $=1(2p – p2) b. En-o sin a. (2 Marks) пп (2 Marks) 2 C. 2m +2 53 Lm=1 3 (2 Marks) [Sub Total 10 Marks]
Find the Laplace Transform of f(t)= - lifts 4; f(t) = 1 if t> 4.
[10] 1. Solve the first-order equation Tư + (x – 2) = -xe-3, > 0.
question 5c 5. Find the Laurent series expansion of: (a) f(x) = 2*1 about i, (b) f(x) = 22 + 1-2, convergent on {2 < 121 <4}, (c)* f(x) = 2,2-33+2, convergent on {j < lz - 11 < 1}.
(1 point) Find the solution of cºy" + 5xy' + (4 - 4x)y = 0, 2 > 0 of the form 00 yi =T n0 where co 1. Enter T= Cn = n = 1,2,3,...
10. What is the dual of the following problem: Minimize x1 + x2, subject to xi 20, x2 > 0, 2xı > 4, x1 + 3x2 > 11? Find the solution to both this problem and its dual, and verify that minimum equals maximum.
3. (10 pts.) X is a Gaussian random variable with E{X} = 2 and Var(X) = 16. Let Y = 3X +1. Determine the probability: Pr(Y > 2)
Let a1 = 3 and an+1 = a + 5 2an for all n > 1 Prove that (an)nen converges and find limn7oo an:
.n= n(n-1)(n+1) for all n > 2. 12. Use induction to prove (1 : 2) +(2-3)+(3-4) +...+(n-1).n [9 points) 3