Question 1.0 [10 marks] Evaluate dz a. z4+z3-272, With C: Izl = 1 b. J2 cos230...
Question 6 (10 marks) In this question you are asked to produce a proof of the following identity: $(2) = (1) n=1 (a) Let N > 1 be a natural number. Evaluate the integral 1 Cos(12) IN = -dz, 2ni lov z2 sin(T2) where Cn is the positively oriented, square contour shown in Figure 1. (N+ 5)(-1+i) (N + 1)(1+i) (N+ 3)(-1 - i) (N + 1)(1 - ) Figure 1: The contour Cn. In your working, the following limit...
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NOT use a calculator. Exact answers only, no decimals.
1. (10 pt each) Evaluate the following integrals: since) dz In(In(x b. dr c. cos(x)(sin(a)2 dz d.2tan (') dr
1. (10 pt each) Evaluate the following integrals: since) dz In(In(x b. dr c. cos(x)(sin(a)2 dz d.2tan (') dr
need help for this question in full answer
2. The deflection along a uniform beam with fexual Yigidity BI- and applied load f (x) = cos (-) satisfies the equation (a) Evaluate the deflection y (x). Hint: /cos(az)dz-asin (as)+C, /sin(as)dz=-a cos(az) +C (b) Find the influence function (Green's function) G (z,f), where 0 < ξ < 2, for this problem. Hint: Since 0 < ξ < 2, H(0-E)=0, H(2-E)=1. (c) Hence write the deflection of this beam as a definite...
Question 3 (Chapter 6) 12+3-3-6 14 marks] Fix p E N and consider the following set: (c) Compute Ci and C2 (d) Show that a 0 is an extreme point of Cp. Hint: you may use (without proof) that the family of functions fe',e", ..., e) is linearly independent.
Question 3 (Chapter 6) 12+3-3-6 14 marks] Fix p E N and consider the following set: (c) Compute Ci and C2 (d) Show that a 0 is an extreme point of...
Question 14 (12 marks) Consider the following separable differential equation. dy cos(z)(-1) dr (a) Find any constant solutions of this differential equation and hence write down the solution with initial value y=- when r=7 (b) Use partial fractions to evaluate 1 dy. 1 (c) Use the method for solving separable differential equations to solve this DE in the case where y 0 when r T. You may assume that the solution does not cross the constant solutions you found in...
10) Calculate the integral zdac dy dz where D is bounded by the planes x = - 0, y = 0, z = 0, z = 1, and the cylinder x2 + y2 = 1 with x > 0 and y> 0. 11) Let y be the boundary of the rectangle with sides x = 1, y = 2, x = 3 and y = 3. Use Green's theorem to evaluate the following integral 2y + sina 1+2 1 +...
Question 5 (20 marks) a) [10 marks] Evaluate: 1 el-X Vx+y(y – 2x)2 dydx. 0 Jo (Hint: consider a change of variable.) b) [10 marks) Find the volume of the solid bounded by the sphere x2 + y2 + z2 = aand the cylinder x2 + y2 = ax, a > 0.
2. Evaluate the following integrals. (a) [5 marks] | el cos 4xdx -1 x (b) [5 marks] / cosdx -x³+3x²-x- dr. 1dx (c) [10 marks] (п -3)(12+2) 4 (d) [5 marks]/ dx V4-5x-2x2 dx cosh x-sinh x (e) [5 marks]] (Give the final answer in terms of e.)
7. Evaluate the following integral. b) SO3(2-1)05 dz c) f (xl dx (15 marks) 8. A shaded region S is bounded by the curve y = the x-axis, the vertical lines x = 1 and x = V2. The x(4-2)1/4 region is rotated through 21 radians about x-axis to form a solid of revolution. Compute the exact volume of the solid of revolution formed. (10 marks) 9. Between the two towers of a suspension bridge that are 2 kilometres apart,...
Problem 7.1 (10 points) Express the unilateral z-transforms of the following functions as rational functions. Find also the ROC. You may use tables. (a) xl[n]-1-0.2)" (b) x2[n] (0.3)" +2(-5) -0.2n Problem 7.2 (10 points) Express the unilateral z-transforms of the following functions as rational functions. Find also the ROC. You may use tables. (a) xl[n] = 3e-j02" (b) x2[n]- 5cos(5n) (c) x3[n] = e-0.gn sin(0.7n) Problem 7.3 (10 points) The signals given are sampled every 0.3 s, beginning att-0. Find...